complex numbers and functions.PDF
- 2. Complex Numbers and Functions ______________________________________________________________________________________________ Natural is the most fertile source of Mathematical Discoveries - Jean Baptiste Joseph Fourier The Complex Number System Definition: A complex number z is a number of the form z a ib = + , where the symbol i = −1 is called imaginary unit and a b R , . ∈ a is called the real part and b the imaginary part of z, written a z = Re and b z = Im . With this notation, we have z z i z = + Re Im . The set of all complex numbers is denoted by { } C a ib a b R = + ∈ , . If b = 0, then z a i a = + = 0 , is a real number. Also if a = 0, then z ib ib = + = 0 , is a imaginary number; in this case, z is called pure imaginary number. Let a ib + and c id + be complex numbers, with a b c d R , , , . ∈ 1. Equality a ib c id + = + if and only if a c b d = = and . Note: In particular, we have z a ib = + = 0 if and only if a b = = 0 0 and . 2. Fundamental Algebraic Properties of Complex Numbers (i). Addition ( ) ( ) ( ) ( ). a ib c id a c i b d + + + = + + + (ii). Subtraction ( ) ( ) ( ) ( ). a ib c id a c i b d + − + = − + − (iii). Multiplication ( )( ) ( ) ( ). a ib c id ac bd i ad bc + + = − + + Remark (a). By using the multiplication formula, one defines the nonnegative integral power of a complex number z as z z z zz z z z z z z n n 1 2 3 2 1 = = = = − , , , , . ! Further for z ≠ 0, we define the zero power of z is 1; that is, z0 1 = . (b). By definition, we have i2 1 = − , i i 3 = − , i4 1 = . (iv). Division If c id + ≠ 0, then a ib c id ac bd c d i bc ad c d + + = + + + − + 2 2 2 2 .
- 3. Remark (a). Observe that if a ib + = 1, then we have 1 2 2 2 2 c id c c d i d c d + = + + − + . (b). For any nonzero complex number z, we define z z − = 1 1 , where z−1 is called the reciprocal of z. (c). For any nonzero complex number z, we now define the negative integral power of a complex number z as z z z z z z z z z z z n n − − − − − − − − − + − = = = = 1 2 1 1 3 2 1 1 1 1 , , , , . ! (d). i i i − = = − 1 1 , i− = − 2 1, i i − = 3 , i− = 4 1. 3. More Properties of Addition and Multiplication For any complex numbers z z z z , , , and 1 2 3 (i). Commutative Laws of Addition and Multiplication: z z z z z z z z 1 2 2 1 1 2 2 1 + = + = ; . (ii) Associative Laws of Addition and Multiplication: z z z z z z z z z z z z 1 2 3 1 2 3 1 2 3 1 2 3 + + = + + = ( ) ( ) ; ( ) ( ) . (iii). Distributive Law: z z z z z z z 1 2 3 1 2 1 3 ( ) . + = + (iv). Additive and Multiplicative identities: z z z z z z + = + = ⋅ = ⋅ = 0 0 1 1 ; . (v). z z z z + − = − + = ( ) ( ) . 0 Complex Conjugate and Their Properties Definition: Let . , , R b a C ib a z ∈ ∈ + = The complex conjugate, or briefly conjugate, of z is defined by z a ib = − . For any complex numbers z z z C , , , 1 2 ∈ we have the following algebraic properties of the conjugate operation: (i). z z z z 1 2 1 2 + = + , (ii). z z z z 1 2 1 2 − = − , (iii). z z z z 1 2 1 2 = ⋅ ,
- 4. (iv). z z z z 1 2 1 2 = , provided z2 0 ≠ , (v). z z = , (vi). ( ) z z n n = , for all n Z ∈ , (vii). z z z = = if and only if Im , 0 (viii). z z z = − = if and only if Re 0, (ix). z z z + = 2 Re , (x). z z i z − = ( Im ), 2 (xi). ( ) ( ) zz z z = + Re Im . 2 2 Modulus and Their Properties Definition: The modulus or absolute value of a complex number z a ib = + , a b R , ∈ is defined as z a b = + 2 2 . That is the positive square root of the sums of the squares of its real and imaginary parts. For any complex numbers z z z C , , , 1 2 ∈ we have the following algebraic properties of modulus: (i). z z ≥ = = 0 0 0 ; , and z if and only if (ii). z z z z 1 2 1 2 = , (iii). z z z z 1 2 1 2 = , provided z2 0 ≠ , (iv). z z z = = − , (v). z zz = , (vi). z z z ≥ ≥ Re Re , (vii). z z z ≥ ≥ Im Im , (viii). z z z z 1 2 1 2 + ≤ + , (triangle inequality) (ix). z z z z 1 2 1 2 − ≤ + . The Geometric Representation of Complex Numbers In analytic geometry, any complex number z a ib a b R = + ∈ , , can be represented by a point z P a b = ( , ) in xy-plane or Cartesian plane. When the xy-plane is used in this way to plot or represent complex numbers, it is called the Argand plane1 or the complex plane. Under these circumstances, the x- or horizontal axis is called the axis of real number or simply, real axis whereas the y- or vertical axis is called the axis of imaginary numbers or simply, imaginary axis. 1 The plane is named for Jean Robert Argand, a Swiss mathematician who proposed the representation of complex numbers in 1806.
- 5. Furthermore, another possible representation of the complex number z in this plane is as a vector OP . We display z a ib = + as a directed line that begins at the origin and terminates at the point P a b ( , ). Hence the modulus of z, that is z , is the distance of z P a b = ( , ) from the origin. However, there are simple geometrical relationships between the vectors for z a ib = + , the negative of z; − z and the conjugate of z; z in the Argand plane. The vector − z is vector for z reflected through the origin, whereas z is the vector z reflected about the real axis. The addition and subtraction of complex numbers can be interpreted as vector addition which is given by the parallelogram law. The ‘triangle inequality’ is derivable from this geometric complex plane. The length of the vector z z 1 2 + is z z 1 2 + , which must be less than or equal to the combined lengths z z 1 2 + . Thus z z z z 1 2 1 2 + ≤ + . Polar Representation of Complex Numbers Frequently, points in the complex plane, which represent complex numbers, are defined by means of polar coordinates. The complex number z x iy = + can be located as polar coordinate ( , ) r θ instead of its rectangular coordinates ( , ), x y it follows that there is a corresponding way to write complex number in polar form. We see that r is identical to the modulus of z; whereas θ is the directed angle from the positive x-axis to the point P. Thus we have x r = cosθ and y r = sinθ , where r z x y y x = = + = 2 2 , tan . θ We called θ the argument of z and write θ = arg . z The angle θ will be expressed in radians and is regarded as positive when measured in the counterclockwise direction and negative when measured clockwise. The distance r is never negative. For a point at the origin; z = 0, r becomes zero. Here θ is undefined since a ray like that cannot be constructed. Consequently, we now defined the polar for m of a complex number z x iy = + as z r i = + (cos sin ) θ θ (1) Clearly, an important feature of arg z = θ is that it is multivalued, which means for a nonzero complex number z, it has an infinite number of distinct arguments (since sin( ) sin , cos( ) cos , θ π θ θ π θ + = + = ∈ 2 2 k k k Z) . Any two distinct arguments of z differ each other by an integral multiple of 2π , thus two nonzero complex number z r i 1 1 1 1 = + (cos sin ) θ θ and z r i 2 2 2 2 = + (cos sin ) θ θ are equal if and only if r r 1 2 = and θ θ π 1 2 2 = + k , where k is some integer. Consequently, in order to specify a unique value of arg , z we
- 6. may restrict its value to some interval of length. For this, we introduce the concept of principle value of the argument (or principle argument) of a nonzero complex number z, denoted as Arg z, is defined to be the unique value that satisfies π π < ≤ − z Arg . Hence, the relation between argz and Arg z is given by arg , . z z k k Z = + ∈ Arg 2 π Multiplication and Division in Polar From The polar description is particularly useful in the multiplication and division of complex number. Consider z r i 1 1 1 1 = + (cos sin ) θ θ and z r i 2 2 2 2 = + (cos sin ). θ θ 1. Multiplication Multiplying z1 and z2 we have ( ) z z r r i 1 2 1 2 1 2 1 2 = + + + cos( ) sin( ) . θ θ θ θ When two nonzero complex are multiplied together, the resulting product has a modulus equal to the product of the modulus of the two factors and an argument equal to the sum of the arguments of the two factors; that is, z z r r z z z z z z 1 2 1 2 1 2 1 2 1 2 1 2 = = = + = + , arg( ) arg( ) arg( ). θ θ 1. Division Similarly, dividing z1 by z2 we obtain ( ) z z r r i 1 2 1 2 1 2 1 2 = − + − cos( ) sin( ) . θ θ θ θ The modulus of the quotient of two complex numbers is the quotient of their modulus, and the argument of the quotient is the argument of the numerator less the argument of the denominator, thus z z r r z z z z z z 1 2 1 2 1 2 1 2 1 2 1 2 = = = − = − , arg arg( ) arg( ). θ θ Euler’s Formula and Exponential Form of Complex Numbers For any real θ, we could recall that we have the familiar Taylor series representation of sinθ , cosθ and eθ : sin ! ! , , cos ! ! , , ! ! , , θ θ θ θ θ θ θ θ θ θ θ θ θ θ = − + − − ∞ < < ∞ = − + − − ∞ < < ∞ = + + + + − ∞ < < ∞ 3 5 2 4 2 3 3 5 1 2 4 1 2 3 ! ! ! e Thus, it seems reasonable to define
- 7. e i i i iθ θ θ θ = + + + + 1 2 3 2 3 ( ) ! ( ) ! . ! In fact, this series approach was adopted by Karl Weierstrass (1815-1897) in his development of the complex variable theory. By (2), we have e i i i i i i i i i i iθ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ = + + + + = + − − + = − + + + − + + = + 1 2 3 4 5 1 2 3 4 5 1 2 4 3 5 2 3 4 5 2 3 4 5 2 4 3 5 ( ) ! ( ) ! ( ) ! ( ) ! ! ! ! ! ! ! ! ! cos sin . + + + + ! ! ! ! Now, we obtain the very useful result known as Euler’s2 formula or Euler’s identity e i iθ θ θ = + cos sin . (2) Consequently, we can write the polar representation (1) more compactly in exponential form as z rei = θ . Moreover, by the Euler’s formula (2) and the periodicity of the trigonometry functions, we get . integer all for 1 , real all for 1 ) 2 ( k e e k i i = = π θ θ Further, if two nonzero complex numbers z r ei 1 1 1 = θ and z r ei 2 2 2 = θ , the multiplication and division of complex numbers z z 1 2 and have exponential forms z z r r e z z r r e i i 1 2 1 2 1 2 1 2 1 2 1 2 = = + − ( ) ( ) , θ θ θ θ respectively. de Moivre’s Theorem In the previous section we learned to multiply two number of complex quantities together by means of polar and exponential notation. Similarly, we can extend this method to obtain the multiplication of any number of complex numbers. Thus, if z r e k n k k i k = = θ , , , , , 1 2 " for any positive integer n, we have z z z r r r e n n i n 1 2 1 2 1 2 ! ! ! = + + + ( ). ( ) θ θ θ In particular, if all values are identical we obtain ( ) z re r e n i n n in = = θ θ for any positive integer n. Taking r = 1 in this expression, we then have ( ) e e i n in θ θ = for any positive integer n. By Euler’s formula (3), we obtain ( ) cos sin cos sin θ θ θ θ + = + i n i n n (3) 2 Leonhard Euler (1707 -1783) is a Swiss mathematician.
- 8. for any positive integer n. By the same argument, it can be shown that (3) is also true for any nonpositive integer n. Which is known as de Moivre’s3 formula, and more precisely, we have the following theorem: Theorem: (de Moivre’s Theorem) For any θ and for any integer n, ( ) cos sin cos sin θ θ θ θ + = + i n i n n . In term of exponential form, it essentially reduces to ( ) e e i n in θ θ = . Roots of Complex Numbers Definition: Let n be a positive integer≥ 2, and let z be nonzero complex number. Then any complex number w that satisfies w z n = is called the n-th root of z, written as w z n = . Theorem: Given any nonzero complex number z rei = θ , the equation w z n = has precisely n solutions given by w r k n i k n k n = + + + cos sin , θ π θ π 2 2 k n = − 01 1 , , , , " or w r e k n i k n = + θ π 2 , k n = − 01 1 , , , , " where r n denotes the positive real n-th root of r z = and θ = Argz. Elementary Complex Functions Let A and B be sets. A function f from A to B, denoted by B A f → : is a rule which assigns to each element A a ∈ one and only one element , B b ∈ we write ) (a f b = and call b the image of a under f. The set A is the domain-set of f, and the set B is the codomain or target-set of f. The set of all images { } : ) ( ) ( A a a f A f ∈ = is called the range or image-set of f. It must be emphasized that both a domain-set and a rule are needed in order for a function to be well defined. When the domain-set is not mentioned, we agree that the largest possible set is to be taken. The Polynomial and Rational Functions 3 This useful formula was discovered by a French mathematician, Abraham de Moivre (1667 - 1754).
- 9. 1. Complex Polynomial Functions are defined by P z a a z a z a z n n n n ( ) , = + + + + − − 0 1 1 1 ! where a a a C n 0 1 , , , " ∈ and n N ∈ . The integer n is called the degree of polynomial P z ( ), provided that an ≠ 0. The polynomial p z az b ( ) = + is called a linear function. 2. Complex Rational Functions are defined by the quotient of two polynomial functions; that is, R z P z Q z ( ) ( ) ( ) , = where P z Q z ( ) ( ) and are polynomials defined for all z C ∈ for which Q z ( ) . ≠ 0 In particular, the ratio of two linear functions: f z az b cz d ( ) = + + with ad bc − ≠ 0, which is called a linear fractional function or Mobius ## transformation. The Exponential Function In defining complex exponential function, we seek a function which agrees with the exponential function of calculus when the complex variable z x iy = + is real; that is we must require that f x i ex ( ) + = 0 for all real numbers x, and which has, by analogy, the following properties: e e e z z z z 1 2 1 2 = + , e e e z z z z 1 2 1 2 = − for all complex numbers z z 1 2 , . Further, in the previous section we know that by Euler’s identity, we get . , sin cos R y y i y eiy ∈ + = Consequently, combining this we adopt the following definition: Definition: Let z x iy = + be complex number. The complex exponential function ez is defined to be the complex number e e e y i y z x iy x = = + + (cos sin ). Immediately from the definition, we have the following properties: For any complex numbers z z z x iy x y R 1 2 , , , , , = + ∈ we have (i). e e e z z z z 1 2 1 2 = + , (ii). e e e z z z z 1 2 1 2 = − , (iii). , real all for 1 y eiy = (iv). , x z e e = (v). e e z z = , (vi). arg( ) , , e y k k Z z = + ∈ 2 π (vii). ez ≠ 0,
- 10. (viii). e z i k k Z z = = ∈ 1 2 if and only if ( ), , π (ix). e e z z i k k Z z z 1 2 1 2 2 = = + ∈ if and only if ( ), . π Remark In calculus, we know that the real exponential function is one-to-one. However ez is not one-to-one on the whole complex plane. In fact, by (ix) it is periodic with period i( ); 2π that is, e e z i k z + = ( ) , 2 π k Z ∈ . The periodicity of the exponential implies that this function is infinitely many to one. Trigonometric Functions From the Euler’s identity we know that e x i x ix = + cos sin , e x i x ix − = − cos sin for every real number x; and it follows from these equations that e e x ix ix + = − 2cos , e e i x ix ix − = − 2 sin . Hence it is natural to define the sine and cosine functions of a complex variable z as follows: Definition: Given any complex number z, the complex trigonometric functions sinz and cosz in terms of complex exponentials are defines to be sin , z e e i iz iz = − − 2 cos . z e e iz iz = + − 2 Let z x iy x y R = + ∈ , , . Then by simple calculations we obtain sin sin cos . ( ) ( ) z e e i x e e i x e e i x iy i x iy y y y y = − = ⋅ + + ⋅ − + − + − − 2 2 2 Hence sin sin cosh cos sinh . z x y i x y = + Similarly, cos cos cosh sin sinh . z x y i x y = − Also sin sin sinh , z x y 2 2 2 = + cos cos sinh . z x y 2 2 2 = + Therefore we obtain (i). sin , ; z z k k Z = = ∈ 0 if and only if π (ii). cos ( ) , . z z k k Z = = + ∈ 0 2 if and only if π π The other four trigonometric functions of complex argument are easily defined in terms of sine and cosine functions, by analogy with real argument functions, that is tan sin cos , z z z = sec cos , z z = 1 where z k k Z ≠ + ∈ ( ) , ; π π 2 and
- 11. cot cos sin , z z z = csc sin , z z = 1 where z k k Z ≠ ∈ π, . As in the case of the exponential function, a large number of the properties of the real trigonometric functions carry over to the complex trigonometric functions. Following is a list of such properties. For any complex numbers w z C , , ∈ we have (i). sin cos , 2 2 1 z z + = 1 2 2 + = tan sec , z z 1 2 2 + = cot csc ; z z (ii). sin( ) sin cos cos sin , w z w z w z ± = ± cos( ) cos cos sin sin , w z w z w z ± = $ tan( ) tan tan tan tan ; w z w z w z ± = ± 1$ (iii). sin( ) sin , − = − z z tan( ) tan , − = − z z csc( ) csc , − = − z z cot( ) cot , − = − z z cos( ) cos , − = z z sec( ) sec ; − = z z (iv). For any k Z ∈ , sin( ) sin , z k z + = 2 π cos( ) cos , z k z + = 2 π sec( ) sec , z k z + = 2 π csc( ) csc , z k z + = 2 π tan( ) tan , z k z + = π cot( ) cot , z k z + = π (v). sin sin , z z = cos cos , z z = tan tan , z z = sec sec , z z = csc csc , z z = cot cot ; z z = Hyperbolic Functions The complex hyperbolic functions are defined by a natural extension of their definitions in the real case. Definition: For any complex number z, we define the complex hyperbolic sine and the complex hyperbolic cosine as sinh , z e e z z = − − 2 cosh . z e e z z = + − 2 Let z x iy x y R = + ∈ , , . It is directly from the previous definition, we obtain the following identities: sinh sinh cos cosh sin , cosh cosh cos sinh sin , z x y i x y z x y i x y = + = + sinh sinh sin , cosh sinh cos . z x y z x y 2 2 2 2 2 2 = + = +
- 12. Hence we obtain (i). sinh ( ), , z z i k k Z = = ∈ 0 if and only if π (ii). cosh , . z z i k k Z = = + ∈ 0 2 if and only if π π Now, the four remaining complex hyperbolic functions are defined by the equations tanh sinh cosh , z z z = sech z z = 1 cosh , for z i k k Z = + ∈ π π 2 , ; coth coth sinh , z z z = cschz z = 1 sinh , for z i k k Z = ∈ ( ), . π Immediately from the definition, we have some of the most frequently use identities: For any complex numbers w z C , , ∈ (i). cosh sinh , 2 2 1 z z − = 1 2 2 − = tanh , z z sech coth ; 2 2 1 z z − = csch (ii). sinh( ) sinh cosh cosh sinh , w z w z w z ± = ± cosh( ) cosh cosh sinh sinh , w z w z w z ± = ± tanh( ) tanh tanh tanh tanh ; w z w z w z ± = ± ± 1 (iii). sinh( ) sinh , − = − z z tanh( ) tanh , − = − z z csch csch ( ) , − = − z z coth( ) coth , − = − z z cosh( ) cosh , − = z z sech sech ( ) ; − = z z (iv). sinh sinh , z z = cosh cosh , z z = tanh tanh , z z = sech sech z z = , csch csch z z = , coth coth ; z z = Remark (i). Complex trigonometric and hyperbolic functions are related: sin sinh , iz i z = cos cosh , iz z = tan tanh , iz i z = sinh sin , iz i z = cosh cos , iz z = tanh tan . iz i z = (i). The above discussion has emphasized the similarity between the real and their complex extensions. However, this analogy should not carried too far. For example, the real sine and cosine functions are bounded by 1, i.e., sin cos x x ≤ ≤ 1 1 and for all x R ∈ , but sin sinh cos cosh iy y iy hy = = and which become arbitrary large as y → ∞. The Logarithm
- 13. One of basic properties of the real-valued exponential function, e x R x , , ∈ which is not carried over to the complex-valued exponential function is that of being one-to- one. As a consequence of the periodicity property of complex exponential e e z C k Z z z i k = ∈ ∈ + ( ) , , , 2 π this function is, in fact, infinitely many-to-one. Obviously, we cannot define a complex logarithmic as a inverse function of complex exponential since ez is not one-to-one. What we do instead of define the complex logarithmic not as a single value ordinary function, but as a multivalued relationship that inverts the complex exponential function; i.e., w z = log if z ew = , or it will preserve the simple relation e z z log = for all nonzero z C ∈ . Definition: Let z be any nonzero complex number. The complex logarithm of a complex variable z, denoted log , z is defined to be any of the infinitely many values log log arg , z z i z = + z ≠ 0. Remark (i). We can write logz in the equivalent forms log log ( ), z z i z k = + + Arg 2 π k Z ∈ . (ii). The complex logarithm of zero will remain undefined. (iii).The logarithm of the real modulus of z is base e (natural) logarithm. (iv). logz has infinitely many values consisting of the unique real part, Re(log ) log z z = and the infinitely many imaginary parts Im(log ) arg , . z z z k k Z = = + ∈ Arg 2 π In general, the logarithm of any nonzero complex number is a multivalued relation. However we can restrict the image values so as to defined a single-value function. Definition: Given any nonzero complex number z, the principle logarithm function or the principle value of log , z denoted Logz , is defined to be Log Arg z z i z = + log , where π π < ≤ − z Arg . Clearly, by the definition of logz and Logz, they are related by logz = Logz i k k Z + ∈ ( ), . 2 π Let w and z be any two nonzero complex numbers, it is straightforward from the definition that we have the following identities of complex logarithm: (i). log( ) log log , w z w z + = + (ii). log log log , w z w z = − (iii). e z z log , = (iv). log ( ) e z i k z = + 2 π , k Z ∈ ,
- 14. (v). ( ) log log z n z n = for any integer positive n. Complex Exponents Definition: For any fixed complex number c, the complex exponent c of a nonzero complex number z is defined to be z e c c z = log for all { } z C ∈ . 0 Observe that we evaluate ec z log by using the complex exponential function, but since the logarithm of z is multivalued. For this reason, depending on the value of c, zc may has more than one numerical value. The principle value of complex exponential c, zc occurs when logz is replaced by principle logarithm function, Logz in the previous definition. That is, z e c c z = Log for all { } z C ∈ , 0 where − ≤ ≤ π π Arg z If z rei = θ with θ = Argz, then we get z e e e c c r i c r i = = + (log ) log . θ θ Inverse Trigonometric and Hyperbolic In general, complex trigonometric and hyperbolic functions are infinite many-to-one functions. Thus, we define the inverse complex trigonometric and hyperbolic as multiple-valued relation. Definition: For z C ∈ , the inverse trigonometric arctrig z or trig−1 z is defined by z w 1 trig− = if z w = trig . Here, ‘trig w’ denotes any of the complex trigonometric functions such as sin , w cos , w etc. In fact, inverses of trigonometric and hyperbolic functions can be described in terms of logarithms. For instance, to obtain the inverse sine, sin , −1 z we write w z = − sin 1 when z w = sin . That is, w z = − sin 1 when z w e e i iw w = = − − sin . 1 2 Therefore we obtain ( ) ( ) , e iz e iw iw 2 2 1 0 − − = that is quadratic in eiw . Hence we find that e iz z iw = + − ( ) , 1 2 1 2 where ( ) 1 2 1 2 − z is a double-valued of z, we arrive at the expression
- 15. ( ) ( ) sin log ( ) log . − = − + − = ± + − 1 2 1 2 2 1 2 1 z i iz z i z z π Here, we have the five remaining inverse trigonometric, as multiple-valued relations which can be expressed in terms of natural logarithms as follows: ( ) cos log , tan log , , cot log , , sec log , , csc log , . − − − − − = ± + − = − + ≠ ± = − − + ≠ ± = ± + − ≠ = ± + − ≠ 1 2 1 1 1 2 1 2 1 1 2 2 1 2 1 1 0 2 1 1 0 z i z z z i i z i z z i z i i z i z z i z i z z z z i z z z π π The principal value of complex trigonometric functions are defined by ( ) ( ) Arc Log Arc Log sin , cos , z i z z z i z z = + + − = + − π 2 1 1 2 2 Arc Log Arc Log Arc Log Arc Log tan , , cot , , sec , , csc , . z i i z i z z i z i i z i z z i z i z z z z i z z z = − + ≠ ± = − − + ≠ ± = + − ≠ = + + − ≠ 1 2 2 1 2 1 1 0 2 1 1 0 2 2 π π Definition: For any complex number z, the inverse hyperbolic, archyp z or hyp z −1 is defined by w z = − hyp 1 if z w = hyp . Here ‘hyp’ denotes any of the complex hyperbolic functions such as sinh , z cosh , z etc. These relations, which are multiple-valued, can be expressed in term of natural logarithms as follows:
- 16. ( ) ( ) ( ) sinh log , log , cosh log , tanh log , , coth log , , log , , log − − − − − − = + + − + + + = ± + − = − − + ≠ ± = − − + − ≠ ± = ± + − ≠ = + + 1 2 2 1 2 1 1 1 2 1 2 1 1 1 1 2 1 1 1 1 2 1 1 1 1 1 0 1 1 z z z z z i z z z z z z z z i z z z z z z z z z sech csch π π z z z z i z ≠ − + + + ≠ , , log , . 0 1 1 0 2 π The principle value of complex hyperbolic functions are defined by ( ) ( ) Arc Log Arc Log Arc Log Arc Log sinh , cosh , tanh , , coth , , z z z z z z z z z z z i z z z = + + = + − = − − + ≠ ±1 = − − + − ≠ ±1 2 2 1 1 1 2 1 1 1 2 1 1 π Arcsech Log Arccsch Log z z z z z z z z = + − ≠ = + + ≠ 1 1 0 1 1 0 2 2 , , , .