![Complex Numbers II 9
Figure 2.4
x
y
0
1 − i
·
−π/4
We observe that any one of the values θ = −(π/4) ± 2nπ, n = 0, 1, · · · ,
can be used here. The number θ is called an argument of z, and we write
θ = argz. Geometrically, argz denotes the angle measured in radians that
the vector corresponds to z makes with the positive real axis. The argument
of 0 is not defined. The pair (r, arg z) is called the polar coordinates of the
complex number z.
The principal value of arg z, denoted by Arg z, is defined as that unique
value of argz such that −π arg z ≤ π.
If we let z1 = r1(cos θ1 + i sin θ1) and z2 = r2(cos θ2 + i sin θ2), then
z1z2 = r1r2[(cos θ1 cos θ2 − sin θ1 sin θ2) + i(sin θ1 cos θ2 + cos θ1 sin θ2)]
= r1r2[cos(θ1 + θ2) + i sin(θ1 + θ2)].
Thus, |z1z2| = |z1||z2|, arg(z1z2) = argz1 + argz2.
Figure 2.5
x
y
0
·
·
·
z1
z2
z1z2
θ1 θ2 θ1+θ2
r1
r2
r1r2
For the division, we have
z1
z2
=
r1
r2
[cos(θ1 − θ2) + i sin(θ1 − θ2)],
z1
z2
=
|z1|
|z2|
, arg
z1
z2
= arg z1 − arg z2.](https://image.slidesharecdn.com/1020158-250521105736-81521d56/85/An-Introduction-To-Complex-Analysis-1st-Edition-Ravi-P-Agarwal-28-320.jpg)
![Complex Numbers III 13
Example 3.2. Express cos 3θ in terms of cos θ. We have
cos 3θ = Re(cos 3θ + i sin 3θ) = Re(cos θ + i sin θ)3
= Re[cos3
θ + 3 cos2
θ(i sin θ) + 3 cos θ(− sin2
θ) − i sin3
θ]
= cos3
θ − 3 cosθ sin2
θ = 4 cos3
θ − 3 cosθ.
Now, let z = reiθ
= r(cos θ + i sin θ). By using the multiplicative prop-
erty of the exponential function, we get
zn
= rn
einθ
(3.5)
for any positive integer n. If n = −1, −2, · · ·, we define zn
by zn
= (z−1
)−n
.
If z = reiθ
, then z−1
= e−iθ
/r. Hence,
zn
= (z−1
)−n
=
1
r
ei(−θ)
−n
=
1
r
−n
ei(−n)(−θ)
= rn
einθ
.
Hence, formula (3.5) is also valid for negative integers n.
Now we shall see if (3.5) holds for n = 1/m. If we let
ξ = m
√
reiθ/m
, (3.6)
then ξ certainly satisfies ξm
= z. But it is well-known that the equation
ξm
= z has more than one solution. To obtain all the mth roots of z, we
must apply formula (3.5) to every polar representation of z. For example,
let us find all the mth roots of unity. Since
1 = e2kπi
, k = 0, ±1, ±2, · · ·,
applying formula (3.5) to every polar representation of 1, we see that the
complex numbers
z = e(2kπi)/m
, k = 0, ±1, ±2, · · ·,
are mth roots of unity. All these roots lie on the unit circle centered at the
origin and are equally spaced around the circle every 2π/m radians.
Figure 3.1
0
m = 6
π/3](https://image.slidesharecdn.com/1020158-250521105736-81521d56/85/An-Introduction-To-Complex-Analysis-1st-Edition-Ravi-P-Agarwal-32-320.jpg)
![18 Lecture 3
(a). If x = 0 and y 0 (y 0), then Arg z = π/2 (−π/2).
(b). If x 0, then Arg z = tan−1
(y/x) ∈ (−π/2, π/2).
(c). If x 0 and y 0 (y 0), then Arg z = tan−1
(y/x)+π (tan−1
(y/x)−
π).
(d). Arg (z1z2) = Arg z1 + Arg z2 + 2mπ for some integer m. This m is
uniquely chosen so that the LHS ∈ (−π, π]. In particular, let z1 = −1, z2 =
−1, so that Arg z1 = Arg z2 = π and Arg (z1z2) = Arg(1) = 0. Thus the
relation holds with m = −1.
(e). Arg(z1/z2) = Arg z1 − Arg z2 + 2mπ for some integer m. This m is
uniquely chosen so that the LHS ∈ (−π, π].
Answers or Hints
3.1. (a). −2i, (b). −1 + 8i, (c). −10i, (d). i, (e). (1 − i)/2, (f). −2/5,
(g). 2−11
(−1 +
√
3i), (h). −8(1 + i), (i). −4.
3.2. (a). Real axis, (b). imaginary axis, (c). perpendicular bisector (pass-
ing through the origin) of the line segment joining the points z0 and z1,
(d). circle center z = 1, radius 1; i.e., (x − 1)2
+ y2
= 1, (e). circle
center (−2/3, 8/3), radius
√
32/3, (f). circle, (g). 0 y 2π, infinite
strip, (h). region interior to parabola y2
= 2(x − 1/2) but below the line
y = 3, (i). ellipse with foci at z1, z2 and major axis 2a (j). circle.
3.3. Use |z|2
= zz.
3.4. (a). z4 = zzzz = z z z z = (z)4
, (b).
z1
z2z3
= z1
z2z3
= z1
z2z3
.
3.5. If |z| = 1, then z = z−1
.
3.6. |Im (1 − z + z2
)| ≤ |1 − z + z2
| ≤ |1| + |z| + |z2
| ≤ 7, |z4
− 4z2
+ 3| =
|z2
− 3||z2
− 1| ≥ (|z2
| − 3)(|z2
| − 1).
3.7. We have
2z − 1
4 + z2
≤
2|z| + 1
|4 − |z|2|
=
2 · 3 + 1
|4 − 32|
=
7
5
and
2z − 1
4 + z2
≥
|2|z| − 1|
|4 + |z|2|
=
2 · 3 − 1
4 + 32
=
5
13
.
3.8. We shall prove that |1 − zw| ≥ |z −w|. We have |1 − zw|2
− |z −w|2
=
(1−zw)(1−zw)−(z−w)(z−w) = 1−zw−zw+zwzw−zz+zw+wz−ww =
1 − |z|2
− |w|2
+|z|2
|w|2
= (1 − |z|2
)(1 − |w|2
) ≥ 0 since |z| ≤ 1 and |w| ≤ 1.
Equality holds when |z| = |w| = 1.
3.9. (a). (z)2
= z2
iff z2
− (z)2
= 0 iff (z + z)(z − z) = 0 iff either
2Re(z) = z + z = 0 or 2iIm(z) = z − z = 0 iff z is purely imaginary or z is
real. (b). Write z = x+iy. Consider 2|z|2
−(|Re z|+|Im z|)2
= 2(x2
+y2
)−
(|x|+|y|)2
= 2x2
+2y2
−(x2
+y2
+2|x|y|) = x2
+y2
−2|x||y| = (|x|−|y|)2
≥ 0.
3.10. Use the triangle inequality.](https://image.slidesharecdn.com/1020158-250521105736-81521d56/85/An-Introduction-To-Complex-Analysis-1st-Edition-Ravi-P-Agarwal-37-320.jpg)
![24 Lecture 4
Let z1 and z2 be two points in the complex plane. The line segment
joining z1 and z2 is the set {w ∈ C : w = z1 + t(z2 − z1), 0 ≤ t ≤ 1}.
Figure 4.7
·
·
Segment [z1, z2]
z1
z2
Now let z1, z2, · · · , zn+1 be n + 1 points in the complex plane. For each
k = 1, 2, · · · , n, let k denote the line segment joining zk to zk+1. Then the
successive line segments 1, 2, · · · , n form a continuous chain known as a
polygonal path joining z1 to zn+1.
Figure 4.8
x
y
0
· ·
·
·
·
·
z1
z2
z3 zn
zn+1
1 2
n
An open set S is said to be connected if every pair of points z1, z2 in
S can be joined by a polygonal path that lies entirely in S. The polygonal
path may contain line segments that are either horizontal or vertical. An
open connected set is called a domain. Clearly, all open disks are domains.
If S is a domain and S = A ∪ B, where A and B are open and disjoint;
i.e., A ∩ B = ∅, then either A = ∅ or B = ∅. A domain together with some,
none, or all of its boundary points is called a region.
· ·
·
·
·
z2
z1
Connected
z2
·
·
z1
Not connected](https://image.slidesharecdn.com/1020158-250521105736-81521d56/85/An-Introduction-To-Complex-Analysis-1st-Edition-Ravi-P-Agarwal-43-320.jpg)
![Set Theory in the Complex Plane 27
(c).
Open connected (ellipse)
•
• (d).
Open connected
· ·
4.2. Points 0 and 2 cannot be connected by a polygonal line.
0 2
· x
y
4.3. (a). Let w ∈ ∩n
k=1Sk. Then w ∈ Sk, k = 1, · · · , n. Since each Sk is
open, there is an rk 0 such that {z : |z − w| rk} ⊂ Sk, k = 1, · · · , n.
Let r = min{r1, · · · , rn}. Then {z : |z − w| r} ⊆ {z : |z − w| rk} ⊂
Sk, k = 1, · · · , n. Thus, {z : |z − w| r} ⊂ ∩n
k=1Sk.
(b). Use the property of open sets.
(c). ∩∞
n=1{z : |z| 1/n} = {0}.
4.4. (a). Verify directly. (b). For a, b, c ≥ 0 and c ≤ a + b, use c
1+c
≤
a
1+a
+ b
1+b
. (c). Verify directly.
4.5. The straight line passing through (x, y, 0) and (0, 0, 1) in parametric
form is (tx, ty, 1 − t). This line also passes through the point (ξ, η, ζ) on
the Riemann sphere, provided t2
x2
+ t2
y2
+ (1 − t)2
= 1. This gives t = 0
and t = 2/(x2
+ y2
+ 1). The value t = 0 gives the north pole, whereas
t = 2/(x2
+ y2
+ 1) gives (ξ, η, ζ) =
2x
x2+y2+1 , 2y
x2+y2+1 , x2
+y2
−1
x2+y2+1
. From
this, it also follows that |z|2
+ 1 = 2
1−ζ
.
4.6. If (ξ1, η1, ζ1) and (ξ2, η2, ζ2) are the points on S corresponding to
z1 and z2, then d(z1, z2) = [(ξ1 − ξ2)2
+ (η1 − η2)2
+ (ζ1 − ζ2)2
]1/2
=
[2 − 2(ξ1ξ2 + η1η2 + ζ1ζ2)]1/2
. Now use Problem 4.5. If z2 = ∞, then again
from Problem 4.5, we have d(z1, ∞) = [ξ2
1 +η2
1 +(ζ1−1)2
]1/2
= [2−2ζ1]1/2
=
[2 − 2(|z1|2
− 1)/(|z1|2
+ 1)]1/2
= 2/(|z1|2
+ 1)1/2
.](https://image.slidesharecdn.com/1020158-250521105736-81521d56/85/An-Introduction-To-Complex-Analysis-1st-Edition-Ravi-P-Agarwal-46-320.jpg)
![30 Lecture 5
(i). Given any 0, we have
|z2
− 2 − (−2 + 2i)| = |z2
− 2i| = |z2 + 2i| = |z2
+ 2i|
= |z − (1 − i)||z + (1 − i)|
≤ |z − (1 − i)|(|z − (1 − i)| + 2|1 − i|)
≤ |z − (1 − i)|(1 + 2
√
2) if |z − (1 − i)| 1
if |z − (1 − i)| min 1,
1 + 2
√
2
.
(ii). Given any 0, from (i) we have
||z2
− 2| −
√
8| = ||z2
− 2| − | − 2 + 2i||
≤ |z2
− 2 − (−2 + 2i)|
if |z − (1 − i)| min 1,
1 + 2
√
2
.
Example 5.4. (i). Clearly, limz→z0 z = z0. (ii). From the inequalities
|Re(z − z0)| ≤ [(Re(z − z0))2
+ (Im(z − z0))2
]1/2
= |z − z0|,
|Im(z − z0)| ≤ |z − z0|,
it follows that limz→z0 Re z = Re z0, limz→z0 Im z = Im z0.
Example 5.5. limz→0(z/z) does not exist. Indeed, we have
lim
z → 0
along x-axis
z
z
= lim
x→0
x + i0
x − i0
= 1,
lim
z → 0
along y-axis
z
z
= lim
y→0
0 + iy
0 − iy
= − 1.
The following result relates real limits of u(x, y) and v(x, y) with the
complex limit of f(z) = u(x, y) + iv(x, y).
Theorem 5.1. Let f(z) = u(x, y) + iv(x, y), z0 = x0 + iy0, and w0 =
u0 + iv0. Then, limz→z0 f(z) = w0 if and only if limx→x0, y→y0 u(x, y) = u0
and limx→x0, y→y0 v(x, y) = v0.
In view of Theorem 5.1 and the standard results in calculus, the follow-
ing theorem is immediate.
Theorem 5.2. If limz→z0 f(z) = A and limz→z0 g(z) = B, then
(i) limz→z0 (f(z) ± g(z)) = A ± B, (ii) limz→z0 f(z)g(z) = AB, and
(iii) limz→z0
f(z)
g(z)
=
A
B
if B = 0.](https://image.slidesharecdn.com/1020158-250521105736-81521d56/85/An-Introduction-To-Complex-Analysis-1st-Edition-Ravi-P-Agarwal-49-320.jpg)
![Complex Functions 35
if |z − (1 − i)| min{1, /5},
(c). |z − z0| = |z − z0| if |z − z0| ,
(d). |2z+1−(5−2i)| = |2z−(4−2i)| = 2|z−(2−i)| if |z−(2−i)| /2.
5.5. (a). −8i, (b). −7i/2, (c). 6i, (d). −1/2, (e). 1, (f). ∞.
5.6. (a). limz→0,z=x(z/z)2
= 1, limz→0,y=x(z/z)2
= −1.
(b). Let 0. Choose δ = . Then, 0 |z − 0| δ implies |z2
/z − 0| =
|z| .
5.7. Since limz→z0 f(z) = 0, given any 0, there exists δ 0 such that
|f(z)−0| /M whenever |z−z0| δ. Thus, |f(z)g(z)−0| = |f(z)||g(z)| ≤
M|f(z)| if |z − z0| δ.
5.8. Use the fact ||f(z)| − |w0|| ≤ |f(z) − w0|.
5.9. Let 0. Since g is continuous at w0, there exists a δ1 0 such that
|w − w0| δ1 implies that |g(w) − g(w0)| . Now, f is continuous at z0,
so there exists a δ2 0 such that |z − z0| δ2 implies |f(z) − f(z0)| δ1.
Combining these, we find that |z − z0| δ2 implies |f(z) − f(z0)| δ1,
which in turn implies |(g ◦ f)(z) − (g ◦ f)(z0)| = |g[f(z)] − g[f(z0)]| .
5.10. Continuous at 1, discontinuous at −1, i, −i.
5.11. f is not continuous at z0 if there exists 0 0 with the following
property: For every δ 0, there exists zδ such that |zδ − z0| δ and
|f(zδ) − f(z0)| ≥ 0. Now let z0 = x0 0. Take 0 = 3π/2. For each δ 0,
let zδ = x0 − i(δ/2). Then, |zδ − z0| = |iδ/2| = δ/2 δ, −π f(zδ)
−π/2, f(z0) = π, so |f(zδ) − f(z0)| 3π/2 = 0, and f is not continuous
at z0. Thus, f is not continuous at every point on the negative real axis. It
is also not continuous at z = 0 because it is not defined there.
5.12. If f is continuous at z0, then given 0 there exists a δ 0 such
that |z1 − z0| δ/2 ⇒ |f(z1) − f(z0)| /2 and |z2 − z0| δ/2 ⇒
|f(z2) − f(z0)| /2. But then |z1 − z2| ≤ |z1 − z0| + |z0 − z2| δ ⇒
|f(z1) − f(z2)| ≤ |f(z1) − f(z0)| + |f(z2) − f(z0)| . For the converse, we
assume that 0 |z−z0| δ, 0 |z
−z0| δ; otherwise, we can take z
= z0
and then there is nothing to prove. Let zn → z0, zn = z0, and 0. There
is a δ 0 such that 0 |z −z0| δ, |z
− z0| δ implies |f(z)−f(z
)| ,
and there is an N such that n ≥ N implies 0 |zn − z0| δ. Then, for
m, n ≥ N, we have |f(zm) − f(zn)| . So, w0 = limn→∞ f(zn) exists. To
see that limz→z0 f(z) = w0, take a δ1 0 such that 0 |z − z0| δ1, 0
|z
− z0| δ1 implies |f(z) − f(z
)| /2, and an N1 such that n ≥ N1
implies 0 |zn − z0| δ1 and |f(zn) − w0| /2. Then, 0 |z − z0| δ1
implies |f(z) − w0| ≤ |f(z) − f(zN1 )| + |f(zN1 ) − w0| .
5.13. (i). Suppose that f : U → C is continuous and U is compact. Con-
sider a covering of f(U) to be open sets V. The inverse images f−1
(V ) are
open and form a covering of U. Since U is compact, by Theorem 4.4 we
can select a finite subcovering such that U ⊂ f−1
(V1) ∪ · · · ∪ f−1
(Vn). It
follows that f(U) ⊂ V1 ∪ · · · ∪ Vn, which in view of Theorem 4.4 implies
that f(U) is compact. (ii). Suppose that f : U → C is continuous and U
is connected. If f(U) = A ∪ B where A and B are open and disjoint, then
U = f−1
(A) ∪ f−1
(B), which is a union of disjoint and open sets. Since U](https://image.slidesharecdn.com/1020158-250521105736-81521d56/85/An-Introduction-To-Complex-Analysis-1st-Edition-Ravi-P-Agarwal-54-320.jpg)
![all people as far as you can). To bring into the lists a combatant
equal, or rather superior to Egoism and Malice combined—this is the
task of all Ethics. Heic Rhodus, heic salta![2]
The division of human duty into two classes has long been
recognised, and no doubt owes its origin to the nature of morality
itself. We have. (1) the duties ordained by law (otherwise called the
—perfect, obligatory, narrower duties), and (2) those prescribed by
virtue (otherwise called imperfect, wider, meritorious, or, preferably,
the duties taught by love). On p. 57 (R., p. 60) we find Kant desiring
to give a further confirmation to the moral principle, which he
propounded, by undertaking to derive this classification from it. But
the attempt turns out to be so forced, and so obviously bad, that it
only testifies in the strongest way against the soundness of his
position. For, according to him, the duties laid down by statutes rest
on a precept, the contrary of which, taken as a general natural law,
is declared to be quite unthinkable without contradiction; while the
duties inculcated by virtue are made to depend on a maxim, the
opposite of which can (he says) be conceived as a general natural
law, but cannot possibly be wished for. I beg the reader to reflect
that the rule of injustice, the reign of might instead of right, which in
the Kantian view is not even thinkable as a natural law, is in reality,
and in point of fact, the dominant order of things not only in the
animal kingdom, but among men as well. It is true that an attempt
has been made among civilised peoples to obviate its injurious
effects by means of all the machinery of state government; but as
soon as this, wherever, or of whatever kind, it be, is suspended or
eluded, the natural law immediately resumes its sway. Indeed
between nation and nation it never ceases to prevail; the customary
jargon about justice is well known to be nothing but diplomacy's
official style; the real arbiter is brute force. On the other hand,
genuine, i.e., voluntary, acts of justice, do occur beyond all doubt,
but always only as exceptions to the rule. Furthermore: wishing to
give instances by way of introducing the above-mentioned
classification, Kant establishes the duties prescribed by law first (p.
53; R., p. 48) through the so-called duty towards oneself,—the duty](https://image.slidesharecdn.com/1020158-250521105736-81521d56/85/An-Introduction-To-Complex-Analysis-1st-Edition-Ravi-P-Agarwal-57-320.jpg)
![of not ending one's life voluntarily, if the pain outweigh the pleasure.
Accordingly, the rule of suicide is held to be not even thinkable as a
general natural law. I, on the contrary, maintain that, since here
there can be no intervention of state control, it is exactly this rule
which is proved to be an actually existing, unchecked natural law.
For it is absolutely certain (as daily experience attests) that men in
the vast majority of cases turn to self-destruction directly the
gigantic strength of the innate instinct of self-preservation is
distinctly overpowered by great suffering. To suppose that there is
any thought whatever that can have a deferring effect, after the fear
of death, which is so strong and so closely bound up with the nature
of every living thing, has shown itself powerless; in other words, to
suppose that there is a thought still mightier than this fear—is a
daring assumption, all the more so, when we see, that it is one
which is so difficult to discover that the moralists are not yet able to
determine it with precision. In any case, it is certain that arguments
against suicide of the sort put forward by Kant in this connection (p.
53: R., p. 48, and p. 67; R., p. 57) have never hitherto restrained
any one tired of life even for a moment. Thus a natural law, which
incontestably exists, and is operative every day, is declared by Kant
to be simply unthinkable without contradiction, and all for the sake
of making his Moral Principle the basis of the classification of duties!
At this point it is, I confess, not without satisfaction that I look
forward to the groundwork which I shall give to Ethics in the sequel.
From it the division of Duty into what is prescribed by law, and what
is taught by love, or, better, into justice and loving-kindness, results
quite naturally though a principle of separation which arises from the
nature of the subject, and which entirely of itself draws a sharp line
of demarkation; so that the foundation of Morals, which I shall
present, has in fact ready to hand that confirmation, to which Kant,
with a view to support his own position, lays a completely
groundless claim.
[1] How rashly do we sanction an unjust law, which will come home to ourselves!
—(Hor., Sat., Lib. I., iii. 67.)](https://image.slidesharecdn.com/1020158-250521105736-81521d56/85/An-Introduction-To-Complex-Analysis-1st-Edition-Ravi-P-Agarwal-58-320.jpg)
![[2] Here is Rhodes, here make your leap! I.e., Here is the place of trial, here let
us see what you can do! This Latin proverb is derived from one of Aesop's fables.
A braggart boasts of having once accomplished a wonderful jump in Rhodes, and
appeals to the evidence of the eye-witnesses. The bystanders then exclaim:
Friend, if this be true, you have no need of witnesses; for this is Rhodes, and
your leap you can make here. The words are: ἀλλ', ὦ ϕίλε, εἰ τοῡtο ἀληθές ἐστιν,
oὐδὲν δεῑ σοι μαρτύρων αὕtη γὰρ 'Rόδος καὶ πήδημα. V. Fabulae Aesopicae
Collectae. Edit. Halm, Leipzig: Teubner. 1875. Nr. 203b, p. 102. The other version
of the fable (Nr. 203, p. 101) gives: ὦ oὗtos, eἰ ἀlêthès τoῡτ ἐstin, oὐdὲn deῑ soi
martyrôn ἰdoὺ ἡ Ρόδος, ἰdoὺ kaὶ τὸ πήδημα.—(Translator.)
CHAPTER VI.
ON THE DERIVED FORMS OF THE LEADING
PRINCIPLE OF THE KANTIAN ETHICS.
It is well known that Kant put the leading principle of his Ethics into
another quite different shape, in which it is expressed directly; the
first being indirect, indeed nothing more than an indication as to
how the principle is to be sought for. Beginning at p. 63 (R., p. 55),
he prepares the way for his second formula by means of very
strange, ambiguous, not to say distorted,[1]
definitions of the
conceptions End and Means, which may be much more simply and
correctly denoted thus: an End is the direct motive of an act of the
Will, a Means the indirect: simplex sigillum veri (simplicity is the
seal of truth). Kant, however, slips through his wonderful
enunciations to the statement: Man, indeed every rational being,
exists as an end in himself. On this I must remark that to exist
as an end in oneself is an unthinkable expression, a contradictio
in adjecto.[2]
To be an end means to be an object of volition. Every
end can only exist in relation to a will, whose end, i.e., (as above
stated), whose direct motive it is. Only thus can the idea, end have
any sense, which is lost as soon as such connection is broken. But](https://image.slidesharecdn.com/1020158-250521105736-81521d56/85/An-Introduction-To-Complex-Analysis-1st-Edition-Ravi-P-Agarwal-59-320.jpg)
![morals they are of course at once outlawed; they are merely
things, simply means to ends of any sort; and so they are good
for vivisection, for deer-stalking, bull-fights, horse-races, etc., and
they may be whipped to death as they struggle along with heavy
quarry carts. Shame on such a morality which is worthy of Pariahs,
Chandalas and Mlechchas[3]
; which fails to recognise the Eternal
Reality immanent in everything that has life, and shining forth with
inscrutable significance from all eyes that see the sun! This is a
morality which knows and values only the precious species that gave
it birth; whose characteristic—reason—it makes the condition under
which a being may be an object of moral regard.
By this rough path, then,—indeed, per fas et nefas (by fair means
and by foul), Kant reaches the second form in which he expresses
the fundamental principle of his Ethics: Act in such a way that you
at all times treat mankind, as much in your own person, as in the
person of every one else, not only as a Means, but also as an End.
Such a statement is a very artificial and roundabout way of saying:
Do not consider yourself alone, but others also; this in turn is a
paraphrase for: Quod tibi fieri non vis, alteri ne feceris (do not to
another what you are unwilling should be done to yourself); and the
latter, as I have said, contains nothing but the premises to the
conclusion, which is the true and final goal of all morals and of all
moralising; Neminem laede, immo omnes, quantum potes juva (do
harm to no one; but rather help all people as far as lies in your
power). Like all beautiful things, this proposition looks best unveiled.
Be it only observed that the alleged duties towards oneself are
dragged into this second Kantian edict intentionally and not without
difficulty. Some place of course had to be found for them.[4]
Another objection that could be raised against the formula is that
the malefactor condemned to be executed is treated merely as an
instrument, and not as an end, and this with perfectly good reason;
for he is the indispensable means of upholding the terror of the law
by its fulfilment, and of thus accomplishing the law's end—the
repression of crime.](https://image.slidesharecdn.com/1020158-250521105736-81521d56/85/An-Introduction-To-Complex-Analysis-1st-Edition-Ravi-P-Agarwal-61-320.jpg)
![interest by the Will when acting from a sense of duty. All previous
moral principles had thus (he says) broken down, because the latter
invariably attributed to human actions at bottom a certain interest,
whether originating in compulsion, or in pleasurable attraction—an
interest which might be one's own, or another's (p. 73; R., p. 62).
(Another's: let this be particularly noticed.) Whereas a universally
legislative Will must prescribe actions which are not based on any
interest at all, but solely on a feeling of duty. I beg the reader to
think what this really means. As a matter of fact, nothing less than
volition without motive, in other words, effect without cause.
Interest and Motive are interchangeable ideas; what is interest but
quod mea interest, that which is of importance to me? And is not
this, in one word, whatever stirs and sets in motion my Will?
Consequently, what is an interest other than the working of a motive
upon the Will? Therefore where a motive moves the Will, there the
latter has an interest; but where the Will is affected by no motive,
there in truth it can be as little active, as a stone is able to leave its
place without being pushed or pulled. No educated person will
require any demonstration of this. It follows that every action,
inasmuch as it necessarily must have a motive, necessarily also
presupposes an interest. Kant, however, propounds a second entirely
new class of actions which are performed without any interest, i.e.,
without motive. And these actions are—all deeds of justice and
loving-kindness! It will be seen that this monstrous assumption, to
be refuted, needed only to be reduced to its real meaning, which
was concealed through the word interest being trifled with.
Meanwhile Kant celebrates (p. 74 sqq.; R., p. 62) the triumph of his
Autonomy of the Will by setting up a moral Utopia called the
Kingdom of Ends, which is peopled with nothing but rational
beings in abstracto. These, one and all, are always willing, without
willing any actual thing (i.e., without interest): the only thing that
they will is that they may all perpetually will in accordance with one
maxim (i.e., Autonomy). Difficile est satiram non scribere[5]
(it is
difficult to refrain from writing a satire).](https://image.slidesharecdn.com/1020158-250521105736-81521d56/85/An-Introduction-To-Complex-Analysis-1st-Edition-Ravi-P-Agarwal-64-320.jpg)
![But there is something else to which Kant is led by his autonomy of
the will; and it involves more serious consequences than the little
innocent Kingdom of Ends, which is perfectly harmless and may be
left in peace. I mean the conception of human dignity. Now this
dignity is made to rest solely on man's autonomy, and to lie in the
fact that the law which he ought to obey is his own work, his
relation to it thus being the same as that of the subjects of a
constitutional government to their statutes. As an ornamental finish
to the Kantian system of morals such a theory might after all be
passed over. Only this expression Human Dignity, once it was
uttered by Kant, became the shibboleth of all perplexed and empty-
headed moralists. For behind that imposing formula they concealed
their lack, not to say, of a real ethical basis, but of any basis at all
which was possessed of an intelligible meaning; supposing cleverly
enough that their readers would be so pleased to see themselves
invested with such a dignity that they would be quite satisfied.[6]
Let us, however, look at this conception a little more carefully, and
submit it to the test of reality. Kant (p. 79; R., p. 66) defines dignity
as an unconditioned, incomparable value. This is an explanation
which makes such an effect by its magnificent sound that one does
not readily summon up courage to examine it at close quarters; else
we should find that it too is nothing but a hollow hyperbole, within
which there lurks like a gnawing worm, the contradictio in adjecto.
Every value is the estimation of one thing compared with another; it
is thus a conception of comparison, and consequently relative; and
this relativity is precisely that which forms the essence of the idea.
According to Diogenes Laertius (Book VII., chap. 106),[7]
this was
already correctly taught by the Stoics. He says: τὴn δὲ ἀξίαν εἶναι
ἀμοιβὴν δοκιμάστου, ἢν ἂν ὁ ἔμπειρος τῶν Πραγμάτων τάξῃ ὅμοιον
εἐπεῑν, ἀμείβεσθαι πυροὺς πρὸς τὰς σὺν ἡμιονô κριθάς.[8]
An
incomparable, unconditioned, absolute value, such as dignity
is declared by Kant to be, is thus, like so much else in Philosophy,
the statement in words of a thought which is really unthinkable; just
as much as the highest number, or the greatest space.](https://image.slidesharecdn.com/1020158-250521105736-81521d56/85/An-Introduction-To-Complex-Analysis-1st-Edition-Ravi-P-Agarwal-65-320.jpg)
![as absolutely necessary assumptions, however tacit and concealed,
all the alleged consequences that had been drawn from it.
At the end of this severe investigation, which must also have been
tiring to my readers, perhaps I may be allowed, by way of diversion,
to make a jesting, indeed frivolous comparison. I would liken Kant,
in his self-mystification, to a man who at a ball has been flirting the
whole evening with a masked beauty, in hopes of making a
conquest; till at last, throwing off her disguise, she reveals herself—
as his wife.
[1] To keep the play of words in geschrobene, verschrobene, we may perhaps
render them: twisted ... mistwisted.—(Translator.)
[2] A contradiction in that which is added. A term applied to two ideas which
cannot be brought into a thinkable relationship.—(Translator.)
[3] A Chaṇḍāla (or Ćaṇḍāla) means one who is born of a Brahman woman by a
Śūdra husband, such a union being an abomination. Hence it is a term applied to a
low common person. Mlechcha (or Mleććha) means a foreigner; one who does not
speak Sanskṛit, and is not subject to Hindu institutions. The transition from a a
barbarian to a bad or wicked man, is easy.—(Translator.)
[4] These so-called duties have been discussed in Chapter III. of this Part.
[5] Juvenal, Sat. I. 30.
[6] It appears that G. W. Block in his Neue Grundlegung der Philosophie der
Sitten, 1802, was the first to make Human Dignity expressly and exclusively the
foundation-stone of Ethics, which he then built up entirely on it.
[7] V. Diogenes Laertius, de Clarorum Philosophorum Vitis, etc., edit. O. Gabr.
Cobet. Paris; Didot, 1862. In this edition the passage quoted is in chap. 105 ad
fin.,, p. 182.—(Translator.)
[8] They teach that worth is the equivalent value of a thing which has been
tested, whatever an expert may fix that value to be; as, for instance, to take
wheat in exchange for barley and a mule.—(Translator.)
CHAPTER VII.](https://image.slidesharecdn.com/1020158-250521105736-81521d56/85/An-Introduction-To-Complex-Analysis-1st-Edition-Ravi-P-Agarwal-68-320.jpg)
![KANT'S DOCTRINE OF CONSCIENCE.
The alleged Practical Reason with its Categorical Imperative, is
manifestly very closely connected with Conscience, although
essentially different from it in two respects. In the first place, the
Categorical Imperative, as commanding, necessarily speaks before
the act, whereas Conscience does not till afterwards. Before the act
Conscience can at best only speak indirectly, that is, by means of
reflection, which holds up to it the recollection of previous cases, in
which similar acts after they were committed received its
disapproval. It is on this that the etymology of the word Gewissen
(Conscience) appears to me to rest, because only what has
already taken place is gewiss[1]
(certain). Undoubtedly, through
external inducement and kindled emotion, or by reason of the
internal discord of bad humour, impure, base thoughts, and evil
desires rise up in all people, even in the best. But for these a man is
not morally responsible, and need not load his conscience with
them; since they only show what the genus homo, not what the
individual, who thinks them, would be capable of doing. Other
motives, if not simultaneously, yet almost immediately, come into his
consciousness, and confronting the unworthy inclinations prevent
them from ever being crystallised into deeds; thus causing them to
resemble the out-voted minority of an acting committee. By deeds
alone each person gains an empirical knowledge no less of himself
than of others, just as it is deeds alone that burden the conscience.
For, unlike thoughts, these are not problematic; on the contrary, they
are certain (gewiss), they are unchangeable, and are not only
thought, but known (gewusst). The Latin conscientia,[2]
and the
Greek συνείδησις[3]
have the same sense. Conscience is thus the
knowledge that a man has about what he has done.
The second point of difference between the alleged Categorical
Imperative and Conscience is, that the latter always draws its
material from experience; which the former cannot do, since it is](https://image.slidesharecdn.com/1020158-250521105736-81521d56/85/An-Introduction-To-Complex-Analysis-1st-Edition-Ravi-P-Agarwal-69-320.jpg)
![purely a priori. Nevertheless, we may reasonably suppose that Kant's
Doctrine of Conscience will throw some light on this new conception
of an absolute Ought which he introduced. His theory is most
completely set forth in the Metaphysische Anfangsgründe zur
Tugendlehre, § 13, and in the following criticism I shall assume that
the few pages which contain it are lying before the reader.
The Kantian interpretation of Conscience makes an exceedingly
imposing effect, before which one used to stand with reverential
awe, and all the less confidence was felt in demurring to it, because
there lay heavy on the mind the ever-present fear of having
theoretical objections construed as practical, and, if the correctness
of Kant's view were denied, of being regarded as devoid of
conscience. I, however, cannot be led astray in this manner, since
the question here is of theory, not of practice; and I am not
concerned with the preaching of Morals, but with the exact
investigation of the ultimate ethical basis.
We notice at once that Kant employs exclusively Latin legal
terminology, which, however, would seem little adapted to reflect the
most secret stirrings of the human heart. Yet this language, this
judicial way of treating the subject, he retains from first to last, as
though it were essential and proper to the matter. And so we find
brought upon the stage of our inner self a complete Court of justice,
with indictment, judge, plaintiff, defendant, and sentence;—nothing
is wanting. Now if this tribunal, as portrayed by Kant, really existed
in our breasts, it would be astonishing if a single person could be
found to be, I do not say, so bad, but so stupid, as to act against
his conscience. For such a supernatural assize, of an entirely special
kind, set up in our consciousness, such a secret court—like another
Fehmgericht[4]
—held in the dark recesses of our inmost being, would
inspire everybody with a terror and fear of the gods strong enough
to really keep him from grasping at short transient advantages, in
face of the dreadful threats of superhuman powers, speaking in
tones so near and so clear. In real life, on the contrary, we find, that
the efficiency of conscience is generally considered such a vanishing
quantity that all peoples have bethought themselves of helping it out](https://image.slidesharecdn.com/1020158-250521105736-81521d56/85/An-Introduction-To-Complex-Analysis-1st-Edition-Ravi-P-Agarwal-70-320.jpg)
![by means of positive religion, or even of entirely replacing it by the
latter. Moreover, if Conscience were indeed of this peculiar nature,
the Royal Society could never have thought of the question put for
the present Prize Essay.
But if we look more closely at Kant's exposition, we shall find that its
imposing effect is mainly produced by the fact that he attributes to
the moral verdict passed on ourselves, as its peculiar and essential
characteristic, a form which in fact is not so at all. This metaphorical
bar of judgment is no more applicable to moral self-examination
than it is to every other reflection as regards what we have done,
and might have done otherwise, where no ethical question is
involved. For it is not only true that the same procedure of
indictment, defence, and sentence is occasionally assumed by that
obviously spurious and artificial conscience which is based on mere
superstition; as, for instance, when a Hindu reproaches himself with
having been the murderer of a cow, or when a Jew remembers that
he has smoked his pipe at home on the Sabbath; but even the self-
questioning which springs from no ethical source, being indeed
rather unmoral than moral, often appears in a shape of this sort, as
the following case may exemplify. Suppose I, good-naturedly, but
thoughtlessly, have made myself surety for a friend, and suppose
there comes with evening the clear perception of the heavy
responsibility I have taken on myself—a responsibility that may
easily involve me in serious trouble, as the wise old saying, ἐγγύα
παρά δ' ἃτα![5]
predicts; then at once there rise up within me the
Accuser and the Counsel for the defence, ready to confront each
other. The latter endeavours to palliate my rashness in giving bail so
hastily, by pointing out the stress of circumstance or of obligation, or,
it may be, the simple straightforwardness of the transaction;
perhaps he even seeks excuse by commending my kind heart. Last
of all comes the Judge who inexorably passes the sentence: A fool's
piece of work! and I am overwhelmed with confusion So much for
this judicial form of which Kant is so fond; his other modes of
expression are, for the most part, open to the same criticism. For
instance, that which he attributes to conscience, at the beginning of](https://image.slidesharecdn.com/1020158-250521105736-81521d56/85/An-Introduction-To-Complex-Analysis-1st-Edition-Ravi-P-Agarwal-71-320.jpg)
![the paragraph, as its peculiar property, applies equally to all other
scruples of an entirely different sort. He says: It (conscience)
follows him like his shadow, try though he may to escape. By
pleasures and distractions he may be stupefied and billed to sleep,
but he cannot avoid occasionally waking up and coming to himself;
and then he is immediately aware of the terrible voice, etc.
Obviously, this may be just as well understood, word for word, of the
secret consciousness of some person of private means, who feels
that his expenses far exceed his income, and that thus his capital is
being affected, and will gradually melt away.
We have seen that Kant represents the use of legal terms as
essential to the subject, and that he keeps to them from beginning
to end; let it now be noted how he employs the same style for the
following finely devised sophism. He says: That a person accused
by his conscience should be identified with the judge is an absurd
way of portraying a court of justice; for in that case the accuser
would invariably lose. And he adds, by way of elucidating this
statement, a very ambiguous and obscure note. His conclusion is
that, if we would avoid falling into a contradiction, we must think of
the judge (in the judicial conscience-drama that is enacted in our
breasts) as different from us, in fact, as another person; nay more,
as one that is an omniscient knower of hearts, whose hests are
obligatory on all, and who is almighty for every purpose of executive
authority.[6]
He thus passes by a perfectly smooth path from
conscience to superstition, making the latter a necessary
consequence of the former; while he is secretly sure that he will be
all the more willingly followed because the reader's earliest training
will have certainly rendered him familiar with such ideas, if not have
made them his second nature. Here, then, Kant finds an easy task,—
a thing he ought rather to have despised; for he should have
concerned himself not only with preaching, but also with practising
truthfulness. I entirely reject the above quoted sentence, and all the
conclusions consequent thereon, and I declare it to be nothing but a
shuffling trick. It is not true that the accuser must always lose,
when the accused is the same person as the judge; at least not in](https://image.slidesharecdn.com/1020158-250521105736-81521d56/85/An-Introduction-To-Complex-Analysis-1st-Edition-Ravi-P-Agarwal-72-320.jpg)
![the court of judgment in our hearts. In the instance I gave of one
man going surety for another, did the accuser lose? Or must we in
this case also, if we wish to avoid a contradiction, really assume a
personification after Kant's fashion, and be driven to view objectively
as another person that voice whose deliverance would have been
those terrible words: A fool's piece of work!? A sort of Mercury,
forsooth, in living flesh? Or perhaps a prosopopoeia of the Μῆτις
(cunning) recommended by Homer (Il. xxiii. 313 sqq.)?[7]
But thus
we should only be landed, as before, on the broad path of
superstition, aye, and pagan superstition too.
It is in this passage that Kant indicates his Moral Theology, briefly
indeed, yet not without all its vital points. The fact that he takes
care, not to attribute to it any objective validity, but rather to present
it merely as a form subjectively unavoidable, does not free him from
the arbitrariness with which he constructs it, even though he only
claims its necessity for human consciousness. His fabric rests, as we
have seen, on a tissue of baseless assumptions.
So much, then, is certain. The entire imagery—that of a judicial
drama—whereby Kant depicts conscience is wholly unessential and
in no way peculiar to it; although he keeps this figure, as if it were
proper to the subject, right through to the end, in order finally to
deduce certain conclusions from it. As a matter of fact it is a
sufficiently common form, which our thoughts easily take when we
consider any circumstance of real life. It is due for the most part to
the conflict of opposing motives which usually spring up, and which
are successively weighed and tested by our reflecting reason. And
no difference is made whether these motives are moral or egoistic in
their nature, nor whether our deliberations are concerned with some
action in the past, or in the future. Now if we strip from Kant's
exposition its dress of legal metaphor, which is only an optional
dramatic appendage, the surrounding nimbus with all its imposing
effect immediately disappears as well, and there remains nothing but
the fact that sometimes, when we think over our actions, we are
seized with a certain self-dissatisfaction, which is marked by a
special characteristic. It is with our conduct per se that we are](https://image.slidesharecdn.com/1020158-250521105736-81521d56/85/An-Introduction-To-Complex-Analysis-1st-Edition-Ravi-P-Agarwal-73-320.jpg)
![discontented, not with its result, and this feeling does not, as in
every other case in which we regret the stupidity of our behaviour,
rest on egoistic grounds. For on these occasions the cause of our
dissatisfaction is precisely because we have been too egoistic,
because we have taken too much thought for ourselves, and not
enough for our neighbour; or perhaps even because, without any
resulting advantage, we have made the misery of others an object in
itself. That we may be dissatisfied with ourselves, and saddened by
reason of sufferings which we have inflicted, not undergone, is a
plain fact and impossible to be denied. The connection of this with
the only ethical basis that can stand an adequate test we shall
examine further on. But Kant, like a clever special pleader, tried by
magnifying and embellishing the original datum to make all that he
possibly could of it, in order to prepare a very broad foundation for
his Ethics and Moral Theology.
[1] Both words are, of course, derived from wissen = scire = εἱδέναι.—
(Translator.)
[2] Cf. Horace's conscire sibi, pallescere culpa: Epist. I. 1, 61. To be conscious of
having done wrong, to turn pale at the thought of the crime.
[3] Συνείδησις = consciousness (of right or wrong done).—(Translator.)
[4] The celebrated Secret Tribunal of Westphalia, which came into prominence
about A.D. 1220. In A.D. 1335 the Archbishop of Cologne was appointed head of
all the Fehme benches in Westphalia by the Emperor Charles IV. The reader will
remember the description of the trial scene in Scott's Anne of Geierstein. Perhaps
the Court of Star Chamber comes nearest to it in English History.—(Translator.)
[5] If you give a pledge, be sure that Ate (the goddess of mischief) is beside you;
i.e., beware of giving pledges.—Thales ap. Plat. Charm. 165 A.
[6] Kant leads up to this position with great ingenuity, by having recourse to the
theory of the two characters coexistent in man—the noumenal (or intelligible) and
the empirical; the one being in time, the other, timeless; the one, fast bound by
the law of causality, the other free.—(Translator.)
[7] Greek: Άλλ' ἄγε δὴ σύ, ϕίλος, μêτιν ἐμβάλλεο θυμῷ, κ.τ.λ.](https://image.slidesharecdn.com/1020158-250521105736-81521d56/85/An-Introduction-To-Complex-Analysis-1st-Edition-Ravi-P-Agarwal-74-320.jpg)
![CHAPTER VIII.
KANT'S DOCTRINE OF THE INTELLIGIBLE [1]
AND EMPIRICAL CHARACTER. THEORY OF
FREEDOM.
The attack I have made, in the cause of truth, on Kant's system of
Morals, does not, like those of my predecessors, touch the surface
only, but penetrates to its deepest roots. It seems, therefore, only
just that, before I leave this part of my subject, I should bring to
remembrance the brilliant and conspicuous service which he
nevertheless rendered to ethical science. I allude to his doctrine of
the co-existence of Freedom and Necessity. We find it first in the
Kritik der Reinen Vernunft (pp. 533-554 of the first, and pp. 561-582
of the fifth, edition); but it is still more clearly expounded in the
Kritik der Praktischen Vernunft (fourth edition, pp. 169-179; R., pp.
224-231).
The strict and absolute necessity of the acts of Will, determined by
motives as they arise, was first shown by Hobbes, then by Spinoza,
and Hume, and also by Dietrich von Holbach in his Système de la
Nature; and lastly by Priestley it was most completely and precisely
demonstrated. This point, indeed, has been so clearly proved, and
placed beyond all doubt, that it must be reckoned among the
number of perfectly established truths, and only crass ignorance
could continue to speak of a freedom, of a liberum arbitrium
indifferentiae (a free and indifferent choice) in the individual acts of
men. Nor did Kant, owing to the irrefutable reasoning of his
predecessors, hesitate to consider the Will as fast bound in the
chains of Necessity, the matter admitting, as he thought, of no
further dispute or doubt. This is proved by all the passages in which
he speaks of freedom only from the theoretical standpoint.
Nevertheless, it is true that our actions are attended with a
consciousness of independence and original initiative, which makes](https://image.slidesharecdn.com/1020158-250521105736-81521d56/85/An-Introduction-To-Complex-Analysis-1st-Edition-Ravi-P-Agarwal-75-320.jpg)
An Introduction To Complex Analysis 1st Edition Ravi P Agarwal
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- 6. An Introduction to Complex Analysis
- 8. Sandra Pinelas Ravi P. Agarwal • Kanishka Perera An Introduction to Complex Analysis
- 9. e-ISBN 978-1-4614-0195-7 DOI 10.1007/978-1-4614-0195-7 Ravi P. Agarwal Department of Mathematics Sandra Pinelas Department of Mathematics Azores University Kanishka Perera Department of Mathematical Sciences Florida Institute of Technology Melbourne FL 32901, USA © Springer Science+Business Media, LLC 2011 subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer All rights reserved. This work may not be translated or copied in whole or in part without the written are not identified as such, is not to be taken as an expression of opinion as to whether or not they are The use in this publication of trade names, trademarks, service marks, and similar terms, even if they permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, Library of Congress Control Number: 2011931536 software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. ISBN 978-1-4614-0194-0 Springer New York Dordrecht Heidelberg London Mathematics Subject Classification (2010): M12074, M12007 , , Apartado 1422 9501-801 Ponta Delgada, Portugal sandra.pinelas@clix.pt kperera@fit.edu Florida Institute of Technology Melbourne FL 32901, USA , agarwal@fit.edu
- 10. Dedicated to our mothers: Godawari Agarwal, Soma Perera, and Maria Pinelas
- 12. Preface Complex analysis is a branch of mathematics that involves functions of complex numbers. It provides an extremely powerful tool with an unex- pectedly large number of applications, including in number theory, applied mathematics, physics, hydrodynamics, thermodynamics, and electrical en- gineering. Rapid growth in the theory of complex analysis and in its appli- cations has resulted in continued interest in its study by students in many disciplines. This has given complex analysis a distinct place in mathematics curricula all over the world, and it is now being taught at various levels in almost every institution. Although several excellent books on complex analysis have been written, the present rigorous and perspicuous introductory text can be used directly in class for students of applied sciences. In fact, in an effort to bring the subject to a wider audience, we provide a compact, but thorough, intro- duction to the subject in An Introduction to Complex Analysis. This book is intended for readers who have had a course in calculus, and hence it can be used for a senior undergraduate course. It should also be suitable for a beginning graduate course because in undergraduate courses students do not have any exposure to various intricate concepts, perhaps due to an inadequate level of mathematical sophistication. The subject matter has been organized in the form of theorems and their proofs, and the presentation is rather unconventional. It comprises 50 class tested lectures that we have given mostly to math majors and en- gineering students at various institutions all over the globe over a period of almost 40 years. These lectures provide flexibility in the choice of ma- terial for a particular one-semester course. It is our belief that the content in a particular lecture, together with the problems therein, provides fairly adequate coverage of the topic under study. A brief description of the topics covered in this book follows: In Lec- ture 1 we first define complex numbers (imaginary numbers) and then for such numbers introduce basic operations–addition, subtraction, multipli- cation, division, modulus, and conjugate. We also show how the complex numbers can be represented on the xy-plane. In Lecture 2, we show that complex numbers can be viewed as two-dimensional vectors, which leads to the triangle inequality. We also express complex numbers in polar form. In Lecture 3, we first show that every complex number can be written in exponential form and then use this form to raise a rational power to a given complex number. We also extract roots of a complex number and prove that complex numbers cannot be totally ordered. In Lecture 4, we collect some essential definitions about sets in the complex plane. We also introduce stereographic projection and define the Riemann sphere. This vii
- 13. ensures that in the complex plane there is only one point at infinity. In Lecture 5, first we introduce a complex-valued function of a com- plex variable and then for such functions define the concept of limit and continuity at a point. In Lectures 6 and 7, we define the differentia- tion of complex functions. This leads to a special class of functions known as analytic functions. These functions are of great importance in theory as well as applications, and constitute a major part of complex analysis. We also develop the Cauchy-Riemann equations, which provide an easier test to verify the analyticity of a function. We also show that the real and imaginary parts of an analytic function are solutions of the Laplace equation. In Lectures 8 and 9, we define the exponential function, provide some of its basic properties, and then use it to introduce complex trigonometric and hyperbolic functions. Next, we define the logarithmic function, study some of its properties, and then introduce complex powers and inverse trigonometric functions. In Lectures 10 and 11, we present graphical representations of some elementary functions. Specially, we study graphical representations of the Möbius transformation, the trigonometric mapping sin z, and the function z1/2 . In Lecture 12, we collect a few items that are used repeatedly in complex integration. We also state Jordan’s Curve Theorem, which seems to be quite obvious; however, its proof is rather complicated. In Lecture 13, we introduce integration of complex-valued functions along a directed contour. We also prove an inequality that plays a fundamental role in our later lectures. In Lecture 14, we provide conditions on functions so that their contour integral is independent of the path joining the initial and terminal points. This result, in particular, helps in computing the contour integrals rather easily. In Lecture 15, we prove that the integral of an analytic function over a simple closed contour is zero. This is one of the fundamental theorems of complex analysis. In Lecture 16, we show that the integral of a given function along some given path can be replaced by the integral of the same function along a more amenable path. In Lecture 17, we present Cauchy’s integral formula, which expresses the value of an analytic function at any point of a domain in terms of the values on the boundary of this domain. This is the most fundamental theorem of complex analysis, as it has numerous applications. In Lecture 18, we show that for an analytic function in a given domain all the derivatives exist and are analytic. Here we also prove Morera’s Theorem and establish Cauchy’s inequality for the derivatives, which plays an important role in proving Liouville’s Theorem. In Lecture 19, we prove the Fundamental Theorem of Algebra, which states that every nonconstant polynomial with complex coefficients has at least one zero. Here, for a given polynomial, we also provide some bounds viii Preface
- 14. on its zeros in terms of the coefficients. In Lecture 20, we prove that a function analytic in a bounded domain and continuous up to and including its boundary attains its maximum modulus on the boundary. This result has direct applications to harmonic functions. In Lectures 21 and 22, we collect several results for complex sequences and series of numbers and functions. These results are needed repeatedly in later lectures. In Lecture 23, we introduce a power series and show how to compute its radius of convergence. We also show that within its radius of convergence a power series can be integrated and differentiated term-by-term. In Lecture 24, we prove Taylor’s Theorem, which expands a given analytic function in an infinite power series at each of its points of analyticity. In Lecture 25, we expand a function that is analytic in an annulus domain. The resulting expansion, known as Laurent’s series, involves positive as well as negative integral powers of (z − z0). From ap- plications point of view, such an expansion is very useful. In Lecture 26, we use Taylor’s series to study zeros of analytic functions. We also show that the zeros of an analytic function are isolated. In Lecture 27, we in- troduce a technique known as analytic continuation, whose principal task is to extend the domain of a given analytic function. In Lecture 28, we define the concept of symmetry of two points with respect to a line or a circle. We shall also prove Schwarz’s Reflection Principle, which is of great practical importance for analytic continuation. In Lectures 29 and 30, we define, classify, characterize singular points of complex functions, and study their behavior in the neighborhoods of singularities. We also discuss zeros and singularities of analytic functions at infinity. The value of an iterated integral depends on the order in which the integration is performed, the difference being called the residue. In Lecture 31, we use Laurent’s expansion to establish Cauchy’s Residue Theorem, which has far-reaching applications. In particular, integrals that have a finite number of isolated singularities inside a contour can be integrated rather easily. In Lectures 32-35, we show how the theory of residues can be applied to compute certain types of definite as well as improper real integrals. For this, depending on the complexity of an integrand, one needs to choose a contour cleverly. In Lecture 36, Cauchy’s Residue Theorem is further applied to find sums of certain series. In Lecture 37, we prove three important results, known as the Argu- ment Principle, Rouché’s Theorem, and Hurwitz’s Theorem. We also show that Rouché’s Theorem provides locations of the zeros and poles of mero- morphic functions. In Lecture 38, we further use Rouché’s Theorem to investigate the behavior of the mapping f generated by an analytic func- tion w = f(z). Then we study some properties of the inverse mapping f−1 . We also discuss functions that map the boundaries of their domains to the ix Preface
- 15. boundaries of their ranges. Such results are very important for constructing solutions of Laplace’s equation with boundary conditions. In Lecture 39, we study conformal mappings that have the angle- preserving property, and in Lecture 40 we employ these mappings to es- tablish some basic properties of harmonic functions. In Lecture 41, we provide an explicit formula for the derivative of a conformal mapping that maps the upper half-plane onto a given bounded or unbounded polygonal region. The integration of this formula, known as the Schwarz-Christoffel transformation, is often applied in physical problems such as heat conduc- tion, fluid mechanics, and electrostatics. In Lecture 42, we introduce infinite products of complex numbers and functions and provide necessary and sufficient conditions for their conver- gence, whereas in Lecture 43 we provide representations of entire functions as finite/infinite products involving their finite/infinite zeros. In Lecture 44, we construct a meromorphic function in the entire complex plane with preassigned poles and the corresponding principal parts. Periodicity of analytic/meromorphic functions is examined in Lecture 45. Here, doubly periodic (elliptic) functions are also introduced. The Riemann zeta function is one of the most important functions of classical mathematics, with a variety of applications in analytic number theory. In Lecture 46, we study some of its elementary properties. Lecture 47 is devoted to Bieberbach’s conjecture (now theorem), which had been a chal- lenge to the mathematical community for almost 68 years. A Riemann surface is an ingenious construct for visualizing a multi-valued function. These surfaces have proved to be of inestimable value, especially in the study of algebraic functions. In Lecture 48, we construct Riemann sur- faces for some simple functions. In Lecture 49, we discuss the geometric and topological features of the complex plane associated with dynamical systems, whose evolution is governed by some simple iterative schemes. This work, initiated by Julia and Mandelbrot, has recently found applica- tions in physical, engineering, medical, and aesthetic problems; specially those exhibiting chaotic behavior. Finally, in Lecture 50, we give a brief history of complex numbers. The road had been very slippery, full of confusions and superstitions; how- ever, complex numbers forced their entry into mathematics. In fact, there is really nothing imaginary about imaginary numbers and complex about complex numbers. Two types of problems are included in this book, those that illustrate the general theory and others designed to fill out text material. The problems form an integral part of the book, and every reader is urged to attempt most, if not all of them. For the convenience of the reader, we have provided answers or hints to all the problems. x Preface
- 16. In writing a book of this nature, no originality can be claimed, only a humble attempt has been made to present the subject as simply, clearly, and accurately as possible. The illustrative examples are usually very simple, keeping in mind an average student. It is earnestly hoped that An Introduction to Complex Analysis will serve an inquisitive reader as a starting point in this rich, vast, and ever-expanding field of knowledge. We would like to express our appreciation to Professors Hassan Azad, Siegfried Carl, Eugene Dshalalow, Mohamed A. El-Gebeily, Kunquan Lan, Radu Precup, Patricia J.Y. Wong, Agacik Zafer, Yong Zhou, and Changrong Zhu for their suggestions and criticisms. We also thank Ms. Vaishali Damle at Springer New York for her support and cooperation. Ravi P Agarwal Kanishka Perera Sandra Pinelas xi Preface
- 18. Contents Preface 1. Complex Numbers I 1 2. Complex Numbers II 6 3. Complex Numbers III 11 4. Set Theory in the Complex Plane 20 5. Complex Functions 28 6. Analytic Functions I 37 7. Analytic Functions II 42 8. Elementary Functions I 52 9. Elementary Functions II 57 10. Mappings by Functions I 64 11. Mappings by Functions II 69 12. Curves, Contours, and Simply Connected Domains 77 13. Complex Integration 83 14. Independence of Path 91 15. Cauchy-Goursat Theorem 96 16. Deformation Theorem 102 17. Cauchy’s Integral Formula 111 18. Cauchy’s Integral Formula for Derivatives 116 19. The Fundamental Theorem of Algebra 125 20. Maximum Modulus Principle 132 21. Sequences and Series of Numbers 138 22. Sequences and Series of Functions 145 23. Power Series 151 24. Taylor’s Series 159 25. Laurent’s Series 169 xiii vii
- 19. 26. Zeros of Analytic Functions 177 27. Analytic Continuation 183 28. Symmetry and Reflection 190 29. Singularities and Poles I 195 30. Singularities and Poles II 200 31. Cauchy’s Residue Theorem 207 32. Evaluation of Real Integrals by Contour Integration I 215 33. Evaluation of Real Integrals by Contour Integration II 220 34. Indented Contour Integrals 229 35. Contour Integrals Involving Multi-valued Functions 235 36. Summation of Series 242 37. Argument Principle and Rouché and Hurwitz Theorems 247 38. Behavior of Analytic Mappings 253 39. Conformal Mappings 258 40. Harmonic Functions 267 41. The Schwarz-Christoffel Transformation 275 42. Infinite Products 281 43. Weierstrass’s Factorization Theorem 287 44. Mittag-Leffler Theorem 293 45. Periodic Functions 298 46. The Riemann Zeta Function 303 47. Bieberbach’s Conjecture 308 48. 312 49. Julia and Mandelbrot Sets 316 50. History of Complex Numbers 321 References for Further Reading 327 Index 329 xiv Contents The Riemann Surfaces
- 20. Lecture 1 Complex Numbers I We begin this lecture with the definition of complex numbers and then introduce basic operations-addition, subtraction, multiplication, and divi- sion of complex numbers. Next, we shall show how the complex numbers can be represented on the xy-plane. Finally, we shall define the modulus and conjugate of a complex number. Throughout these lectures, the following well-known notations will be used: I N = {1, 2, · · ·}, the set of all natural numbers; Z = {· · · , −2, −1, 0, 1, 2, · · ·}, the set of all integers; Q = {m/n : m, n ∈ Z, n = 0}, the set of all rational numbers; I R = the set of all real numbers. A complex number is an expression of the form a + ib, where a and b ∈ I R, and i (sometimes j) is just a symbol. C = {a + ib : a, b ∈ I R}, the set of all complex numbers. It is clear that I N ⊂ Z ⊂ Q ⊂ I R ⊂ C. For a complex number, z = a + ib, Re(z) = a is the real part of z, and Im(z) = b is the imaginary part of z. If a = 0, then z is said to be a purely imaginary number. Two complex numbers, z and w are equal; i.e., z = w, if and only if, Re(z) = Re(w) and Im(z) = Im(w). Clearly, z = 0 is the only number that is real as well as purely imaginary. The following operations are defined on the complex number system: (i). Addition: (a + bi) + (c + di) = (a + c) + (b + d)i. (ii). Subtraction: (a + bi) − (c + di) = (a − c) + (b − d)i. (iii). Multiplication: (a + bi)(c + di) = (ac − bd) + (bc + ad)i. As in real number system, 0 = 0 + 0i is a complex number such that z + 0 = z. There is obviously a unique complex number 0 that possesses this property. From (iii), it is clear that i2 = −1, and hence, formally, i = √ −1. Thus, except for zero, positive real numbers have real square roots, and negative real numbers have purely imaginary square roots. 1 R.P. Agarwal et al., An Introduction to Complex Analysis, DOI 10.1007/978-1-4614-0195-7_1, © Springer Science+Business Media, LLC 2011
- 21. 2 Lecture 1 For complex numbers z1, z2, z3 we have the following easily verifiable properties: (I). Commutativity of addition: z1 + z2 = z2 + z1. (II). Commutativity of multiplication: z1z2 = z2z1. (III). Associativity of addition: z1 + (z2 + z3) = (z1 + z2) + z3. (IV). Associativity of multiplication: z1(z2z3) = (z1z2)z3. (V). Distributive law: (z1 + z2)z3 = z1z3 + z2z3. As an illustration, we shall show only (I). Let z1 = a1 +b1i, z2 = a2+b2i then z1 + z2 = (a1 + a2) + (b1 + b2)i = (a2 + a1) + (b2 + b1)i = (a2 + b2i) + (a1 + b1i) = z2 + z1. Clearly, C with addition and multiplication forms a field. We also note that, for any integer k, i4k = 1, i4k+1 = i, i4k+2 = − 1, i4k+3 = − i. The rule for division is derived as a + bi c + di = a + bi c + di · c − di c − di = ac + bd c2 + d2 + bc − ad c2 + d2 i, c2 + d2 = 0. Example 1.1. Find the quotient (6 + 2i) − (1 + 3i) −1 + i − 2 . (6 + 2i) − (1 + 3i) −1 + i − 2 = 5 − i −3 + i = (5 − i) (−3 + i) (−3 − i) (−3 − i) = −15 − 1 − 5i + 3i 9 + 1 = − 8 5 − 1 5 i. Geometrically, we can represent complex numbers as points in the xy- plane by associating to each complex number a + bi the point (a, b) in the xy-plane (also known as an Argand diagram). The plane is referred to as the complex plane. The x-axis is called the real axis, and the y-axis is called the imaginary axis. The number z = 0 corresponds to the origin of the plane. This establishes a one-to-one correspondence between the set of all complex numbers and the set of all points in the complex plane.
- 22. Complex Numbers I 3 Figure 1.1 x y 1 2 3 4 -1 -2 -3 -4 i 2i -i -2i 0 ·2 + i ·−3 − 2i We can justify the above representation of complex numbers as follows: Let A be a point on the real axis such that OA = a. Since i·i a = i2 a = −a, we can conclude that twice multiplication of the real number a by i amounts to the rotation of OA through two right angles to the position OA . Thus, it naturally follows that the multiplication by i is equivalent to the rotation of OA through one right angle to the position OA . Hence, if y Oy is a line perpendicular to the real axis x Ox, then all imaginary numbers are represented by points on y Oy. Figure 1.2 x y 0 x y × × × A A A The absolute value or modulus of the number z = a + ib is denoted by |z| and given by |z| = √ a2 + b2. Since a ≤ |a| = √ a2 ≤ √ a2 + b2 and b ≤ |b| = √ b2 ≤ √ a2 + b2, it follows that Re(z) ≤ |Re(z)| ≤ |z| and Im(z) ≤ |Im(z)| ≤ |z|. Now, let z1 = a1 + b1i and z2 = a2 + b2i then |z1 − z2| = (a1 − a2)2 + (b1 − b2)2. Hence, |z1 − z2| is just the distance between the points z1 and z2. This fact is useful in describing certain curves in the plane.
- 23. 4 Lecture 1 Figure 1.3 x y 0 ·z · · z1 z2 |z| |z1 − z2| Example 1.2. The equation |z −1+3i| = 2 represents the circle whose center is z0 = 1 − 3i and radius is R = 2. Figure 1.4 x y −3i 0 · 1 − 3i 2 Example 1.3. The equation |z + 2| = |z − 1| represents the perpendic- ular bisector of the line segment joining −2 and 1; i.e., the line x = −1/2. Figure 1.5 x y 0 1 -1 -2 - 1 2 |z + 2| |z − 1|
- 24. Complex Numbers I 5 The complex conjugate of the number z = a + bi is denoted by z and given by z = a − bi. Geometrically, z is the reflection of the point z about the real axis. Figure 1.6 x y 0 a + ib a − ib · · The following relations are immediate: 1. |z1z2| = |z1||z2|, z1 z2 = |z1| |z2| , (z2 = 0). 2. |z| ≥ 0, and |z| = 0, if and only if z = 0. 3. z = z, if and only if z ∈ I R. 4. z = −z, if and only if z = bi for some b ∈ I R. 5. z1 ± z2 = z1 ± z2. 6. z1z2 = (z1)(z2). 7. z1 z2 = z1 z2 , z2 = 0. 8. Re(z) = z + z 2 , Im(z) = z − z 2i . 9. z = z. 10. |z| = |z|, zz = |z|2 . As an illustration, we shall show only relation 6. Let z1 = a1 +b1i, z2 = a2 + b2i. Then z1z2 = (a1 + b1i)(a2 + b2i) = (a1a2 − b1b2) + i(a1b2 + b1a2) = (a1a2 − b1b2) − i(a1b2 + b1a2) = (a1 − b1i)(a2 − b2i) = (z1)(z2).
- 25. Lecture 2 Complex Numbers II In this lecture, we shall first show that complex numbers can be viewed as two-dimensional vectors, which leads to the triangle inequality. Next, we shall express complex numbers in polar form, which helps in reducing the computation in tedious expressions. For each point (number) z in the complex plane, we can associate a vector, namely the directed line segment from the origin to the point z; i.e., z = a + bi ←→ − → v = (a, b). Thus, complex numbers can also be interpreted as two-dimensional ordered pairs. The length of the vector associated with z is |z|. If z1 = a1 + b1i ←→ − → v 1 = (a1, b1) and z2 = a2 +b2i ←→ − → v 2 = (a2, b2), then z1 + z2 ←→ − → v 1 + − → v 2. Figure 2.1 x y 0 z1 z2 z1 + z2 − → v 1 − → v 2 − → v 1 +− → v 2 Using this correspondence and the fact that the length of any side of a triangle is less than or equal to the sum of the lengths of the two other sides, we have |z1 + z2| ≤ |z1| + |z2| (2.1) for any two complex numbers z1 and z2. This inequality also follows from |z1 + z2|2 = (z1 + z2)(z1 + z2) = (z1 + z2)(z1 + z2) = z1z1 + z1z2 + z2z1 + z2z2 = |z1|2 + (z1z2 + z1z2) + |z2|2 = |z1|2 + 2Re(z1z2) + |z2|2 ≤ |z1|2 + 2|z1z2| + |z2|2 = (|z1| + |z2|)2 . Applying the inequality (2.1) to the complex numbers z2 − z1 and z1, R.P. Agarwal et al., An Introduction to Complex Analysis, DOI 10.1007/978-1-4614-0195-7_2, © Springer Science+Business Media, LLC 2011 6
- 26. Complex Numbers II 7 we get |z2| = |z2 − z1 + z1| ≤ |z2 − z1| + |z1|, and hence |z2| − |z1| ≤ |z2 − z1|. (2.2) Similarly, we have |z1| − |z2| ≤ |z1 − z2|. (2.3) Combining inequalities (2.2) and (2.3), we obtain ||z1| − |z2|| ≤ |z1 − z2|. (2.4) Each of the inequalities (2.1)-(2.4) will be called a triangle inequality. In- equality (2.4) tells us that the length of one side of a triangle is greater than or equal to the difference of the lengths of the two other sides. From (2.1) and an easy induction, we get the generalized triangle inequality |z1 + z2 + · · · + zn| ≤ |z1| + |z2| + · · · + |zn|. (2.5) From the demonstration above, it is clear that, in (2.1), equality holds if and only if Re(z1z2) = |z1z2|; i.e., z1z2 is real and nonnegative. If z2 = 0, then since z1z2 = z1|z2|2 /z2, this condition is equivalent to z1/z2 ≥ 0. Now we shall show that equality holds in (2.5) if and only if the ratio of any two nonzero terms is positive. For this, if equality holds in (2.5), then, since |z1 + z2 + z3 + · · · + zn| = |(z1 + z2) + z3 + · · · + zn| ≤ |z1 + z2| + |z3| + · · · + |zn| ≤ |z1| + |z2| + |z3| + · · · + |zn|, we must have |z1 + z2| = |z1| + |z2|. But, this holds only when z1/z2 ≥ 0, provided z2 = 0. Since the numbering of the terms is arbitrary, the ratio of any two nonzero terms must be positive. Conversely, suppose that the ratio of any two nonzero terms is positive. Then, if z1 = 0, we have |z1 + z2 + · · · + zn| = |z1| 1 + z2 z1 + · · · + zn z1 = |z1| 1 + z2 z1 + · · · + zn z1 = |z1| 1 + |z2| |z1| + · · · + |zn| |z1| = |z1| + |z2| + · · · + |zn|. Example 2.1. If |z| = 1, then, from (2.5), it follows that |z2 + 2z + 6 + 8i| ≤ |z|2 + 2|z| + |6 + 8i| = 1 + 2 + √ 36 + 64 = 13.
- 27. 8 Lecture 2 Similarly, from (2.1) and (2.4), we find 2 ≤ |z2 − 3| ≤ 4. Note that the product of two complex numbers z1 and z2 is a new complex number that can be represented by a vector in the same plane as the vectors for z1 and z2. However, this product is neither the scalar (dot) nor the vector (cross) product used in ordinary vector analysis. Now let z = x + yi, r = |z| = x2 + y2, and θ be a number satisfying cos θ = x r and sin θ = y r . Then, z can be expressed in polar (trigonometric) form as z = r(cos θ + i sin θ). Figure 2.2 x y 0 x y z = x + iy θ r To find θ, we usually compute tan−1 (y/x) and adjust the quadrant prob- lem by adding or subtracting π when appropriate. Recall that tan−1 (y/x) ∈ (−π/2, π/2). Figure 2.3 x y 0 π/6 −π/6 √ 3 + i − √ 3 − i √ 3 − i − √ 3 + i tan−1 (y/x) + π tan−1 (y/x) − π Example 2.2. Express 1−i in polar form. Here r = √ 2 and θ = −π/4, and hence 1 − i = √ 2 cos − π 4 + i sin − π 4 .
- 28. Complex Numbers II 9 Figure 2.4 x y 0 1 − i · −π/4 We observe that any one of the values θ = −(π/4) ± 2nπ, n = 0, 1, · · · , can be used here. The number θ is called an argument of z, and we write θ = argz. Geometrically, argz denotes the angle measured in radians that the vector corresponds to z makes with the positive real axis. The argument of 0 is not defined. The pair (r, arg z) is called the polar coordinates of the complex number z. The principal value of arg z, denoted by Arg z, is defined as that unique value of argz such that −π arg z ≤ π. If we let z1 = r1(cos θ1 + i sin θ1) and z2 = r2(cos θ2 + i sin θ2), then z1z2 = r1r2[(cos θ1 cos θ2 − sin θ1 sin θ2) + i(sin θ1 cos θ2 + cos θ1 sin θ2)] = r1r2[cos(θ1 + θ2) + i sin(θ1 + θ2)]. Thus, |z1z2| = |z1||z2|, arg(z1z2) = argz1 + argz2. Figure 2.5 x y 0 · · · z1 z2 z1z2 θ1 θ2 θ1+θ2 r1 r2 r1r2 For the division, we have z1 z2 = r1 r2 [cos(θ1 − θ2) + i sin(θ1 − θ2)], z1 z2 = |z1| |z2| , arg z1 z2 = arg z1 − arg z2.
- 29. 10 Lecture 2 Example 2.3. Write the quotient 1 + i √ 3 − i in polar form. Since the polar forms of 1 + i and √ 3 − i are 1+i = √ 2 cos π 4 + i sin π 4 and √ 3−i = 2 cos − π 6 + i sin − π 6 , it follows that 1 + i √ 3 − i = √ 2 2 cos π 4 − − π 6 + i sin π 4 − − π 6 = √ 2 2 cos 5π 12 + i sin 5π 12 . Recall that, geometrically, the point z is the reflection in the real axis of the point z. Hence, arg z = −argz.
- 30. Lecture 3 Complex Numbers III In this lecture, we shall first show that every complex number can be written in exponential form, and then use this form to raise a rational power to a given complex number. We shall also extract roots of a complex number. Finally, we shall prove that complex numbers cannot be ordered. If z = x + iy, then ez is defined to be the complex number ez = ex (cos y + i sin y). (3.1) This number ez satisfies the usual algebraic properties of the exponential function. For example, ez1 ez2 = ez1+z2 and ez1 ez2 = ez1−z2 . In fact, if z1 = x1 + iy1 and z2 = x2 + iy2, then, in view of Lecture 2, we have ez1 ez2 = ex1 (cos y1 + i sin y1)ex2 (cos y2 + i sin y2) = ex1+x2 (cos(y1 + y2) + i sin(y1 + y2)) = e(x1+x2)+i(y1+y2) = ez1+z2 . In particular, for z = iy, the definition above gives one of the most impor- tant formulas of Euler eiy = cos y + i sin y, (3.2) which immediately leads to the following identities: cos y = Re(eiy ) = eiy + e−iy 2 , sin y = Im(eiy ) = eiy − e−iy 2i . When y = π, formula (3.2) reduces to the amazing equality eπi = −1. In this relation, the transcendental number e comes from calculus, the tran- scendental number π comes from geometry, and i comes from algebra, and the combination eπi gives −1, the basic unit for generating the arithmetic system for counting numbers. Using Euler’s formula, we can express a complex number z = r(cos θ + i sin θ) in exponential form; i.e., z = r(cos θ + i sin θ) = reiθ . (3.3) R.P. Agarwal et al., An Introduction to Complex Analysis, DOI 10.1007/978-1-4614-0195-7_3, © Springer Science+Business Media, LLC 2011 11
- 31. 12 Lecture 3 The rules for multiplying and dividing complex numbers in exponential form are given by z1z2 = (r1eiθ1 )(r2eiθ2 ) = (r1r2)ei(θ1+θ2) , z1 z2 = r1eiθ1 r2eiθ2 = r1 r2 ei(θ1−θ2) . Finally, the complex conjugate of the complex number z = reiθ is given by z = re−iθ . Example 3.1. Compute (1). 1 + i √ 3 − i and (2). (1 + i)24 . (1). We have 1 + i = √ 2eiπ/4 , √ 3 − i = 2e−iπ/6 , and therefore 1 + i √ 3 − i = √ 2eiπ/4 2e−iπ/6 = √ 2 2 ei5π/12 . (2). (1 + i)24 = ( √ 2eiπ/4 )24 = 212 ei6π = 212 . From the exponential representation of complex numbers, De Moivre’s formula (cos θ + i sin θ)n = cos nθ + i sin nθ, n = 1, 2, · · ·, (3.4) follows immediately. In fact, we have (cos θ + i sin θ)n = (eiθ )n = eiθ · eiθ · · · eiθ = eiθ+iθ+···+iθ = einθ = cos nθ + i sin nθ. From (3.4), it is immediate to deduce that 1 + i tan θ 1 − i tan θ n = 1 + i tan nθ 1 − i tan nθ . Similarly, since 1 + sin θ ± i cos θ = 2 cos π 4 − θ 2 cos π 4 − θ 2 ± i sin π 4 − θ 2 , it follows that 1 + sin θ + i cos θ 1 + sin θ − i cos θ n = cos nπ 2 − nθ + i sin nπ 2 − nθ .
- 32. Complex Numbers III 13 Example 3.2. Express cos 3θ in terms of cos θ. We have cos 3θ = Re(cos 3θ + i sin 3θ) = Re(cos θ + i sin θ)3 = Re[cos3 θ + 3 cos2 θ(i sin θ) + 3 cos θ(− sin2 θ) − i sin3 θ] = cos3 θ − 3 cosθ sin2 θ = 4 cos3 θ − 3 cosθ. Now, let z = reiθ = r(cos θ + i sin θ). By using the multiplicative prop- erty of the exponential function, we get zn = rn einθ (3.5) for any positive integer n. If n = −1, −2, · · ·, we define zn by zn = (z−1 )−n . If z = reiθ , then z−1 = e−iθ /r. Hence, zn = (z−1 )−n = 1 r ei(−θ) −n = 1 r −n ei(−n)(−θ) = rn einθ . Hence, formula (3.5) is also valid for negative integers n. Now we shall see if (3.5) holds for n = 1/m. If we let ξ = m √ reiθ/m , (3.6) then ξ certainly satisfies ξm = z. But it is well-known that the equation ξm = z has more than one solution. To obtain all the mth roots of z, we must apply formula (3.5) to every polar representation of z. For example, let us find all the mth roots of unity. Since 1 = e2kπi , k = 0, ±1, ±2, · · ·, applying formula (3.5) to every polar representation of 1, we see that the complex numbers z = e(2kπi)/m , k = 0, ±1, ±2, · · ·, are mth roots of unity. All these roots lie on the unit circle centered at the origin and are equally spaced around the circle every 2π/m radians. Figure 3.1 0 m = 6 π/3
- 33. 14 Lecture 3 Hence, all of the distinct m roots of unity are obtained by writing z = e(2kπi)/m , k = 0, 1, · · ·, m − 1. (3.7) In the general case, the m distinct roots of a complex number z = reiθ are given by z1/m = m √ rei(θ+2kπ)/m , k = 0, 1, · · · , m − 1. Example 3.3. Find all the cube roots of √ 2 + i √ 2. In polar form, we have √ 2 + i √ 2 = 2eiπ/4 . Hence, ( √ 2 + i √ 2)1/3 = 3 √ 2ei( π 12 + 2kπ 3 ), k = 0, 1, 2; i.e., 3 √ 2 cos π 12 + i sin π 12 , 3 √ 2 cos 3π 4 + i sin 3π 4 , 3 √ 2 cos 17π 12 + i sin 17π 12 , are the cube roots of √ 2 + i √ 2. Example 3.4. Solve the equation (z+1)5 = z5 . We rewrite the equation as z + 1 z 5 = 1. Hence, z + 1 z = e2kπi/5 , k = 0, 1, 2, 3, 4, or z = 1 e2kπi/5 − 1 = − 1 2 1 + i cot πk 5 , k = 0, 1, 2, 3, 4. Similarly, for any natural number n, the roots of the equation (z +1)n + zn = 0 are z = − 1 2 1 + i cot π + 2kπ n , k = 0, 1, · · · , n − 1. We conclude this lecture by proving that complex numbers cannot be ordered. (Recall that the definition of the order relation denoted by in the real number system is based on the existence of a subset P (the positive reals) having the following properties: (i) For any number α = 0, either α or −α (but not both) belongs to P. (ii) If α and β belong to P, so does α+β. (iii) If α and β belong to P, so does α·β. When such a set P exists, we write α β if and only if α − β belongs to P.) Indeed, suppose there is a nonempty subset P of the complex numbers satisfying (i), (ii), and (iii). Assume that i ∈ P. Then, by (iii), i2 = −1 ∈ P and (−1)i = −i ∈ P. This
- 34. Complex Numbers III 15 violates (i). Similarly, (i) is violated by assuming −i ∈ P. Therefore, the words positive and negative are never applied to complex numbers. Problems 3.1. Express each of the following complex numbers in the form x+iy : (a). ( √ 2 − i) − i(1 − √ 2i), (b). (2 − 3i)(−2 + i), (c). (1 − i)(2 − i)(3 − i), (d). 4 + 3i 3 − 4i , (e). 1 + i i + i 1 − i , (f). 1 + 2i 3 − 4i + 2 − i 5i , (g). (1 + √ 3 i)−10 , (h). (−1 + i)7 , (i). (1 − i)4 . 3.2. Describe the following loci or regions: (a). |z − z0| = |z − z0|, where Im z0 = 0, (b). |z − z0| = |z + z0|, where Re z0 = 0, (c). |z − z0| = |z − z1|, where z0 = z1, (d). |z − 1| = 1, (e). |z − 2| = 2|z − 2i|, (f). z − z0 z − z1 = c, where z0 = z1 and c = 1, (g). 0 Im z 2π, (h). Re z |z − 1| 1, Im z 3, (i). |z − z1| + |z − z2| = 2a, (j). azz + kz + kz + d = 0, k ∈ C, a, d ∈ I R, and |k|2 ad. 3.3. Let α, β ∈ C. Prove that |α + β|2 + |α − β|2 = 2(|α|2 + |β|2 ), and deduce that |α + α2 − β2| + |α − α2 − β2| = |α + β| + |α − β|. 3.4. Use the properties of conjugates to show that (a). (z)4 = (z4), (b). z1 z2z3 = z1 z2z3 . 3.5. If |z| = 1, then show that az + b bz + a = 1
- 35. 16 Lecture 3 for all complex numbers a and b. 3.6. If |z| = 2, use the triangle inequality to show that |Im(1 − z + z2 )| ≤ 7 and |z4 − 4z2 + 3| ≥ 3. 3.7. Prove that if |z| = 3, then 5 13 ≤ 2z − 1 4 + z2 ≤ 7 5 . 3.8. Let z and w be such that zw = 1, |z| ≤ 1, and |w| ≤ 1. Prove that z − w 1 − zw ≤ 1. Determine when equality holds. 3.9. (a). Prove that z is either real or purely imaginary if and only if (z)2 = z2 . (b). Prove that √ 2|z| ≥ |Re z| + |Im z|. 3.10. Show that there are complex numbers z satisfying |z−a|+|z+a| = 2|b| if and only if |a| ≤ |b|. If this condition holds, find the largest and smallest values of |z|. 3.11. Let z1, z2, · · · , zn and w1, w2, · · · , wn be complex numbers. Estab- lish Lagrange’s identity n k=1 zkwk 2 = n k=1 |zk|2 n k=1 |wk|2 − k |zkw − zwk|2 , and deduce Cauchy’s inequality n k=1 zkwk 2 ≤ n k=1 |zk|2 n k=1 |wk|2 . 3.12. Express the following in the form r(cos θ + i sin θ), − π θ ≤ π : (a). (1 − i)( √ 3 + i) (1 + i)( √ 3 − i) , (b). −8 + 4 i + 25 3 − 4i . 3.13. Find the principal argument (Arg) for each of the following com- plex numbers: (a). 5 cos π 8 − i sin π 8 , (b). −3 + √ 3i, (c). − 2 1 + √ 3i , (d). ( √ 3 − i)6 .
- 36. Complex Numbers III 17 3.14. Given z1z2 = 0, prove that Re z1z2 = |z1||z2| if and only if Arg z1 = Arg z2. Hence, show that |z1 + z2| = |z1| + |z2| if and only if Arg z1 = Arg z2. 3.15. What is wrong in the following? 1 = √ 1 = (−1)(−1) = √ −1 √ −1 = i i = − 1. 3.16. Show that (1 − i)49 cos π 40 + i sin π 40 10 (8i − 8 √ 3)6 = − √ 2. 3.17. Let z = reiθ and w = Reiφ , where 0 r R. Show that Re w + z w − z = R2 − r2 R2 − 2Rr cos(θ − φ) + r2 . 3.18. Solve the following equations: (a). z2 = 2i, (b). z2 = 1 − √ 3i, (c). z4 = −16, (d). z4 = −8 − 8 √ 3i. 3.19. For the root of unity z = e2πi/m , m 1, show that 1 + z + z2 + · · · + zm−1 = 0. 3.20. Let a and b be two real constants and n be a positive integer. Prove that all roots of the equation 1 + iz 1 − iz n = a + ib are real if and only if a2 + b2 = 1. 3.21. A quarternion is an ordered pair of complex numbers; e.g., ((1, 2), (3, 4)) and (2+i, 1−i). The sum of quarternions (A, B) and (C, D) is defined as (A + C, B + D). Thus, ((1, 2), (3, 4)) + ((5, 6), (7, 8)) = ((6, 8), (10, 12)) and (1 − i, 4 + i) + (7 + 2i, −5 + i) = (8 + i, −1 + 2i). Similarly, the scalar multiplication by a complex number A of a quaternion (B, C) is defined by the quadternion (AB, AC). Show that the addition and scalar multiplica- tion of quaternions satisfy all the properties of addition and multiplication of real numbers. 3.22. Observe that:
- 37. 18 Lecture 3 (a). If x = 0 and y 0 (y 0), then Arg z = π/2 (−π/2). (b). If x 0, then Arg z = tan−1 (y/x) ∈ (−π/2, π/2). (c). If x 0 and y 0 (y 0), then Arg z = tan−1 (y/x)+π (tan−1 (y/x)− π). (d). Arg (z1z2) = Arg z1 + Arg z2 + 2mπ for some integer m. This m is uniquely chosen so that the LHS ∈ (−π, π]. In particular, let z1 = −1, z2 = −1, so that Arg z1 = Arg z2 = π and Arg (z1z2) = Arg(1) = 0. Thus the relation holds with m = −1. (e). Arg(z1/z2) = Arg z1 − Arg z2 + 2mπ for some integer m. This m is uniquely chosen so that the LHS ∈ (−π, π]. Answers or Hints 3.1. (a). −2i, (b). −1 + 8i, (c). −10i, (d). i, (e). (1 − i)/2, (f). −2/5, (g). 2−11 (−1 + √ 3i), (h). −8(1 + i), (i). −4. 3.2. (a). Real axis, (b). imaginary axis, (c). perpendicular bisector (pass- ing through the origin) of the line segment joining the points z0 and z1, (d). circle center z = 1, radius 1; i.e., (x − 1)2 + y2 = 1, (e). circle center (−2/3, 8/3), radius √ 32/3, (f). circle, (g). 0 y 2π, infinite strip, (h). region interior to parabola y2 = 2(x − 1/2) but below the line y = 3, (i). ellipse with foci at z1, z2 and major axis 2a (j). circle. 3.3. Use |z|2 = zz. 3.4. (a). z4 = zzzz = z z z z = (z)4 , (b). z1 z2z3 = z1 z2z3 = z1 z2z3 . 3.5. If |z| = 1, then z = z−1 . 3.6. |Im (1 − z + z2 )| ≤ |1 − z + z2 | ≤ |1| + |z| + |z2 | ≤ 7, |z4 − 4z2 + 3| = |z2 − 3||z2 − 1| ≥ (|z2 | − 3)(|z2 | − 1). 3.7. We have 2z − 1 4 + z2 ≤ 2|z| + 1 |4 − |z|2| = 2 · 3 + 1 |4 − 32| = 7 5 and 2z − 1 4 + z2 ≥ |2|z| − 1| |4 + |z|2| = 2 · 3 − 1 4 + 32 = 5 13 . 3.8. We shall prove that |1 − zw| ≥ |z −w|. We have |1 − zw|2 − |z −w|2 = (1−zw)(1−zw)−(z−w)(z−w) = 1−zw−zw+zwzw−zz+zw+wz−ww = 1 − |z|2 − |w|2 +|z|2 |w|2 = (1 − |z|2 )(1 − |w|2 ) ≥ 0 since |z| ≤ 1 and |w| ≤ 1. Equality holds when |z| = |w| = 1. 3.9. (a). (z)2 = z2 iff z2 − (z)2 = 0 iff (z + z)(z − z) = 0 iff either 2Re(z) = z + z = 0 or 2iIm(z) = z − z = 0 iff z is purely imaginary or z is real. (b). Write z = x+iy. Consider 2|z|2 −(|Re z|+|Im z|)2 = 2(x2 +y2 )− (|x|+|y|)2 = 2x2 +2y2 −(x2 +y2 +2|x|y|) = x2 +y2 −2|x||y| = (|x|−|y|)2 ≥ 0. 3.10. Use the triangle inequality.
- 38. Complex Numbers III 19 3.11. We have n k=1 zkwk 2 = n k=1 zkwk n =1 zw = n k=1 |zk|2 |wk|2 + k= zkwkzw = n k=1 |zk|2 n k=1 |wk|2 − k= |zk|2 |w|2 + k= zkwkzw = n k=1 |zk|2 n k=1 |wk|2 − k |zkw − zwk|2 . 3.12. (a). cos(−π/6) + i sin(−π/6), (b). 5(cos π + i sin π). 3.13. (a). −π/8, (b). 5π/6, (c). 2π/3, (d). π. 3.14. Let z1 = r1eiθ1 , z2 = r2eiθ2 . Then, z1z2 = r1r2ei(θ1−θ2) . Re(z1z2) = r1r2 cos(θ1 − θ2) = r1r2 if and only if θ1 − θ2 = 2kπ, k ∈ Z. Thus, if and only if Arg z1-Arg z2 = 2kπ, k ∈ Z. But for −π Arg z1, Arg z2 ≤ π, the only possibility is Arg z1 = Arg z2. Conversely, if Arg z1 = Arg z2, then Re (z1z2) = r1r2 = |z1||z2|. Now, |z1 + z2| = |z1| + |z2| ⇐⇒ z1z1 + z2z2 + z1z2 + z2z1 = |z1|2 + |z2|2 + 2|z1|z2| ⇐⇒ z1z2 + z2z1 = 2|z1||z2| ⇐⇒ Re(z1z2 +z2z1) = Re(z1z2)+Re(z2z1) = 2|z1||z2| ⇐⇒ Re(z1z2) = |z1||z2| and Re(z1z2) = |z1||z2| ⇐⇒ Arg (z1) = Arg (z2). 3.15. If a is a positive real number, then √ a denotes the positive square root of a. However, if w is a complex number, what is the meaning of √ w? Let us try to find a reasonable definition of √ w. We know that the equation z2 = w has two solutions, namely z = ± |w|ei(Argw)/2 . If we want √ −1 = i, then we need to define √ w = |w|ei(Argw)/2 . However, with this definition, the expression √ w √ w = √ w2 will not hold in general. In particular, this does not hold for w = −1. 3.16. Use 1 − i = √ 2 cos −π 4 + i sin −π 4 and 8i − 8 √ 3 = 16 cos 5π 6 +i sin 5π 6 . 3.17. Use |w − z|2 = (w − z)(w − z). 3.18. (a). z2 = 2i = 2eiπ/2 , z = √ 2eiπ/4 , √ 2 exp i 2 π 2 + 2π , (b). z2 = 1 − √ 3i = 2e−iπ/3 , z = √ 2e−iπ/6 , √ 2ei5π/6 , (c). z4 = −16 = 24 eiπ , z = 2 exp i π+2kπ 4 , k = 0, 1, 2, 3, (d). z4 = −8 − 8 √ 3i = 16ei4π/3 , z = 2 exp i 4 4π 3 + 2kπ , k = 0, 1, 2, 3. 3.19. Multiply 1 + z + z2 + · · · + zm−1 by 1 − z. 3.20. Suppose all the roots are real. Let z = x be a real root. Then a + ib = 1+ix 1−ix n implies that |a + ib|2 = 1+ix 1−ix 2n = 1+x2 1+x2 n = 1, and hence a2 + b2 = 1. Conversely, suppose a2 + b2 = 1. Let z = x + iy be a root. Then we have 1 = a2 + b2 = |a+ ib|2 = (1−y)+ix (1+y)−ix 2n = (1−y)2 +x2 (1+y)2+x2 n , and hence (1 + y)2 + x2 = (1 − y)2 + x2 , which implies that y = 0. 3.21. Verify directly.
- 39. Lecture 4 Set Theory in the Complex Plane In this lecture, we collect some essential definitions about sets in the complex plane. These definitions will be used throughout without further mention. The set S of all points that satisfy the inequality |z −z0| , where is a positive real number, is called an open disk centered at z0 with radius and denoted as B(z0, ). It is also called the -neighborhood of z0, or simply a neighborhood of z0. In Figure 4.1, the dashed boundary curve means that the boundary points do not belong to the set. The neighborhood |z| 1 is called the open unit disk. Figure 4.1 z0 · z · Dotted boundary curve means the boundary points do not belong to S A point z0 that lies in the set S is called an interior point of S if there is a neighborhood of z0 that is completely contained in S. Example 4.1. Every point z in an open disk B(z0, ) is an interior point. Example 4.2. If S is the right half-plane Re(z) 0 and z0 = 0.01, then z0 is an interior point of S. Figure 4.2 z0 · · .02 R.P. Agarwal et al., An Introduction to Complex Analysis, DOI 10.1007/978-1-4614-0195-7_4, © Springer Science+Business Media, LLC 2011 20
- 40. Set Theory in the Complex Plane 21 Example 4.3. If S = {z : |z| ≤ 1}, then every complex number z such that |z| = 1 is not an interior point, whereas every complex number z such that |z| 1 is an interior point. If every point of a set S is an interior point of S, we say that S is an open set. Note that the empty set and the set of all complex numbers are open, whereas a finite set of points is not open. It is often convenient to add the element ∞ to C. The enlarged set C ∪ {∞} is called the extended complex plane. Unlike the extended real line, there is no −∞. For this, we identify the complex plane with the xy- plane of I R3 , let S denote the sphere with radius 1 centered at the origin of I R3 , and call the point N = (0, 0, 1) on the sphere the north pole. Now, from a point P in the complex plane, we draw a line through N. Then, the point P is mapped to the point P on the surface of S, where this line intersects the sphere. This is clearly a one-to-one and onto (bijective) correspondence between points on S and the extended complex plane. In fact, the open disk B(0, 1) is mapped onto the southern hemisphere, the circle |z| = 1 onto the equator, the exterior |z| 1 onto the northern hemisphere, and the north pole N corresponds to ∞. Here, S is called the Riemann sphere and the correspondence is called a stereographic projection (see Figure 4.3). Thus, the sets of the form {z : |z − z0| r 0} are open and called neighborhoods of ∞. In what follows we shall make the following conventions: z1 + ∞ = ∞ + z1 = ∞ for all z1 ∈ C, z2 × ∞ = ∞ × z2 = ∞ for all z2 ∈ C but z2 = 0, z1/0 = ∞ for all z1 = 0, and z2/∞ = 0 for z2 = ∞. Figure 4.3 • (0, 0, 1) • • (ξ, η, ζ) z = x + iy ξ2 + η2 + ζ2 = 1 N P P S A point z0 is called an exterior point of S if there is some neighborhood of z0 that does not contain any points of S. A point z0 is said to be a
- 41. 22 Lecture 4 boundary point of a set S if every neighborhood of z0 contains at least one point of S and at least one point not in S. Thus, a boundary point is neither an interior point nor an exterior point. The set of all boundary points of S denoted as ∂S is called the boundary or frontier of S. In Figure 4.4, the solid boundary curve means the boundary points belong to S. Figure 4.4 · S z0 ∂S Solid boundary curve means the boundary points belong to S Example 4.4. Let 0 ρ1 ρ2 and S = {z : ρ1 |z| ≤ ρ2}. Clearly, the circular annulus S is neither open nor closed. The boundary of S is the set {z : |z| = ρ2} ∪ {z : |z| = ρ1}. Figure 4.5 ρ2 ρ1 · A set S is said to be closed if it contains all of its boundary points; i.e., ∂S ⊆ S. It follows that S is open if and only if its complement C − S is closed. The sets C and ∅ are both open and closed. The closure of S is the set S = S ∪ ∂S. For example, the closure of the open disk B(z0, r) is the closed disk B(z0, r) = {z : |z − z0| ≤ r}. A point z∗ is said to be an accumulation point (limit point) of the set S if every neighborhood of z∗ contains infinitely many points of the set S. It follows that a set S is closed if it contains all its accumulation points. A set of points S is said to be bounded if there exists a positive real number R such that |z| R for every z in S. An unbounded set is one that is not bounded.
- 42. Set Theory in the Complex Plane 23 Figure 4.6 Unbounded S S Bounded Let S be a subset of complex numbers. The diameter of S, denoted as diam S, is defined as diam S = sup z,w∈S |z − w|. Clearly, S is bounded if and only if diam S ∞. The following result, known as the Nested Closed Sets Theorem, is very useful. Theorem 4.1 (Cantor). Suppose that S1, S2, · · · is a sequence of nonempty closed subsets of C satisfying 1. Sn ⊃ Sn+1, n = 1, 2, · · · , 2. diam Sn → 0 as n → ∞. Then, ∞ n=1 Sn contains precisely one point. Theorem 4.1 is often used to prove the following well-known result. Theorem 4.2 (Bolzano-Weierstrass). If S is an infinite bounded set of complex numbers, then S has at least one accumulation point. A set is called compact if it is closed and bounded. Clearly, all closed disks B(z0, r) are compact, whereas every open disk B(z0, r) is not compact. For compact sets, the following result is fundamental. Theorem 4.3. Let S be a compact set and r 0. Then, there exists a finite number of open disks of radius r whose union contains S. Let S ⊂ C and {Sα : α ∈ Λ} be a family of open subsets of C, where Λ is any indexing set. If S ⊆ α∈Λ Sα, we say that the family {Sα : α ∈ Λ} covers S. If Λ ⊂ Λ, we call the family {Sα : α ∈ Λ } a subfamily, and if it covers S, we call it a subcovering of S. Theorem 4.4. Let S ⊂ C be a compact set, and let {Sα : α ∈ Λ} be an open covering of S. Then, there exists a finite subcovering; i.e., a finite number of open sets S1, · · · , Sn whose union covers S. Conversely, if every open covering of S has a finite subcovering, then S is compact.
- 43. 24 Lecture 4 Let z1 and z2 be two points in the complex plane. The line segment joining z1 and z2 is the set {w ∈ C : w = z1 + t(z2 − z1), 0 ≤ t ≤ 1}. Figure 4.7 · · Segment [z1, z2] z1 z2 Now let z1, z2, · · · , zn+1 be n + 1 points in the complex plane. For each k = 1, 2, · · · , n, let k denote the line segment joining zk to zk+1. Then the successive line segments 1, 2, · · · , n form a continuous chain known as a polygonal path joining z1 to zn+1. Figure 4.8 x y 0 · · · · · · z1 z2 z3 zn zn+1 1 2 n An open set S is said to be connected if every pair of points z1, z2 in S can be joined by a polygonal path that lies entirely in S. The polygonal path may contain line segments that are either horizontal or vertical. An open connected set is called a domain. Clearly, all open disks are domains. If S is a domain and S = A ∪ B, where A and B are open and disjoint; i.e., A ∩ B = ∅, then either A = ∅ or B = ∅. A domain together with some, none, or all of its boundary points is called a region. · · · · · z2 z1 Connected z2 · · z1 Not connected
- 44. Set Theory in the Complex Plane 25 z1 z2 · · Connected · Not connected z1 z2 · Figure 4.9 A set S is said to be convex if each pair of points P and Q can be joined by a line segment PQ such that every point in the line segment also lies in S. For example, open disks and closed disks are convex; however, the union of two intersecting discs, while neither lies inside the other, is not convex. Clearly, every convex set is necessarily connected. Furthermore, it follows that the intersection of two or more convex sets is also convex. Problems 4.1. Shade the following regions and determine whether they are open and connected: (a). {z ∈ C : −π/3 ≤ arg z π/2}, (b). {z ∈ C : |z − 1| |z + 1|}, (c). {z ∈ C : |z − 1| + |z − i| 2 √ 2}, (d). {z ∈ C : 1/2 |z − 1| √ 2} {z ∈ C : 1/2 |z + 1| √ 2}. 4.2. Let S be the open set consisting of all points z such that |z| 1 or |z − 2| 1. Show that S is not connected. 4.3. Show that: (a). If S1, · · · , Sn are open sets in C, then so is n k=1 Sk. (b). If {Sα : α ∈ Λ} is a collection of open sets in C, where Λ is any indexing set, then S = α∈Λ Sα is also open. (c). The intersection of an arbitrary family of open sets in C need not be open. 4.4. Let S be a nonempty set. Suppose that to each ordered pair (x, y) ∈ S × S a nonnegative real number d(x, y) is assigned that satisfies the following conditions: (i). d(x, y) ≥ 0 and d(x, y) = 0 if and only if x = y,
- 45. 26 Lecture 4 (ii). d(x, y) = d(y, x) for all x, y ∈ S, (iii). d(x, z) ≤ d(x, y) + d(y, z) for all x, y, z ∈ S. Then, d(x, y) is called a metric on S. The set S with metric d is called a metric space and is denoted as (S, d). Show that in C the following are metrics: (a). d(z, w) = |z − w|, (b). d(z, w) = |z − w| 1 + |z − w| , (c). d(z, w) = 0 if z = w 1 if z = w. 4.5. Let the point z = x + iy correspond to the point (ξ, η, ζ) on the Riemann sphere (see Figure 4.3). Show that ξ = 2 Re z |z|2 + 1 , η = 2 Im z |z|2 + 1 , ζ = |z|2 − 1 |z|2 + 1 , and Re z = ξ 1 − ζ , Im = η 1 − ζ . 4.6. Show that if z1 and z2 are finite points in the complex plane C, then the distance between their stereographic projection is given by d(z1, z2) = 2|z1 − z2| 1 + |z1|2 1 + |z2|2 . This distance is called the spherical distance or chordal distance between z1 and z2. Also, show that if z2 = ∞, then the corresponding distance is given by d(z1, ∞) = 2 1 + |z1|2 . Answers or Hints 4.1. (a). Not open −π/3 (b). Open connected
- 46. Set Theory in the Complex Plane 27 (c). Open connected (ellipse) • • (d). Open connected · · 4.2. Points 0 and 2 cannot be connected by a polygonal line. 0 2 · x y 4.3. (a). Let w ∈ ∩n k=1Sk. Then w ∈ Sk, k = 1, · · · , n. Since each Sk is open, there is an rk 0 such that {z : |z − w| rk} ⊂ Sk, k = 1, · · · , n. Let r = min{r1, · · · , rn}. Then {z : |z − w| r} ⊆ {z : |z − w| rk} ⊂ Sk, k = 1, · · · , n. Thus, {z : |z − w| r} ⊂ ∩n k=1Sk. (b). Use the property of open sets. (c). ∩∞ n=1{z : |z| 1/n} = {0}. 4.4. (a). Verify directly. (b). For a, b, c ≥ 0 and c ≤ a + b, use c 1+c ≤ a 1+a + b 1+b . (c). Verify directly. 4.5. The straight line passing through (x, y, 0) and (0, 0, 1) in parametric form is (tx, ty, 1 − t). This line also passes through the point (ξ, η, ζ) on the Riemann sphere, provided t2 x2 + t2 y2 + (1 − t)2 = 1. This gives t = 0 and t = 2/(x2 + y2 + 1). The value t = 0 gives the north pole, whereas t = 2/(x2 + y2 + 1) gives (ξ, η, ζ) = 2x x2+y2+1 , 2y x2+y2+1 , x2 +y2 −1 x2+y2+1 . From this, it also follows that |z|2 + 1 = 2 1−ζ . 4.6. If (ξ1, η1, ζ1) and (ξ2, η2, ζ2) are the points on S corresponding to z1 and z2, then d(z1, z2) = [(ξ1 − ξ2)2 + (η1 − η2)2 + (ζ1 − ζ2)2 ]1/2 = [2 − 2(ξ1ξ2 + η1η2 + ζ1ζ2)]1/2 . Now use Problem 4.5. If z2 = ∞, then again from Problem 4.5, we have d(z1, ∞) = [ξ2 1 +η2 1 +(ζ1−1)2 ]1/2 = [2−2ζ1]1/2 = [2 − 2(|z1|2 − 1)/(|z1|2 + 1)]1/2 = 2/(|z1|2 + 1)1/2 .
- 47. Lecture 5 Complex Functions In this lecture, first we shall introduce a complex-valued function of a complex variable, and then for such a function define the concept of limit and continuity at a point. Let S be a set of complex numbers. A complex function (complex-valued of a complex variable) f defined on S is a rule that assigns to each z = x+iy in S a unique complex number w = u + iv and written as f : S → C. The number w is called the value of f at z and is denoted by f(z); i.e., w = f(z). The set S is called the domain of f, the set W = {f(z) : z ∈ S}, often denoted as f(S), is called the range or image of f, and f is said to map S onto W. The function w = f(z) is said to be from S into W if the range of S under f is a subset of W. When a function is given by a formula and the domain is not specified, the domain is taken to be the largest set on which the formula is defined. A function f is called one-to-one (or univalent, or injective) on a set S if the equation f(z1) = f(z2), where z1 and z2 are in S, implies that z1 = z2. The function f(z) = iz is one-to-one, but f(z) = z2 is not one-to-one since f(i) = f(−i) = −1. A one-to-one and onto function is called bijective. We shall also consider multi-valued functions: a multi- valued function is a rule that assigns a finite or infinite non-empty subset of C for each element of its domain S. In Lecture 2, we have already seen that the function f(z) = argz is multi-valued. As every complex number z is characterized by a pair of real numbers x and y, a complex function f of the complex variable z can be specified by two real functions u = u(x, y) and v = v(x, y). It is customary to write w = f(z) = u(x, y)+iv(x, y). The functions u and v, respectively, are called the real and imaginary parts of f. The common domain of the functions u and v corresponds to the domain of the function f. Example 5.1. For the function w = f(z) = 3z2 + 7z, we have f(x + iy) = 3(x + iy)2 + 7(x + iy) = (3x2 − 3y2 + 7x) + i(6xy + 7y), and hence u = 3x2 − 3y2 + 7x and v = 6xy + 7y. Similarly, for the function w = f(z) = |z|2 , we find f(x + iy) = |x + iy|2 = x2 + y2 , and hence u = x2 + y2 and v = 0. Thus, this function is a real-valued func- tion of a complex variable. Clearly, the domain of both of these functions is R.P. Agarwal et al., An Introduction to Complex Analysis, DOI 10.1007/978-1-4614-0195-7_5, © Springer Science+Business Media, LLC 2011 28
- 48. Complex Functions 29 C. For the function w = f(z) = z/|z|, the domain is C{0}, and its range is |z| = 1. Example 5.2. The complex exponential function f(z) = ez is defined by the formula (3.1). Clearly, for this function, u = ex cos y and v = ex sin y, which are defined for all (x, y) ∈ I R2 . Thus, for the function ez the domain is C. The exponential function provides a basic tool for the application of complex variables to electrical circuits, control systems, wave propagation, and time-invariant physical systems. Recall that a vector-valued function of two real variables F(x, y) = (P(x, y), Q(x, y)) is also called a two-dimensional vector filed. Using the standard orthogonal unit basis vectors i and j, we can express this vector field as F(x, y) = P(x, y)i + Q(x, y)j. There is a natural way to represent this vector field with a complex function f(z). In fact, we can use the functions P and Q as the real and imaginary parts of f, in which case we say that the complex function f(z) = P(x, y) + iQ(x, y) is the complex representation of the vector field F(x, y) = P(x, y)i + Q(x, y)j. Conversely, any complex function f(z) = u(x, y) + iv(x, y) has an associated vector field F(x, y) = u(x, y)i + v(x, y)j. From this point of view, both F(x, y) = P(x, y)i + Q(x, y)j and f(z) = u(x, y) + iv(x, y) can be called vector fields. This interpretation is often used to study various applications of complex functions in applied mathematical problems. Let f be a function defined in some neighborhood of z0, with the possible exception of the point z0 itself. We say that the limit of f(z) as z approaches z0 (independent of the path) is the number w0 if |f(z)−w0| → 0 as |z−z0| → 0 and we write limz→z0 f(z) = w0. Hence, f(z) can be made arbitrarily close to w0 if we choose z sufficiently close to z0. Equivalently, we say that w0 is the limit of f as z approaches z0 if, for any given 0, there exists a δ 0 such that 0 |z − z0| δ =⇒ |f(z) − w0| . Figure 5.1 x y z0 · · 0 z u v 0 f(z) w0 · Example 5.3. By definition, we shall show that (i) limz→1−i(z2 −2) = −2 + 2i and (ii) limz→1−i |z2 − 2| = √ 8.
- 49. 30 Lecture 5 (i). Given any 0, we have |z2 − 2 − (−2 + 2i)| = |z2 − 2i| = |z2 + 2i| = |z2 + 2i| = |z − (1 − i)||z + (1 − i)| ≤ |z − (1 − i)|(|z − (1 − i)| + 2|1 − i|) ≤ |z − (1 − i)|(1 + 2 √ 2) if |z − (1 − i)| 1 if |z − (1 − i)| min 1, 1 + 2 √ 2 . (ii). Given any 0, from (i) we have ||z2 − 2| − √ 8| = ||z2 − 2| − | − 2 + 2i|| ≤ |z2 − 2 − (−2 + 2i)| if |z − (1 − i)| min 1, 1 + 2 √ 2 . Example 5.4. (i). Clearly, limz→z0 z = z0. (ii). From the inequalities |Re(z − z0)| ≤ [(Re(z − z0))2 + (Im(z − z0))2 ]1/2 = |z − z0|, |Im(z − z0)| ≤ |z − z0|, it follows that limz→z0 Re z = Re z0, limz→z0 Im z = Im z0. Example 5.5. limz→0(z/z) does not exist. Indeed, we have lim z → 0 along x-axis z z = lim x→0 x + i0 x − i0 = 1, lim z → 0 along y-axis z z = lim y→0 0 + iy 0 − iy = − 1. The following result relates real limits of u(x, y) and v(x, y) with the complex limit of f(z) = u(x, y) + iv(x, y). Theorem 5.1. Let f(z) = u(x, y) + iv(x, y), z0 = x0 + iy0, and w0 = u0 + iv0. Then, limz→z0 f(z) = w0 if and only if limx→x0, y→y0 u(x, y) = u0 and limx→x0, y→y0 v(x, y) = v0. In view of Theorem 5.1 and the standard results in calculus, the follow- ing theorem is immediate. Theorem 5.2. If limz→z0 f(z) = A and limz→z0 g(z) = B, then (i) limz→z0 (f(z) ± g(z)) = A ± B, (ii) limz→z0 f(z)g(z) = AB, and (iii) limz→z0 f(z) g(z) = A B if B = 0.
- 50. Complex Functions 31 For the composition of two functions f and g denoted and defined as (f ◦ g)(z) = f(g(z)), we have the following result. Theorem 5.3. If limz→z0 g(z) = w0 and limw→w0 f(w) = A, then lim z→z0 f(g(z)) = A = f lim z→z0 g(z) . Now we shall define limits that involve ∞. For this, we note that z → ∞ means |z| → ∞, and similarly, f(z) → ∞ means |f(z)| → ∞. The statement limz→z0 f(z) = ∞ means that for any M 0 there is a δ 0 such that 0 |z − z0| δ implies |f(z)| M and is equivalent to limz→z0 1/f(z) = 0. The statement limz→∞ f(z) = w0 means that for any 0 there is an R 0 such that |z| R implies |f(z) − w0| , and is equivalent to limz→0 f(1/z) = w0. The statement limz→∞ f(z) = ∞ means that for any M 0 there is an R 0 such that |z| R implies |f(z)| M. Example 5.6. Since 2z + 3 3z + 2 = 2 + 3/z 3 + 2/z , limz→∞(2z + 3)/(3z + 2) = 2/3. Similarly, limz→∞(2z + 3)/(3z2 + 2) = 0 and limz→∞(2z2 + 3)/(3z + 2) = ∞. Let f be a function defined in a neighborhood of z0. Then, f is contin- uous at z0 if limz→z0 f(z) = f(z0). Equivalently, f is continuous at z0 if for any given 0, there exists a δ 0 such that |z − z0| δ =⇒ |f(z) − f(z0)| . A function f is said to be continuous on a set S if it is continuous at each point of S. Example 5.7. The functions f(z) = Re (z) and g(z) = Im (z) are continuous for all z. Example 5.8. The function f(z) = |z| is continuous for all z. For this, let z0 be given. Then lim z→z0 |z| = lim z→z0 (Re z)2 + (Im z)2 = (Re z0)2 + (Im z0)2 = |z0|. Hence, f(z) is continuous at z0. Since z0 is arbitrary, we conclude that f(z) is continuous for all z.
- 51. 32 Lecture 5 It follows from Theorem 5.1 that a function f(z) = u(x, y) + iv(x, y) of a complex variable is continuous at a point z0 = x0 + iy0 if and only if u(x, y) and v(x, y) are continuous at (x0, y0). Example 5.9. The exponential function f(z) = ez is continuous on the whole complex plane since ex cos y and ex sin y both are continuous for all (x, y) ∈ I R2 . The following result is an immediate consequence of Theorem 5.2. Theorem 5.4. If f(z) and g(z) are continuous at z0, then so are (i) f(z) ± g(z), (ii) f(z)g(z), and (iii) f(z)/g(z) provided g(z0) = 0. Now let f : S → W, S1 ⊂ S, and W1 ⊂ W. The inverse image denoted as f−1 (W1) consists of all z ∈ S such that f(z) ∈ W1. It follows that f(f−1 (W1)) ⊂ W1 and f−1 (f(S1)) ⊃ S1. By definition, in terms of inverse image continuous functions can be characterized as follows: A function is continuous if and only if the inverse image of every open set is open. Similarly, a function is continuous if and only if the inverse image of every closed set is closed. For continuous functions we also have the following result. Theorem 5.5. Let f : S → C be continuous. Then, (i). a compact set of S is mapped onto a compact set in f(S), and (ii). a connected set of S is mapped onto a connected set of f(S). It is easy to see that the constant function and the function f(z) = z are continuous on the whole plane. Thus, from Theorem 5.4, we deduce that the polynomial functions; i.e., functions of the form P(z) = a0 + a1z + a2z2 + · · · + anzn , (5.1) where ai, 0 ≤ i ≤ n are constants, are also continuous on the whole plane. Rational functions in z, which are defined as quotients of polynomials; i.e., P(z) Q(z) = a0 + a1z + · · · + anzn b0 + b1z + · · · + bmzm , (5.2) are therefore continuous at each point where the denominator does not vanish. Example 5.10. We shall find the limits as z → 2i of the functions f1(z) = z2 − 2z + 1, f2(z) = z + 2i z , f3(z) = z2 + 4 z(z − 2i) . Since f1(z) and f2(z) are continuous at z = 2i, we have limz→2i f1(z) = f1(2i) = −3 − 4i, limz→2i f2(z) = f2(2i) = 2. Since f3(z) is not defined at
- 52. Complex Functions 33 z = 2i, it is not continuous. However, for z = 2i and z = 0, we have f3(z) = (z + 2i)(z − 2i) z(z − 2i) = z + 2i z = f2(z) and so limz→2i f3(z) = limz→2i f2(z) = 2. Thus, the discontinuity of f3(z) at z = 2i can be removed by setting f2(2i) = 2. The function f3(z) is said to have a removable discontinuity at z = 2i. Problems 5.1. For each of the following functions, describe the domain of defini- tion that is understood: (a). f(z) = z z2 + 3 , (b). f(z) = z z + z , (c). f(z) = 1 1 − |z|2 . 5.2. (a). Write the function f(z) = z3 + 2z + 1 in the form f(z) = u(x, y) + iv(x, y). (b). Suppose that f(z) = x2 −y2 −2y + i(2x−2xy). Express f(z) in terms of z. 5.3. Show that when a limit of a function f(z) exists at a point z0, it is unique. 5.4. Use the definition of limit to prove that: (a). lim z→z0 (z2 + 5) = z2 0 + 5, (b). lim z→1−i z2 = (1 + i)2 , (c). lim z→z0 z = z0, (d). lim z→2−i (2z + 1) = 5 − 2i. 5.5. Find each of the following limits: (a). lim z→2+3i (z − 5i)2 , (b). lim z→2 z2 + 3 iz , (c). lim z→3i z2 + 9 z − 3i , (d). lim z→i z2 + 1 z4 − 1 , (e). lim z→∞ z2 + 1 z2 + z + 1 − i , (f). lim z→∞ z3 + 3iz2 + 7 z2 − i . 5.6. Prove that: (a). lim z→0 z z 2 does not exist, (b). lim z→0 z2 z = 0. 5.7. Show that if lim z→z0 f(z) = 0 and there exists a positive num- ber M such that |g(z)| ≤ M for all z in some neighborhood of z0, then lim z→z0 f(z)g(z) = 0. Use this result to show that limz→0 zei/|z| = 0.
- 53. 34 Lecture 5 5.8. Show that if lim z→z0 f(z) = w0, then lim z→z0 |f(z)| = |w0|. 5.9. Suppose that f is continuous at z0 and g is continuous at w0 = f(z0). Prove that the composite function g ◦ f is continuous at z0. 5.10. Discuss the continuity of the function f(z) = ⎧ ⎨ ⎩ z3 − 1 z − 1 , |z| = 1 3, |z| = 1 at the points 1, − 1, i, and −i. 5.11. Prove that the function f(z) = Arg(z) is discontinuous at each point on the nonpositive real axis. 5.12 (Cauchy’s Criterion). Show that limz→z0 f(z) = w0 if and only if for a given 0 there exists a δ 0 such that for any z, z satisfying |z − z0| δ, |z − z0| δ, the inequality |f(z) − f(z )| holds. 5.13. Prove Theorem 5.5. 5.14. The function f : S → C is said to be uniformly continuous on S if for every given 0 there exists a δ = δ() 0 such that |f(z1)−f(z2)| for all z1, z2 ∈ S with |z1 − z2| δ. Show that on a compact set every continuous function is uniformly continuous. Answers or Hints 5.1. (a). z2 = −3 ⇐⇒ z = ± √ 3i, (b). z + z = 0 ⇐⇒ z is not purely imaginary; i.e., Re(z) = 0, (c). |z|2 = 1 ⇐⇒ |z| = 1. 5.2. (a). (x+iy)3 +2(x+iy)+1 = (x3 −3xy2 +2x+1)+i(3x2 y −y3 +2y). (b). Use x = (z + z)/2, y = (z − z)/2i to obtain f(z) = z2 + 2iz. 5.3. Suppose that limz→z0 f(z) = w0 and limz→z0 f(z) = w1. Then, for any positive number , there are positive numbers δ0 and δ1 such that |f(z) − w0| whenever 0 |z − z0| δ0 and |f(z) − w1| whenever 0 |z − z0| δ1. So, if 0 |z − z0| δ = min{δ0, δ1}, then |w0 − w1| = | − (f(z) − w0) + (f(z) − w1)| ≤ |f(z) − w0| + |f(z) − w1| 2; i.e., |w0 − w1| 2. But, can be chosen arbitrarily small. Hence, w0 − w1 = 0, or w0 = w1. 5.4. (a). |z2 + 5 − (z2 0 + 5)| = |z − z0||z + z0| ≤ |z − z0|(|z − z0| + 2|z0|) ≤ (1 + 2|z0|)|z − z0| if |z − z0| 1 if 0 |z − z0| min 1+2|z0| , 1 , (b). |z2 − (1 + i)2 | = |z2 − (1 − i)2 | ≤ |z − (1 − i)||z + (1 − i)| ≤ |z−(1−i)|(|z−(1−i)|+2|1−i|) 5|z−(1−i)| if |z−(1−i)| 1
- 54. Complex Functions 35 if |z − (1 − i)| min{1, /5}, (c). |z − z0| = |z − z0| if |z − z0| , (d). |2z+1−(5−2i)| = |2z−(4−2i)| = 2|z−(2−i)| if |z−(2−i)| /2. 5.5. (a). −8i, (b). −7i/2, (c). 6i, (d). −1/2, (e). 1, (f). ∞. 5.6. (a). limz→0,z=x(z/z)2 = 1, limz→0,y=x(z/z)2 = −1. (b). Let 0. Choose δ = . Then, 0 |z − 0| δ implies |z2 /z − 0| = |z| . 5.7. Since limz→z0 f(z) = 0, given any 0, there exists δ 0 such that |f(z)−0| /M whenever |z−z0| δ. Thus, |f(z)g(z)−0| = |f(z)||g(z)| ≤ M|f(z)| if |z − z0| δ. 5.8. Use the fact ||f(z)| − |w0|| ≤ |f(z) − w0|. 5.9. Let 0. Since g is continuous at w0, there exists a δ1 0 such that |w − w0| δ1 implies that |g(w) − g(w0)| . Now, f is continuous at z0, so there exists a δ2 0 such that |z − z0| δ2 implies |f(z) − f(z0)| δ1. Combining these, we find that |z − z0| δ2 implies |f(z) − f(z0)| δ1, which in turn implies |(g ◦ f)(z) − (g ◦ f)(z0)| = |g[f(z)] − g[f(z0)]| . 5.10. Continuous at 1, discontinuous at −1, i, −i. 5.11. f is not continuous at z0 if there exists 0 0 with the following property: For every δ 0, there exists zδ such that |zδ − z0| δ and |f(zδ) − f(z0)| ≥ 0. Now let z0 = x0 0. Take 0 = 3π/2. For each δ 0, let zδ = x0 − i(δ/2). Then, |zδ − z0| = |iδ/2| = δ/2 δ, −π f(zδ) −π/2, f(z0) = π, so |f(zδ) − f(z0)| 3π/2 = 0, and f is not continuous at z0. Thus, f is not continuous at every point on the negative real axis. It is also not continuous at z = 0 because it is not defined there. 5.12. If f is continuous at z0, then given 0 there exists a δ 0 such that |z1 − z0| δ/2 ⇒ |f(z1) − f(z0)| /2 and |z2 − z0| δ/2 ⇒ |f(z2) − f(z0)| /2. But then |z1 − z2| ≤ |z1 − z0| + |z0 − z2| δ ⇒ |f(z1) − f(z2)| ≤ |f(z1) − f(z0)| + |f(z2) − f(z0)| . For the converse, we assume that 0 |z−z0| δ, 0 |z −z0| δ; otherwise, we can take z = z0 and then there is nothing to prove. Let zn → z0, zn = z0, and 0. There is a δ 0 such that 0 |z −z0| δ, |z − z0| δ implies |f(z)−f(z )| , and there is an N such that n ≥ N implies 0 |zn − z0| δ. Then, for m, n ≥ N, we have |f(zm) − f(zn)| . So, w0 = limn→∞ f(zn) exists. To see that limz→z0 f(z) = w0, take a δ1 0 such that 0 |z − z0| δ1, 0 |z − z0| δ1 implies |f(z) − f(z )| /2, and an N1 such that n ≥ N1 implies 0 |zn − z0| δ1 and |f(zn) − w0| /2. Then, 0 |z − z0| δ1 implies |f(z) − w0| ≤ |f(z) − f(zN1 )| + |f(zN1 ) − w0| . 5.13. (i). Suppose that f : U → C is continuous and U is compact. Con- sider a covering of f(U) to be open sets V. The inverse images f−1 (V ) are open and form a covering of U. Since U is compact, by Theorem 4.4 we can select a finite subcovering such that U ⊂ f−1 (V1) ∪ · · · ∪ f−1 (Vn). It follows that f(U) ⊂ V1 ∪ · · · ∪ Vn, which in view of Theorem 4.4 implies that f(U) is compact. (ii). Suppose that f : U → C is continuous and U is connected. If f(U) = A ∪ B where A and B are open and disjoint, then U = f−1 (A) ∪ f−1 (B), which is a union of disjoint and open sets. Since U
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- 56. heartedness should be the general rule, and consequently order the world Upon the simple plan, That they should take, who have the power, And they should keep, who can. —(WORDSWORTH.) In the foregoing chapter we showed that the Kantian leading principle of Ethics is devoid of all real foundation. It is now clear that to this singular defect must be added, notwithstanding Kant's express assertion to the contrary, its concealed hypothetical nature, whereby its basis turns out to be nothing else than Egoism, the latter being the secret interpreter of the direction which it contains. Furthermore, regarding it solely as a formula, we find that it is only a periphrasis, an obscure and disguised mode of expressing the well- known rule: Quod tibi fieri non vis, alteri ne feceris (do not to another what you are unwilling should be done to yourself); if, that is, by omitting the non and ne, we remove the limitation, and include the duties taught by love as well as those prescribed by law. For it is obvious that this is the only precept which I can wish should regulate the conduct of all men (speaking, of course, from the point of view of the possibly passive part I may play, where my Egoism is touched). This rule, Quod tibi fieri, etc., is, however, in its turn, merely a circumlocution for, or, if it be preferred, a premise of, the proposition which I have laid down as the simplest and purest definition of the conduct required by the common consent of all ethical systems; namely, Neminem laede, immo omnes, quantum potes, juva (do harm to no one; but rather help all people, as far as lies in your power). The true and real substance of Morals is this, and never can be anything else. But on what is it based? What is it that lends force to this command? This is the old and difficult problem with which man is still to-day confronted. For, on the other side, we hear Egoism crying with a loud voice: Neminem juva, immo omnes, si forte conducit, laede (help nobody, but rather injure all people, if it brings you any advantage); nay more, Malice gives us the variant: Immo omnes, quantum potes, laede (but rather injure
- 57. all people as far as you can). To bring into the lists a combatant equal, or rather superior to Egoism and Malice combined—this is the task of all Ethics. Heic Rhodus, heic salta![2] The division of human duty into two classes has long been recognised, and no doubt owes its origin to the nature of morality itself. We have. (1) the duties ordained by law (otherwise called the —perfect, obligatory, narrower duties), and (2) those prescribed by virtue (otherwise called imperfect, wider, meritorious, or, preferably, the duties taught by love). On p. 57 (R., p. 60) we find Kant desiring to give a further confirmation to the moral principle, which he propounded, by undertaking to derive this classification from it. But the attempt turns out to be so forced, and so obviously bad, that it only testifies in the strongest way against the soundness of his position. For, according to him, the duties laid down by statutes rest on a precept, the contrary of which, taken as a general natural law, is declared to be quite unthinkable without contradiction; while the duties inculcated by virtue are made to depend on a maxim, the opposite of which can (he says) be conceived as a general natural law, but cannot possibly be wished for. I beg the reader to reflect that the rule of injustice, the reign of might instead of right, which in the Kantian view is not even thinkable as a natural law, is in reality, and in point of fact, the dominant order of things not only in the animal kingdom, but among men as well. It is true that an attempt has been made among civilised peoples to obviate its injurious effects by means of all the machinery of state government; but as soon as this, wherever, or of whatever kind, it be, is suspended or eluded, the natural law immediately resumes its sway. Indeed between nation and nation it never ceases to prevail; the customary jargon about justice is well known to be nothing but diplomacy's official style; the real arbiter is brute force. On the other hand, genuine, i.e., voluntary, acts of justice, do occur beyond all doubt, but always only as exceptions to the rule. Furthermore: wishing to give instances by way of introducing the above-mentioned classification, Kant establishes the duties prescribed by law first (p. 53; R., p. 48) through the so-called duty towards oneself,—the duty
- 58. of not ending one's life voluntarily, if the pain outweigh the pleasure. Accordingly, the rule of suicide is held to be not even thinkable as a general natural law. I, on the contrary, maintain that, since here there can be no intervention of state control, it is exactly this rule which is proved to be an actually existing, unchecked natural law. For it is absolutely certain (as daily experience attests) that men in the vast majority of cases turn to self-destruction directly the gigantic strength of the innate instinct of self-preservation is distinctly overpowered by great suffering. To suppose that there is any thought whatever that can have a deferring effect, after the fear of death, which is so strong and so closely bound up with the nature of every living thing, has shown itself powerless; in other words, to suppose that there is a thought still mightier than this fear—is a daring assumption, all the more so, when we see, that it is one which is so difficult to discover that the moralists are not yet able to determine it with precision. In any case, it is certain that arguments against suicide of the sort put forward by Kant in this connection (p. 53: R., p. 48, and p. 67; R., p. 57) have never hitherto restrained any one tired of life even for a moment. Thus a natural law, which incontestably exists, and is operative every day, is declared by Kant to be simply unthinkable without contradiction, and all for the sake of making his Moral Principle the basis of the classification of duties! At this point it is, I confess, not without satisfaction that I look forward to the groundwork which I shall give to Ethics in the sequel. From it the division of Duty into what is prescribed by law, and what is taught by love, or, better, into justice and loving-kindness, results quite naturally though a principle of separation which arises from the nature of the subject, and which entirely of itself draws a sharp line of demarkation; so that the foundation of Morals, which I shall present, has in fact ready to hand that confirmation, to which Kant, with a view to support his own position, lays a completely groundless claim. [1] How rashly do we sanction an unjust law, which will come home to ourselves! —(Hor., Sat., Lib. I., iii. 67.)
- 59. [2] Here is Rhodes, here make your leap! I.e., Here is the place of trial, here let us see what you can do! This Latin proverb is derived from one of Aesop's fables. A braggart boasts of having once accomplished a wonderful jump in Rhodes, and appeals to the evidence of the eye-witnesses. The bystanders then exclaim: Friend, if this be true, you have no need of witnesses; for this is Rhodes, and your leap you can make here. The words are: ἀλλ', ὦ ϕίλε, εἰ τοῡtο ἀληθές ἐστιν, oὐδὲν δεῑ σοι μαρτύρων αὕtη γὰρ 'Rόδος καὶ πήδημα. V. Fabulae Aesopicae Collectae. Edit. Halm, Leipzig: Teubner. 1875. Nr. 203b, p. 102. The other version of the fable (Nr. 203, p. 101) gives: ὦ oὗtos, eἰ ἀlêthès τoῡτ ἐstin, oὐdὲn deῑ soi martyrôn ἰdoὺ ἡ Ρόδος, ἰdoὺ kaὶ τὸ πήδημα.—(Translator.) CHAPTER VI. ON THE DERIVED FORMS OF THE LEADING PRINCIPLE OF THE KANTIAN ETHICS. It is well known that Kant put the leading principle of his Ethics into another quite different shape, in which it is expressed directly; the first being indirect, indeed nothing more than an indication as to how the principle is to be sought for. Beginning at p. 63 (R., p. 55), he prepares the way for his second formula by means of very strange, ambiguous, not to say distorted,[1] definitions of the conceptions End and Means, which may be much more simply and correctly denoted thus: an End is the direct motive of an act of the Will, a Means the indirect: simplex sigillum veri (simplicity is the seal of truth). Kant, however, slips through his wonderful enunciations to the statement: Man, indeed every rational being, exists as an end in himself. On this I must remark that to exist as an end in oneself is an unthinkable expression, a contradictio in adjecto.[2] To be an end means to be an object of volition. Every end can only exist in relation to a will, whose end, i.e., (as above stated), whose direct motive it is. Only thus can the idea, end have any sense, which is lost as soon as such connection is broken. But
- 60. this relation, which is essential to the thing, necessarily excludes every in itself. End in oneself is exactly like saying: Friend in oneself;—enemy in oneself;—uncle in oneself;—north or east in itself;—above or below in itself; and so on. At bottom the end in itself is in the same case as the absolute ought; the same thought —the theological—secretly, indeed, unconsciously lies at the root of each as its condition. Nor is the absolute worth, which is supposed to be attached to this alleged, though unthinkable, end in itself, at all better circumstanced. It also must be characterised, without pity, as a contradictio in adjecto. Every worth is a valuation by comparison, and its bearing is necessarily twofold. First, it is relative, since it exists for some one; and secondly, it is comparative, as being compared with something else, and estimated accordingly. Severed from these two conditions, the conception, worth, loses all sense and meaning, and so obviously, that further demonstration is needless. But more: just as the phrases end in itself and absolute worth outrage logic, so true morality is outraged by the statement on p. 65 (R., p. 56), that irrational beings (that is, animals) are things, and should therefore be treated simply as means, which are not at the same time ends. In harmony with this, it is expressly declared in the Metaphysische Anfanggründe der Tugendlehre, § 16: A man can have no duties towards any being, except towards his fellow-men; and then, § 17, we read: To treat animals cruelly runs counter to the duty of man towards himself; because it deadens the feeling of sympathy for them in their sufferings, and thus weakens a natural tendency which is very serviceable to morality in relation to other men. So one is only to have compassion on animals for the sake of practice, and they are as it were the pathological phantom on which to train one's sympathy with men! In common with the whole of Asia that is not tainted by Islâm (which is tantamount to Judaism), I regard such tenets as odious and revolting. Here, once again, we see withal how entirely this philosophical morality, which is, as explained above, only a theological one in disguise, depends in reality on the biblical Ethics. Thus, because Christian morals leave animals out of consideration (of which more later on); therefore in philosophical
- 61. morals they are of course at once outlawed; they are merely things, simply means to ends of any sort; and so they are good for vivisection, for deer-stalking, bull-fights, horse-races, etc., and they may be whipped to death as they struggle along with heavy quarry carts. Shame on such a morality which is worthy of Pariahs, Chandalas and Mlechchas[3] ; which fails to recognise the Eternal Reality immanent in everything that has life, and shining forth with inscrutable significance from all eyes that see the sun! This is a morality which knows and values only the precious species that gave it birth; whose characteristic—reason—it makes the condition under which a being may be an object of moral regard. By this rough path, then,—indeed, per fas et nefas (by fair means and by foul), Kant reaches the second form in which he expresses the fundamental principle of his Ethics: Act in such a way that you at all times treat mankind, as much in your own person, as in the person of every one else, not only as a Means, but also as an End. Such a statement is a very artificial and roundabout way of saying: Do not consider yourself alone, but others also; this in turn is a paraphrase for: Quod tibi fieri non vis, alteri ne feceris (do not to another what you are unwilling should be done to yourself); and the latter, as I have said, contains nothing but the premises to the conclusion, which is the true and final goal of all morals and of all moralising; Neminem laede, immo omnes, quantum potes juva (do harm to no one; but rather help all people as far as lies in your power). Like all beautiful things, this proposition looks best unveiled. Be it only observed that the alleged duties towards oneself are dragged into this second Kantian edict intentionally and not without difficulty. Some place of course had to be found for them.[4] Another objection that could be raised against the formula is that the malefactor condemned to be executed is treated merely as an instrument, and not as an end, and this with perfectly good reason; for he is the indispensable means of upholding the terror of the law by its fulfilment, and of thus accomplishing the law's end—the repression of crime.
- 62. But if this second definition helps nothing towards laying a foundation for Ethics, if it cannot even pass muster as its leading principle, that is, as an adequate and direct summary of ethical precepts; it has nevertheless the merit of containing a fine aperçu of moral psychology, for it marks egoism by an exceedingly characteristic token, which is quite worth while being here more closely considered. This egoism, then, of which each of us is full, and to conceal which, as our partie honteuse, we have invented politeness, is perpetually peering through every veil cast over it, and may especially be detected in the fact that our dealings with all those, who come across our path, are directed by the one object of trying to find, before everything else, and as if by instinct, a possible means to any of the numerous ends with which we are always engrossed. When we make a new acquaintance, our first thought, as a rule, is whether the man can be useful to us in some way. If he can do nothing for our benefit, then as soon as we are convinced of this, he himself generally becomes nothing to us. To seek in all other people a possible means to our ends, in other words, to make them our instruments, is almost part of the very nature of human eyes; and whether the instrument will have to suffer more or less in the using, is a thought which comes much later, sometimes not at all. That we assume others to be similarly disposed is shown in many ways; e.g., by the fact that, when we ask any one for information or advice, we lose all confidence in his words directly we discover that he may have some interest in the matter, however small or remote. For then we immediately take for granted that he will make us a means to his ends, and hence give his advice not in accordance with his discernment, but with his desire, and this, no matter how exact the former may be, or how little the latter seem involved; since we know only too well that a cubic inch of desire weighs much more than a cubic yard of discernment. Conversely, when we ask in such cases: What ought I to do? as a rule, nothing else will occur to our counsellor, but how we should shape our action to suit his own ends; and to this effect he will give his reply immediately, and as it were mechanically, without so much as bestowing a thought on our ends; because it is his Will that directly dictates the answer, or
- 63. ever the question can come before the bar of his real judgment. Hence he tries to mould our conduct to his own benefit, without even being conscious of it, and while he supposes that he is speaking out of the abundance of his discernment, in reality he is nothing but the mouth-piece of his own desire; indeed, such self- deception may lead him so far as to utter lies, without being aware of it. So greatly does the influence of the Will preponderate that of the Intelligence. Consequently, it is not the testimony of our own consciousness, but rather, for the most part, that of our interest, which avails to determine whether our language be in accordance with what we discern, or what we desire. To take another case. Let us suppose that a man pursued by enemies and in danger of life, meets a pedlar and inquires for some by-way of escape; it may happen that the latter will answer him by the question: Do you need any of my wares? It is not of course meant that matters are always like this. On the contrary, many a man is found to show a direct and real participation in another's weal and woe, or (in Kant's language) to regard him as an end and not as a means. How far it seems natural, or the reverse, to each one to treat his neighbour for once in the way as an end, instead of (as usual) a means,—this is the criterion of the great ethical difference existing between character and character; and that on which the mental attitude of sympathy rests in the last resort will be the true basis of Ethics, and will form the subject of the third part of this Essay. Thus, in his second formula, Kant distinguishes Egoism and its opposite by a very characteristic trait; and this point of merit I have all the more gladly brought out into strong light and illustrated, because in other respects there is little in the groundwork of his Ethics that I can admit. The third and last form in which Kant put forward his Moral Principle is the Autonomy of the Will: The Will of every rational being is universally legislative for all rational beings. This of course follows from the first form. As a consequence of the third, however, we are asked to believe (see p. 71; R., p. 60) that the specific characteristic of the Categorical Imperative lies in the renunciation of all
- 64. interest by the Will when acting from a sense of duty. All previous moral principles had thus (he says) broken down, because the latter invariably attributed to human actions at bottom a certain interest, whether originating in compulsion, or in pleasurable attraction—an interest which might be one's own, or another's (p. 73; R., p. 62). (Another's: let this be particularly noticed.) Whereas a universally legislative Will must prescribe actions which are not based on any interest at all, but solely on a feeling of duty. I beg the reader to think what this really means. As a matter of fact, nothing less than volition without motive, in other words, effect without cause. Interest and Motive are interchangeable ideas; what is interest but quod mea interest, that which is of importance to me? And is not this, in one word, whatever stirs and sets in motion my Will? Consequently, what is an interest other than the working of a motive upon the Will? Therefore where a motive moves the Will, there the latter has an interest; but where the Will is affected by no motive, there in truth it can be as little active, as a stone is able to leave its place without being pushed or pulled. No educated person will require any demonstration of this. It follows that every action, inasmuch as it necessarily must have a motive, necessarily also presupposes an interest. Kant, however, propounds a second entirely new class of actions which are performed without any interest, i.e., without motive. And these actions are—all deeds of justice and loving-kindness! It will be seen that this monstrous assumption, to be refuted, needed only to be reduced to its real meaning, which was concealed through the word interest being trifled with. Meanwhile Kant celebrates (p. 74 sqq.; R., p. 62) the triumph of his Autonomy of the Will by setting up a moral Utopia called the Kingdom of Ends, which is peopled with nothing but rational beings in abstracto. These, one and all, are always willing, without willing any actual thing (i.e., without interest): the only thing that they will is that they may all perpetually will in accordance with one maxim (i.e., Autonomy). Difficile est satiram non scribere[5] (it is difficult to refrain from writing a satire).
- 65. But there is something else to which Kant is led by his autonomy of the will; and it involves more serious consequences than the little innocent Kingdom of Ends, which is perfectly harmless and may be left in peace. I mean the conception of human dignity. Now this dignity is made to rest solely on man's autonomy, and to lie in the fact that the law which he ought to obey is his own work, his relation to it thus being the same as that of the subjects of a constitutional government to their statutes. As an ornamental finish to the Kantian system of morals such a theory might after all be passed over. Only this expression Human Dignity, once it was uttered by Kant, became the shibboleth of all perplexed and empty- headed moralists. For behind that imposing formula they concealed their lack, not to say, of a real ethical basis, but of any basis at all which was possessed of an intelligible meaning; supposing cleverly enough that their readers would be so pleased to see themselves invested with such a dignity that they would be quite satisfied.[6] Let us, however, look at this conception a little more carefully, and submit it to the test of reality. Kant (p. 79; R., p. 66) defines dignity as an unconditioned, incomparable value. This is an explanation which makes such an effect by its magnificent sound that one does not readily summon up courage to examine it at close quarters; else we should find that it too is nothing but a hollow hyperbole, within which there lurks like a gnawing worm, the contradictio in adjecto. Every value is the estimation of one thing compared with another; it is thus a conception of comparison, and consequently relative; and this relativity is precisely that which forms the essence of the idea. According to Diogenes Laertius (Book VII., chap. 106),[7] this was already correctly taught by the Stoics. He says: τὴn δὲ ἀξίαν εἶναι ἀμοιβὴν δοκιμάστου, ἢν ἂν ὁ ἔμπειρος τῶν Πραγμάτων τάξῃ ὅμοιον εἐπεῑν, ἀμείβεσθαι πυροὺς πρὸς τὰς σὺν ἡμιονô κριθάς.[8] An incomparable, unconditioned, absolute value, such as dignity is declared by Kant to be, is thus, like so much else in Philosophy, the statement in words of a thought which is really unthinkable; just as much as the highest number, or the greatest space.
- 66. Doch eben wo Begriffe fehlen, Da stellt ein WORT zu rechter Zeit sich ein. (But where conceptions fail, Just there a WORD comes in to fill the blank.) So it was with this expression, Human Dignity. A most acceptable phrase was brought into currency. Thereon every system of Morals, that was spun out through all classes of duty, and all forms of casuistry, found a broad basis; from which serene elevation it could comfortably go on preaching. At the end of his exposition (p. 124; E., p. 97), Kant says: But how it is that Pure Reason without other motives, that may have their derivation elsewhere, can by itself be practical; that is, how, without there being any object for the Will to take an antecedent interest in, the simple principle of the universal validity of all the precepts of Pure Reason, as laws, can of itself provide a motive and bring about an interest which may be called purely moral; or, in other words, how it is that Pure Reason can be practical;—to explain this problem, all human reason is inadequate, and all trouble and work spent on it are vain. Now it should be remembered that, if any one asserts the existence of a thing which cannot even be conceived as possible, it is incumbent on him to prove that it is an actual reality; whereas the Categorical Imperative of Practical Reason is expressly not put forward as a fact of consciousness, nor otherwise founded on experience. Rather are we frequently cautioned not to attempt to explain it by having recourse to empirical anthropology. (Cf. e.g., p. vi. of the preface; R., p. 5; and pp. 59, 60; R., p. 52). Moreover, we are repeatedly (e.g., p. 48; R., p. 44) assured that no instance can show, and consequently there can be no empirical proof, that an Imperative of this sort exists everywhere. And further, on p. 49 (R., p. 45), we read, that the reality of the Categorical Imperative is not a fact of experience. Now if we put all this together, we can hardly avoid the suspicion that Kant is jesting at his readers' expense. But although this practice may be allowed by the present philosophical public of Germany, and seem good in their eyes, yet in Kant's time it was not so much in vogue; and besides,
- 67. Ethics, then, as always, was precisely the subject that least of all could lend itself to jokes. Hence we must continue to hold the conviction that what can neither be conceived as possible, nor proved as actual, is destitute of all credentials to attest its existence. And if, by a strong effort of the imagination, we try to picture to ourselves a man, possessed, as it were, by a daemon, in the form of an absolute Ought, that speaks only in Categorical Imperatives, and, confronting his wishes and inclinations, claims to be the perpetual controller of his actions; in this figure we see no true portrait of human nature, or of our inner life; what we do discern is an artificial substitute for theological Morals, to which it stands in the same relation as a wooden leg to a living one. Our conclusion, therefore, is, that the Kantian Ethics, like all anterior systems, is devoid of any sure foundation. As I showed at the outset, in my examination of its imperative Form, the structure is at bottom nothing but an inversion of theological Morals, cloaked in very abstract formulae of an apparently a priori origin. That this disguise was most artificial and unrecognisable is the more certain, from the fact that Kant, in all good faith, was actually himself deceived by it, and really believed that he could establish, independently of all theology, and on the basis of pure intelligence a priori, those conceptions of the Law and of the hests of Duty, which obviously have no meaning except in theological Ethics; whereas I have sufficiently proved that with him they are destitute of all real foundation, and float loosely in mid air. However, the mask at length falls away in his own workshop, and theological Ethics stands forth unveiled, as witness his doctrine of the Highest Good, the Postulates of Practical Reason; and lastly, his Moral Theology. But this revelation freed neither Kant nor the public from their illusion as to the real state of things; on the contrary, both he and they rejoiced to see all those precepts, which hitherto had been sanctioned by Faith, now ratified and established by Ethics (although only idealiter, and for practical purposes). The truth is that they, in all sincerity, put the effect for the cause, and the cause for the effect, inasmuch as they failed to perceive that at the root of this system of Morals there lay,
- 68. as absolutely necessary assumptions, however tacit and concealed, all the alleged consequences that had been drawn from it. At the end of this severe investigation, which must also have been tiring to my readers, perhaps I may be allowed, by way of diversion, to make a jesting, indeed frivolous comparison. I would liken Kant, in his self-mystification, to a man who at a ball has been flirting the whole evening with a masked beauty, in hopes of making a conquest; till at last, throwing off her disguise, she reveals herself— as his wife. [1] To keep the play of words in geschrobene, verschrobene, we may perhaps render them: twisted ... mistwisted.—(Translator.) [2] A contradiction in that which is added. A term applied to two ideas which cannot be brought into a thinkable relationship.—(Translator.) [3] A Chaṇḍāla (or Ćaṇḍāla) means one who is born of a Brahman woman by a Śūdra husband, such a union being an abomination. Hence it is a term applied to a low common person. Mlechcha (or Mleććha) means a foreigner; one who does not speak Sanskṛit, and is not subject to Hindu institutions. The transition from a a barbarian to a bad or wicked man, is easy.—(Translator.) [4] These so-called duties have been discussed in Chapter III. of this Part. [5] Juvenal, Sat. I. 30. [6] It appears that G. W. Block in his Neue Grundlegung der Philosophie der Sitten, 1802, was the first to make Human Dignity expressly and exclusively the foundation-stone of Ethics, which he then built up entirely on it. [7] V. Diogenes Laertius, de Clarorum Philosophorum Vitis, etc., edit. O. Gabr. Cobet. Paris; Didot, 1862. In this edition the passage quoted is in chap. 105 ad fin.,, p. 182.—(Translator.) [8] They teach that worth is the equivalent value of a thing which has been tested, whatever an expert may fix that value to be; as, for instance, to take wheat in exchange for barley and a mule.—(Translator.) CHAPTER VII.
- 69. KANT'S DOCTRINE OF CONSCIENCE. The alleged Practical Reason with its Categorical Imperative, is manifestly very closely connected with Conscience, although essentially different from it in two respects. In the first place, the Categorical Imperative, as commanding, necessarily speaks before the act, whereas Conscience does not till afterwards. Before the act Conscience can at best only speak indirectly, that is, by means of reflection, which holds up to it the recollection of previous cases, in which similar acts after they were committed received its disapproval. It is on this that the etymology of the word Gewissen (Conscience) appears to me to rest, because only what has already taken place is gewiss[1] (certain). Undoubtedly, through external inducement and kindled emotion, or by reason of the internal discord of bad humour, impure, base thoughts, and evil desires rise up in all people, even in the best. But for these a man is not morally responsible, and need not load his conscience with them; since they only show what the genus homo, not what the individual, who thinks them, would be capable of doing. Other motives, if not simultaneously, yet almost immediately, come into his consciousness, and confronting the unworthy inclinations prevent them from ever being crystallised into deeds; thus causing them to resemble the out-voted minority of an acting committee. By deeds alone each person gains an empirical knowledge no less of himself than of others, just as it is deeds alone that burden the conscience. For, unlike thoughts, these are not problematic; on the contrary, they are certain (gewiss), they are unchangeable, and are not only thought, but known (gewusst). The Latin conscientia,[2] and the Greek συνείδησις[3] have the same sense. Conscience is thus the knowledge that a man has about what he has done. The second point of difference between the alleged Categorical Imperative and Conscience is, that the latter always draws its material from experience; which the former cannot do, since it is
- 70. purely a priori. Nevertheless, we may reasonably suppose that Kant's Doctrine of Conscience will throw some light on this new conception of an absolute Ought which he introduced. His theory is most completely set forth in the Metaphysische Anfangsgründe zur Tugendlehre, § 13, and in the following criticism I shall assume that the few pages which contain it are lying before the reader. The Kantian interpretation of Conscience makes an exceedingly imposing effect, before which one used to stand with reverential awe, and all the less confidence was felt in demurring to it, because there lay heavy on the mind the ever-present fear of having theoretical objections construed as practical, and, if the correctness of Kant's view were denied, of being regarded as devoid of conscience. I, however, cannot be led astray in this manner, since the question here is of theory, not of practice; and I am not concerned with the preaching of Morals, but with the exact investigation of the ultimate ethical basis. We notice at once that Kant employs exclusively Latin legal terminology, which, however, would seem little adapted to reflect the most secret stirrings of the human heart. Yet this language, this judicial way of treating the subject, he retains from first to last, as though it were essential and proper to the matter. And so we find brought upon the stage of our inner self a complete Court of justice, with indictment, judge, plaintiff, defendant, and sentence;—nothing is wanting. Now if this tribunal, as portrayed by Kant, really existed in our breasts, it would be astonishing if a single person could be found to be, I do not say, so bad, but so stupid, as to act against his conscience. For such a supernatural assize, of an entirely special kind, set up in our consciousness, such a secret court—like another Fehmgericht[4] —held in the dark recesses of our inmost being, would inspire everybody with a terror and fear of the gods strong enough to really keep him from grasping at short transient advantages, in face of the dreadful threats of superhuman powers, speaking in tones so near and so clear. In real life, on the contrary, we find, that the efficiency of conscience is generally considered such a vanishing quantity that all peoples have bethought themselves of helping it out
- 71. by means of positive religion, or even of entirely replacing it by the latter. Moreover, if Conscience were indeed of this peculiar nature, the Royal Society could never have thought of the question put for the present Prize Essay. But if we look more closely at Kant's exposition, we shall find that its imposing effect is mainly produced by the fact that he attributes to the moral verdict passed on ourselves, as its peculiar and essential characteristic, a form which in fact is not so at all. This metaphorical bar of judgment is no more applicable to moral self-examination than it is to every other reflection as regards what we have done, and might have done otherwise, where no ethical question is involved. For it is not only true that the same procedure of indictment, defence, and sentence is occasionally assumed by that obviously spurious and artificial conscience which is based on mere superstition; as, for instance, when a Hindu reproaches himself with having been the murderer of a cow, or when a Jew remembers that he has smoked his pipe at home on the Sabbath; but even the self- questioning which springs from no ethical source, being indeed rather unmoral than moral, often appears in a shape of this sort, as the following case may exemplify. Suppose I, good-naturedly, but thoughtlessly, have made myself surety for a friend, and suppose there comes with evening the clear perception of the heavy responsibility I have taken on myself—a responsibility that may easily involve me in serious trouble, as the wise old saying, ἐγγύα παρά δ' ἃτα![5] predicts; then at once there rise up within me the Accuser and the Counsel for the defence, ready to confront each other. The latter endeavours to palliate my rashness in giving bail so hastily, by pointing out the stress of circumstance or of obligation, or, it may be, the simple straightforwardness of the transaction; perhaps he even seeks excuse by commending my kind heart. Last of all comes the Judge who inexorably passes the sentence: A fool's piece of work! and I am overwhelmed with confusion So much for this judicial form of which Kant is so fond; his other modes of expression are, for the most part, open to the same criticism. For instance, that which he attributes to conscience, at the beginning of
- 72. the paragraph, as its peculiar property, applies equally to all other scruples of an entirely different sort. He says: It (conscience) follows him like his shadow, try though he may to escape. By pleasures and distractions he may be stupefied and billed to sleep, but he cannot avoid occasionally waking up and coming to himself; and then he is immediately aware of the terrible voice, etc. Obviously, this may be just as well understood, word for word, of the secret consciousness of some person of private means, who feels that his expenses far exceed his income, and that thus his capital is being affected, and will gradually melt away. We have seen that Kant represents the use of legal terms as essential to the subject, and that he keeps to them from beginning to end; let it now be noted how he employs the same style for the following finely devised sophism. He says: That a person accused by his conscience should be identified with the judge is an absurd way of portraying a court of justice; for in that case the accuser would invariably lose. And he adds, by way of elucidating this statement, a very ambiguous and obscure note. His conclusion is that, if we would avoid falling into a contradiction, we must think of the judge (in the judicial conscience-drama that is enacted in our breasts) as different from us, in fact, as another person; nay more, as one that is an omniscient knower of hearts, whose hests are obligatory on all, and who is almighty for every purpose of executive authority.[6] He thus passes by a perfectly smooth path from conscience to superstition, making the latter a necessary consequence of the former; while he is secretly sure that he will be all the more willingly followed because the reader's earliest training will have certainly rendered him familiar with such ideas, if not have made them his second nature. Here, then, Kant finds an easy task,— a thing he ought rather to have despised; for he should have concerned himself not only with preaching, but also with practising truthfulness. I entirely reject the above quoted sentence, and all the conclusions consequent thereon, and I declare it to be nothing but a shuffling trick. It is not true that the accuser must always lose, when the accused is the same person as the judge; at least not in
- 73. the court of judgment in our hearts. In the instance I gave of one man going surety for another, did the accuser lose? Or must we in this case also, if we wish to avoid a contradiction, really assume a personification after Kant's fashion, and be driven to view objectively as another person that voice whose deliverance would have been those terrible words: A fool's piece of work!? A sort of Mercury, forsooth, in living flesh? Or perhaps a prosopopoeia of the Μῆτις (cunning) recommended by Homer (Il. xxiii. 313 sqq.)?[7] But thus we should only be landed, as before, on the broad path of superstition, aye, and pagan superstition too. It is in this passage that Kant indicates his Moral Theology, briefly indeed, yet not without all its vital points. The fact that he takes care, not to attribute to it any objective validity, but rather to present it merely as a form subjectively unavoidable, does not free him from the arbitrariness with which he constructs it, even though he only claims its necessity for human consciousness. His fabric rests, as we have seen, on a tissue of baseless assumptions. So much, then, is certain. The entire imagery—that of a judicial drama—whereby Kant depicts conscience is wholly unessential and in no way peculiar to it; although he keeps this figure, as if it were proper to the subject, right through to the end, in order finally to deduce certain conclusions from it. As a matter of fact it is a sufficiently common form, which our thoughts easily take when we consider any circumstance of real life. It is due for the most part to the conflict of opposing motives which usually spring up, and which are successively weighed and tested by our reflecting reason. And no difference is made whether these motives are moral or egoistic in their nature, nor whether our deliberations are concerned with some action in the past, or in the future. Now if we strip from Kant's exposition its dress of legal metaphor, which is only an optional dramatic appendage, the surrounding nimbus with all its imposing effect immediately disappears as well, and there remains nothing but the fact that sometimes, when we think over our actions, we are seized with a certain self-dissatisfaction, which is marked by a special characteristic. It is with our conduct per se that we are
- 74. discontented, not with its result, and this feeling does not, as in every other case in which we regret the stupidity of our behaviour, rest on egoistic grounds. For on these occasions the cause of our dissatisfaction is precisely because we have been too egoistic, because we have taken too much thought for ourselves, and not enough for our neighbour; or perhaps even because, without any resulting advantage, we have made the misery of others an object in itself. That we may be dissatisfied with ourselves, and saddened by reason of sufferings which we have inflicted, not undergone, is a plain fact and impossible to be denied. The connection of this with the only ethical basis that can stand an adequate test we shall examine further on. But Kant, like a clever special pleader, tried by magnifying and embellishing the original datum to make all that he possibly could of it, in order to prepare a very broad foundation for his Ethics and Moral Theology. [1] Both words are, of course, derived from wissen = scire = εἱδέναι.— (Translator.) [2] Cf. Horace's conscire sibi, pallescere culpa: Epist. I. 1, 61. To be conscious of having done wrong, to turn pale at the thought of the crime. [3] Συνείδησις = consciousness (of right or wrong done).—(Translator.) [4] The celebrated Secret Tribunal of Westphalia, which came into prominence about A.D. 1220. In A.D. 1335 the Archbishop of Cologne was appointed head of all the Fehme benches in Westphalia by the Emperor Charles IV. The reader will remember the description of the trial scene in Scott's Anne of Geierstein. Perhaps the Court of Star Chamber comes nearest to it in English History.—(Translator.) [5] If you give a pledge, be sure that Ate (the goddess of mischief) is beside you; i.e., beware of giving pledges.—Thales ap. Plat. Charm. 165 A. [6] Kant leads up to this position with great ingenuity, by having recourse to the theory of the two characters coexistent in man—the noumenal (or intelligible) and the empirical; the one being in time, the other, timeless; the one, fast bound by the law of causality, the other free.—(Translator.) [7] Greek: Άλλ' ἄγε δὴ σύ, ϕίλος, μêτιν ἐμβάλλεο θυμῷ, κ.τ.λ.
- 75. CHAPTER VIII. KANT'S DOCTRINE OF THE INTELLIGIBLE [1] AND EMPIRICAL CHARACTER. THEORY OF FREEDOM. The attack I have made, in the cause of truth, on Kant's system of Morals, does not, like those of my predecessors, touch the surface only, but penetrates to its deepest roots. It seems, therefore, only just that, before I leave this part of my subject, I should bring to remembrance the brilliant and conspicuous service which he nevertheless rendered to ethical science. I allude to his doctrine of the co-existence of Freedom and Necessity. We find it first in the Kritik der Reinen Vernunft (pp. 533-554 of the first, and pp. 561-582 of the fifth, edition); but it is still more clearly expounded in the Kritik der Praktischen Vernunft (fourth edition, pp. 169-179; R., pp. 224-231). The strict and absolute necessity of the acts of Will, determined by motives as they arise, was first shown by Hobbes, then by Spinoza, and Hume, and also by Dietrich von Holbach in his Système de la Nature; and lastly by Priestley it was most completely and precisely demonstrated. This point, indeed, has been so clearly proved, and placed beyond all doubt, that it must be reckoned among the number of perfectly established truths, and only crass ignorance could continue to speak of a freedom, of a liberum arbitrium indifferentiae (a free and indifferent choice) in the individual acts of men. Nor did Kant, owing to the irrefutable reasoning of his predecessors, hesitate to consider the Will as fast bound in the chains of Necessity, the matter admitting, as he thought, of no further dispute or doubt. This is proved by all the passages in which he speaks of freedom only from the theoretical standpoint. Nevertheless, it is true that our actions are attended with a consciousness of independence and original initiative, which makes
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