15 Analytic functions

TABLE OF CONTENTS

1.  Polynomials and rational functions

2.  Power series

2.1 Cauchy’s nth root test

2.2 Power series and analytic functions

3. Common functions

3.1 The exponential function

3.2 The trigonometric functions

3.2.1 The hyperbolic functions

3.2.2 Zeros of sin z and cos z

3.3 Periodicity of exp z

3.4 The logarithmic function

3.4.1 Power series for log(1+z)

3.5 The function

LEARNING OBJECTIVES

A n a l y t i c  f u n c t i o n s

1. Polynomials and rational functions

We now wish to study some special functions with which we are familiar from study of functions of a real variable. We would like to extend their definition to the entire complex plane and study their analyticity properties.

The power function zⁿ, where n is a positive integer, is one of the simplest analytic function. It is an entire function. That means it has no singularity in the finite complex plane. The derivative of the function is

Since positive integer powers are entire functions, it follows that a polynomial of order n is also an entire function. That is, the function

A function of the form

2. Power series

An infinite series whose terms consist of positive integer powers of z (in ascending order) is called a power series:

2.1 Cauchy’s n^th root test

Cauchy’s root test is a very useful test for the convergence of an infinite series. Consider the series

Let

where “lim sup” stands for the superior or upper limit.  The root test states that

if M < 1, then the series converges absolutely,

if M > 1, then the series diverges,

if M = 1, and the limit approaches strictly from above, then the series diverges.

In the cases excluded, the test is inconclusive – the series may diverge, converge absolutely or may converge conditionally.

There are three cases to be considered: (i) R = 0, (ii) R finite and (iii) R infinite. The first case, R = 0, is a trivial case – the series converges only for z = 0. In the third case, R infinite, the series converges for all z.

2.2 Power series and analytic functions

We have the theorem that if a power series has a nonzero radius of convergence then its sum is an analytic function regular within its circle of convergence.

Proof

The radius of convergence of each of these series is R.

3. Common functions

3.1 The exponential function

We will now introduce some of the elementary functions with which we are already familiar. However, now we will introduce them as functions of a complex variable via the power series and study their properties; some of these properties may be different and others extensions of those of functions of a real variable. We begin with the well known exponential function.

The exponential function exp(z) is defined by the power series

Hence the exponential function is an entire function, that is, it has no singularities in the entire finite complex plane. When z is a real number x, this function is the same as the function of the elementary algebra. The exponential function exp is also often written as e^z. This is because like the function of real variable, the exponential function of the complex variable also satisfies the relation

Proof

3.2 The trigonometric functions

The elementary definition of trigonometric functions is via geometry where they are defined as ratios of various sides of right-angled triangles. It can then be shown that if x is the measure of an angle in radians, then

for all values of the real variable x. We now define the trigonometric functions “sine” and “cosine” of a complex variable z by

From the ratio test it follows immediately that both these series have an infinite radius of convergence. Hence these represent functions that are analytic and have no singularities in the entire finite complex plane. We can obtain the derivatives of these functions by term by term differentiation of the terms of the series; the result is

The other usual trigonometric functions can be defined by

From these definitions we can find the derivatives of these functions:

The functions tan z and sec z are analytic in any finite domain in which cos z does not vanish; cot z and cosec z in any finite domain in which sin z does not vanish.

We can immediately realize, on comparing series for the exponential and the trigonometric functions, that

On using the addition formula for the exponential function we can easily prove the following well known relations

All these, in fact most, relations that are true for real x are also true for complex z.

3.2.1 The hyperbolic functions

For real x the hyperbolic functions are defined by the relations

We define the hyperbolic functions for all real and complex variables by the same relations:

The other hyperbolic functions are also defined in analogy with the case of real variables:

These are all analytic functions. Whereas sinh z and cosh z are entire functions having no singularities in any finite domain, tanh z and sech z have singularities at the points where cosh z vanishes and cot z and cosech z have singularities where sinh z vanishes.

The following relations follow directly from the definitions of the trigonometric and hyperbolic functions:

Through these equations we can relate the properties of the hyperbolic functions to those of the trigonometric functions.

3.2.2. Zeros of sin z and cos z

We will now use the above relations (16) and (17) and the addition formulae for sines and cosines, equations (12) and (13), to find the zeros of the sine and cosine functions for complex z.

Let z = x + iy. Then

The arguments of all the functions are now real variables; thus we have separated the function sin z into its real and imaginary parts. Therefore, it follows that sin z will be zero if, and only if,

3.3 Periodicity of exp z

When written in the exponent form this gives

Equating real and imaginary parts of this equation, we have

On squaring an adding these two equations we have

Thus we have proved that the exponential is a periodic function with a fundamental period 2πi.

3.4 The logarithmic function

Since the argument of a complex number has infinitely many values, differing from each other by multiples of 2π, a complex number has infinite number of logarithms. We write all these together as

For each z, Log z is an infinite valued function. Each value of Log z obtained by choosing a specific value of the argument is called a branch of the logarithm. The most important branch is the one corresponding to the principal value of the argument and is called the principal value of the logarithm of z. This principal value is denoted by log z. Thus

3.4.1 Power series for log(1+z)

provided the series for f(z) converges.  Now consider the function

This function is analytic and regular for |z| < 1 and has zero derivative. Hence it is a constant, independent of z. By putting z = 0 in the above equation, this constant is found to be zero. Hence

The series converges for |z| < 1.

3.5 The function

If m is an integer then the function

Though Log z is an infinite valued function, exp(m Log z) is single valued.  If n is any other integer, the function

The other (n – 1) solutions are then

Each of these solutions can be reached from others by going around the branch point at z = 0.

The law of indices for real variables is valid for this definition as well:

As we have already seen, log z is discontinuous along the negative real axis. Its value changes by 2πi in going from y⁺ to y^–. Hence

SUMMARY

• We now study some elementary and well known functions as functions of a complex variable. We begin with polynomials and rationals.
• Next we state the Cauchy’s nth root test for convergence of an infinite power series and prove the analyticity of an infinite power series.
• Next we study the analyticity of common functions like the exponential function, the trigonometric functions and the hyperbolic functions.
• Then we obtain the zeros of sin z and cos z and periodicity of the exponential function as functions of complex z.
• Next we study the logarithmic function, introduce branch point and branch cut and obtain the power series for log(1+z).
• Finally we study the functionfor complex z and α.