Complex Functions

A function / defined on an open domain V in the complex plane is said to be analytic, if for any ZQ e T> we can write / ( z ) as the sum of a convergent power series in a neighborhood of ZQ; that is,

f(^) = X^Cn(z-2:o)'',

n=0

where the coefficients c^ (n = 0,1,2,...) are complex numbers.

A point such as ZQ in the neighborhood of which a function / can be written as the sum of a convergent power series is said to be regular. Any point that is not regular is singular.

A power series converges in an open disk, that is, a domain whose bound-ary is a circle of radius r centered at ZQ. The largest value of r is the radius of convergence of the power series. It is denoted by i?, hence, a power se-ries is convergent if its radius of convergence is positive. For example, the power series representing the exponential e^ in a neighborhood of the origin, is X ] ^ o ^^ 1'^^" Its radius of convergence is infinite. In other words, the expo-nential function is analytic in the whole complex plane. This is, in general, not the case. Given a complex function / analytic in a neighborhood of a point zo the power series X l ^ o ^^('^ "" -^o)^ ^ ^ ^ finite radius of convergence whose value is limited by the existence of the singularities of the function

/ . The value of the radius of convergence can be determined by the Cauchy rule which says that the radius of convergence of a power series of the form X^^o ^rii^ ~ ^o)^ is given by the relation:

- =limitsup^_^|cn|^/''.

Consider for example, the function 1/(1 — 2:). Its power series expansion in a neighborhood of the origin is given by Yl^=o ^^ ~ I -\- z -\- z^ -\- -- -. This function has only one singular point at finite distance, namely 2: = 1, and we can easily verify that its radius of convergence is equal to 1, precisely the distance from the origin to this singular point. In fact,

limitsup^_^l^/^ = lim 1^^ = 1.

n—>cxD

Analytic functions are differentiable; that is, if f{z) = Yl^=o ^n{^ — ZQY is a power series converging in an open disk Z)(0, R) of radius i? > 0 centered at the origin, then its derivative defined in the complex plane by

•^ ^ ^

c

exists for all z in J9(0, R). If a complex function / is difl'erentiable at a point zo, it is infinitely differentiable at that point. Differentiable complex functions are also called holomorphic. In the case of functions of a complex variable, the adjectives analytic and holomorphic are synonymous.

The singular point of the function 1/(1 — z) is isolated. That is, there exists, centered at that point, an open disk with a nonzero radius in which 2: = 1 is the only singular point. If, starting from any point ZQ^ where the function is equal to 1/(1 — 2:0)^ we follow a circular path around the point z = 1, when we come back to the point ZQ^ the function is again equal to 1/(1 — ZQ) • We can therefore say that the function is well defined in the whole complex plane except ai z = 1. Again, this property is not shared by all complex functions.

Consider the function y/z. The origin is a singular point because y/z cannot be represented by the sum of a convergent power series in any neighborhood of the origin. This singular point is, however, not isolated. The squares of the two complex numbers zi = pe^^/^ and 22 = p^^^^l'^^'^) are equal, therefore the function ^/z cannot be defined without precaution. If, for instance, we choose y/z to be the function equal to 1 when z = 1, then if we write z = pe^*^, and consider that the value z = 1 corresponds to the choice p = 1 and 0 = 0, following a circular path of radius 1 centered at the origin, the argument 6 increases continuously from 0 to 27r and, consequently y/z = y/pe^^^^ is found to be equal to —1. Such a behavior is not acceptable for a function. Thus, to correctly define the function y/z we should not be able to follow a continuous path around the singular point z = 0. The domain in which the function y/z is well-defined could be, for instance, the set

132 4 Analysis

{z\ z = pe'^, p > 0, -TT < ^ < TT}

that is, the complement in the complex plane of the negative real semi-axis.

Along this so-called branch cut the function y/z is discontinuous. All the points of the branch cut are singular. For the function y/z the origin is not an isolated singular point. Such a singular point is called a branch point The branch cut could have been chosen in many different ways. For instance, any semi-axis whose equation is pe^^ where a is a given argument between 0 and 27r, and p is a parameter varying from 0 to oo, is an acceptable branch cut. Our choice, which corresponds to a = TT is the traditional one, also adopted by Mathematica.

The tridimensional plots of the real and imaginary parts of y/z clearly show the discontinuity along the negative real semi-axis.

PlotSD[Re[Sqrt [x + I y ] ] , {x, - 3 , 3 } , {y, - 3 , 3 } ] ;

Fig. 4.8. Plot of the real part of y/x -h iy in the domain {x, y} G [—3,3] x [—3,3].

Plot3D[Im[Sqrt[x + I y]] , {x, - 3 , 3 } , {y, - 3 , 3 } ] ;

Fig. 4.9. Plot of the imaginary part of y/x + iy in the domain {x, y} G [—3,3] x [-3,3].

More generally, if a function g defined in a domain T> of the complex plane is not injective, the equation g{z) = u may have more than one solution and g has, in this case, more than one inverse function: / i , /2, / s , • • • such that, for k = 1 , 2 , 3 , . . . , g{fk{u)) = u, and these functions cannot be continuous in g{V). This is exactly what happened for the noninjective function g{z) = z^, defined in the whole complex plane and having two noncontinuous inverse functions in the whole complex plane.

Here is another interesting result concerning the representation of functions in the vicinity of an isolated singularity. Let S be the open annulus {z \ ri <

k "" -^ol, ^2}; if / is analytic in S then, for all z G S, we have

0 0

n = —00

where the power series J2^=o ^n{z — ZQ)'^ is convergent for \z — zo\ < r2 and the series X l ^ i ^-n{z — ZQ)"^ is convergent for Iz — 2:o| > ri. This particular expansion is called the Laurent series of / in S.

If the Laurent series is defined in {z \ 0 < \z — zo\ < r}, called the punctured neighborhood of the singular point ZQ , the point ZQ is a pole if the series with negative exponents has a finite number of terms, and the greatest value of n is the order of the pole; if the series with negative exponents has an infinite number of terms, the point ZQ is an essential singularity.

If zo is an isolated singularity of f{z), the coefficient c_i of the Laurent series is called the residue of / at z = ZQ. The Mathematica command Residue [f [ z ] , {z, zO}] finds the residue of / at the point zO.

134 4 Analysis

Residue [Exp [z] / Sin[z]'^2, {z, Pi}]

EPi

The residue theorem allows, in particular, the evaluation of contour integrals.

It states:

Let V he an open simply connected domain in the complex plane and {2:1,..., Zn} a set of n different isolated points. For any analytic function f in X>\{2:i,..., Zn}, we have

j f{z) dz = 2m Y^ Ind(7, Zk)Residue(f, Zk),

•^'^ k=i

where ^ is a closed path in V\{zi^Z2->...,

z^i}-The symbol Ind(7, z^) is the index of the point path Zk with respect to the path 7. It is defined by

This formula is a consequence of the relation dz

J ^

2i7r, where 7 is the circular path 11-^ e^^^* with t G [0,1].

If we integrate along the circular path 7 , the result above is obvious because replacing z by e^^^* yields 2i7r /^ dt = 2m. But, as a consequence of Cauchy's theorem, we can change the circle into the square with vertices l,i, — 1,—i without changing the result. Using Mathematica we can evaluate the contour integral directly:

I n t e g r a t e d / z , {z, 1, I , - 1, - I , 1}]

(2 I ) Pi

For instance, the real integral / ^ f{x) dx of the function / whose only singu-larities, as a function of the complex variable z are poles, none of them being real, can be easily evaluated using the residue theorem. Consider the contour integral/ f{z) dz where 7 is the closed path equal to the union of the paths 7i : 11-^ R{2t — 1) and 72 :11-^ Re^'^^ for t varying from 0 to 1.

This integral is equal to

/ f{x)dx+ I f{z)dz= —-Y\Residue(/,Zk), J-R Ji2 2i7r ^

where the sum is over all the residues of the poles of / in the upper half plane such that \zk\ < R. When i? ^ CXD, the first integral becomes the real integral we have to evaluate and the second one tends to zero if lima^^oo ^ / ( ^ ) = 0-When this condition is satisfied, the real integral / ^ f{x) dx is simply equal to the sum of the residues of / at all poles of the upper half-plane. For instance,

f

because the index of i with respect to the path 7 is 1. The residue is given by Residue[1 / (1 + x ^ 2 ) , {x, I}]

I 2

We thus obtain the classical result: / ^ dx/{l -f- x^) = TT.

We can also evaluate very simply integrals of the form JQ ^ /(cos(x), sin(a;)) dx, where / is a rational fraction in cos(a:) and sin(x). If we put z == e^^, the in-tegral becomes 1/iz/ / ( ( z -h z~^)/2, {z — z~^)/2\) dz, where 7 is the circular path of radius 1 0 y-^ &^ for 6 varying from 0 to 2n. Therefore,

/ /(cos(a:0, szn[x)) dx = ZTT > Residue I - / I , — — — j ,^k j , where the sum is over the residues of all the poles inside the unit circle. For instance, if a > |6|,

/•^^ de _ /•

^0 a - h 6 s i n ^ J 2dz bz'^ -f 2aiz — b

The complex function has only one pole inside the unit circle, namely i(—a + y/s? — b2)/b. Because

Residue [2 / (b z'^2 + 2 a I z - b) , {z, I (- a + SqrtCa'^2 - b'^2]) / b}]

S q r t [ a 2 - b 2 ]

136 4 Analysis we find

I

For a detailed study with historical references to the properties of functions of a complex variable see [7].