multivariate-CalculusCHAP1_Complex_Analysis.pdf

Department of Electronic Engineering
The Islamia University of Bahawalpur, Pakistan
Complex Numbers
MATH-00208 Multivariate Calculus & Complex Analysis

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Chapter Outline
1.1 - The Real Number System
1.2 - Graphical Representation of Real Numbers
1.3 - The Complex Number System
1.4 - Fundamental Operations with Complex Numbers
1.5 - Absolute Value
1.6 - Axiomatic Foundation of the Complex Number System
1.7 - Graphical Representation of Complex Numbers
1.8 - Polar Form of Complex Numbers
1.9 - De Moivre’s Theorem
1.10 - Roots of Complex Numbers
1.11 - Euler’s Formula
1.12 - Polynomial Equations
1.13 - The nth Roots of Unity
1.14 - Vector Interpretation of Complex Numbers
1.15 - Stereographic Projection
1.16 - Dot and Cross Product
1.17 - Complex Conjugate Coordinates

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1.1 The Real Number System
 The number system as we know it today is a result of gradual
development as indicated in the following list.
1) Natural numbers 1, 2, 3, 4,..., also called positive integers, were first
used in counting.
 If a and b are natural numbers, the sum a + b and product a‧b, (a)(b)
or ab are also natural numbers.
 For this reason, the set of natural numbers is said to be closed under
the operations of addition and multiplication or to satisfy the closure
property with respect to these operations.
2) Negative integers and zero, denoted by -1, -2, -3,... and 0,
respectively, permit solutions of equations such as x + b = a where a
and b are any natural numbers.
 This leads to the operation of subtraction, or inverse of addition, and
we write x = a - b.
 The set of positive and negative integers and zero is called the set of
integers and is closed under the operations of addition, multiplication,
and subtraction.

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1.1 The Real Number System
3) Rational numbers or fractions such as 3/4, -8/3,... permit solutions of
equations such as bx = a for all integers a and b where b ≠ 0.
 This leads to the operation of division or inverse of multiplication, and
we write x = a/b or a ¼ b (called the quotient of a and b) where a is
the numerator and b is the denominator.
 The set of integers is a part or subset of the rational numbers, since
integers correspond to rational numbers a/b where b = 1.
 The set of rational numbers is closed under the operations of addition,
subtraction, multiplication, and division, so long as division by zero is
excluded.
4) Irrational numbers such as and are numbers that cannot be
expressed as a/b where a and b are integers and b ≠ 0.
 The set of rational and irrational numbers is called the set of real
numbers.

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1.2 Graphical Representation of Real Numbers
 Real numbers can be represented by points on a line called the real
axis, as indicated in Fig. 1-1.
 The point corresponding to zero is called the origin.
 Conversely, to each point on the line there is one and only one real
number.
 If a point A corresponding to a real number a lies to the right of a
point B corresponding to a real number b, we say that a is greater than
b or b is less than a and write a > b or b < a, respectively.
 The set of all values of x such that a < x < b is called an open interval
on the real axis while a ≤ x ≤ b, which also includes the endpoints a
and b, is called a closed interval.

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1.2 Graphical Representation of Real Numbers
 The symbol x, which can stand for any real number, is called a real
variable.
 The absolute value of a real number a, denoted by |a|, is equal to a if a
> 0, to -a if a < 0 and to 0 if a = 0.
 The distance between two points a and b on the real axis is |a – b|.

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 There is no real number x that satisfies the polynomial equation x2 + 1
= 0.
 To permit solutions of this and similar equations, the set of complex
numbers is introduced.
 We can consider a complex number as having the form a + bi where a
and b are real numbers and i, which is called the imaginary unit, has
the property that i2 = -1.
 If z = a + bi, then a is called the real part of z and b is called the
imaginary part of z and are denoted by Re{z} and Im{z}, respectively.
 The symbol z, which can stand for any complex number, is called a
complex variable.
 Two complex numbers a + bi and c + di are equal if and only if a = c
and b = d.
 We can consider real numbers as a subset of the set of complex
numbers with b = 0.
1.3 The Complex Number System

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 Accordingly the complex numbers 0 + 0i and -3 + 0i represent the real
numbers 0 and -3, respectively.
 If a = 0, the complex number 0 + bi or bi is called a pure imaginary
number.
 The complex conjugate, or briefly conjugate, of a complex number a
+ bi is a - bi.
 The complex conjugate of a complex number z is often indicated by
or z*.
1.3 The Complex Number System

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 In performing operations with complex numbers, we can proceed as in
the algebra of real numbers, replacing i2 by -1 when it occurs.
1.4 Fundamental Operations with Complex Numbers

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 The absolute value or modulus of a complex number a + bi is defined
as
1.5 Absolute Value
 If z1, z2, z3,..., zm are complex numbers, the following properties hold.

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 From a strictly logical point of view, it is desirable to define a
complex number as an ordered pair (a, b) of real numbers a and b
subject to certain operational definitions, which turn out to be
equivalent to those defined in last topics.
 These definitions are as follows, where all letters represent real
numbers.
1.6 Axiomatic Foundation of the Complex Number System
 From these we can show [Problem 1.14] that
(a, b) = a(1, 0) + b(0, 1)
and we associate this with a + bi where i is the symbol for (0, 1) and
has the property that i2 = (0, 1)(0, 1) = (-1, 0)
and (1, 0) can be considered equivalent to the real number 1.
 The ordered pair (0, 0) corresponds to the real number 0.

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 From the above, we can prove the following.
 In general, any set such as S, whose members satisfy the above, is
called a field.
0 is called the identity with respect to addition, 1 is called the identity with respect
to multiplication.

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1.7 Graphical Representation of Complex Numbers
 Suppose real scales are chosen on two mutually perpendicular axes
X΄OX and Y΄OY [called the x and y axes, respectively] as in Fig. 1-2.
 We can locate any point in the plane determined by these lines by the
ordered pair of real numbers (x, y) called rectangular coordinates of
the point.
 Examples of the location of such points are indicated by P, Q, R, S,
and T in Fig. 1-2.

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1.7 Graphical Representation of Complex Numbers
 Since a complex number x + iy can be considered as an ordered pair of
real numbers, we can represent such numbers by points in an xy plane
called the complex plane or Argand diagram.
 The complex number represented by P, for example, could then be
read as either (3, 4) or 3 + 4i.
 To each complex number there corresponds one and only one point in
the plane, and conversely to each point in the plane there corresponds
one and only one complex number.
 Because of this we often refer to the complex number z as the point z.
 Sometimes, we refer to the x and y axes as the real and imaginary
axes, respectively, and to the complex plane as the z plane.
 The distance between two points, z1 = x1 + iy1 and z2 = x2 + iy2, in the
complex plane is given by

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1.8 Polar Form of Complex Numbers
 Let P be a point in the complex plane corresponding to the complex
number (x, y) or x + iy.
 Then we see from Fig. 1-3 that
where
is called the
modulus or absolute value of z = x + iy
[denoted by mod z or |z|]
θ, called the amplitude or argument of
z = x + iy [denoted by arg z], is the
angle that line OP makes with the
positive x axis.
which is called the polar form of the complex number, and r and θ are
called polar coordinates.
 It follows that

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1.8 Polar Form of Complex Numbers
 It is sometimes convenient to write the abbreviation cisθ for cosθ +
isinθ.
 For any complex number z ≠ 0 there corresponds only one value of θ
in 0 ≤ θ ≤ 2π.
 However, any other interval of length 2π, for example -π ≤ θ < π can
be used.
 Any particular choice, decided upon in advance, is called the principal
range, and the value of θ is called its principal value.

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1.9 De Moivre’s Theorem
 Let
then we can show that [see Problem 1.19]
 A generalization of (1.2) leads to
and if z1 = z2 = … = zn = z this becomes
which is often called De Moivre’s theorem.

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1.10 Roots of Complex Numbers
 A number w is called an nth root of a complex number z if wn = z, and
we write w = z1/n.
 From De Moivre’s theorem we can show that if n is a positive integer,
from which it follows that there are n different values for z1/n, i.e., n
different nth roots of z, provided z ≠ 0.
De Moivre’s theorem

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Comparing above two equations

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 By considering k = 5, 6,... as well as negative values, -1, -2,...
repetitions of the above five values of z are obtained.
 Hence, these are the only solutions or roots of the given equation.
 These five roots are called the fifth roots of -32 and are collectively
denoted by (-32)1/5.
 In general, a1/n represents the nth roots of a and there are n such roots.

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1.11 Euler’s Formula
 By assuming that the infinite series expansion
of elementary calculus holds when x = iθ, we can arrive at the result
which is called Euler’s formula.
 In general, we define
 In the special case where y = 0 this reduces to ex.
 Note that in terms of (1.7) De Moivre’s theorem reduces to (eiθ)n =
einθ.
 It is more convenient, however, simply to take (1.7) as a definition of
eiθ.
De Moivre’s theorem

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1.12 Polynomial Equations
 Often in practice we require solutions of polynomial equations having
the form
 Such solutions are also called zeros of the polynomial on the left of
(1.9) or roots of the equation.
 A very important theorem called the fundamental theorem of algebra
states that every polynomial equation of the form (1.9) has at least one
root in the complex plane.
 From this we can show that it has in fact n complex roots, some or all
of which may be identical.
 If z1, z2, . . . , zn are the n roots, then (1.9) can be written
where a0 ≠ 0, a1,..., an are given complex numbers and n is a positive
integer called the degree of the equation.
which is called the factored form of the polynomial equation.

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1.13 The nth Roots of Unity
 The solutions of the equation zn = 1 where n is a positive integer are
called the nth roots of unity and are given by
 If we let
 Geometrically, they represent the n vertices of a regular polygon of n
sides inscribed in a circle of radius one with center at the origin.
 This circle has the equation |z| = 1 and is often called the unit circle.
the n roots are 1, ω, ω2, . . . , ωn-1.

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1.14 Vector Interpretation of Complex Numbers
 A complex number z = x + iy can be considered as a vector OP
whose initial point is the origin O and whose terminal point P is the
point (x, y) as in Fig. 1-4.
 We sometimes call OP = x + iy the position vector of P.
 Two vectors having the same length or magnitude and direction but
different initial points, such as OP and AB in Fig. 1-4, are considered
equal.
 Hence we write OP = AB = x + iy.

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 Addition of complex numbers corresponds to the parallelogram law
for addition of vectors [see Fig. 1-5].
 Thus to add the complex numbers z1 and z2, we complete the
parallelogram OABC whose sides OA and OC correspond to z1 and z2.
 The diagonal OB of this parallelogram corresponds to z1 + z2. See
Problem 1.5.

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1.15 Stereographic Projection
 Let ƿ [Fig. 1-6] be the complex plane and consider a sphere S tangent
to P at z = 0.
 The diameter NS is perpendicular to ƿ and we call points N and S the
north and south poles of S.
 Corresponding to any point A on ƿ we can construct line NA
intersecting S at point A΄.
 Thus to each point of the complex plane ƿ there corresponds one and
only one point of the sphere S, and we can represent any complex
number by a point on the sphere.

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1.15 Stereographic Projection
 For completeness we say that the point N itself corresponds to the
“point at infinity” of the plane.
 The set of all points of the complex plane including the point at
infinity is called the entire complex plane, the entire z plane, or the
extended complex plane.
 The above method for mapping the plane on to the sphere is called
stereographic projection.
 The sphere is sometimes called the Riemann sphere.
 When the diameter of the Riemann sphere is chosen to be unity, the
equator corresponds to the unit circle of the complex plane.

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1.16 Dot and Cross Product
 Let z1 = x1 + iy1 and z2 = x2 + iy2 be two complex numbers [vectors].
 The dot product [also called the scalar product] of z1 and z2 is defined
as the real number
where θ is the angle between z1 and z2 which lies between 0 and π.
 The cross product of z1 and z2 is defined as the vector z1  z2 = (0, 0,
x1y2 - y1x2) perpendicular to the complex plane having magnitude
1) A necessary and sufficient condition that z1 and z2 be
perpendicular is that z1‧z2 = 0.
2) A necessary and sufficient condition that z1 and z2 be parallel is
that |z1z2| = 0.
3) The magnitude of the projection of z1 on z2 is |z1‧z2|/|z2|.
4) The area of a parallelogram having sides z1 and z2 is |z1z2|.

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1.17 Complex Conjugate Coordinates
 A point in the complex plane can be located by rectangular
coordinates (x, y) or polar coordinates (r, θ).
 Many other possibilities exist.
 One such possibility uses the fact that
 The coordinates (z, ) that locate a point are called complex
conjugate coordinates or briefly conjugate coordinates of the point
[see Problem 1.43].
where

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1.17 Complex Conjugate Coordinates

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