14 Functions of a complex variable

TABLE OF CONTENTS

1. Introduction

1.1 Complex Numbers

1.2 An alternative route

1.3 Geometrical interpretation

1.3.1 Polar coordinates

1.4 The point at infinity

1.4.1 Stereographic projection

2. Preliminaries

2.1 Neighbourhood and limit points

2.2 Jordan curves

2.3 Bounded set

2.4 Domain

2.5 The Jordan theorem

2.6 The Bolzano-Weierstrass theorem

3.   Functions of a complex variable

3.1 Continuous functions

3.2 Existence of a derivative

3.3 Cauchy-Riemann equations

3.3.1 Sufficient conditions

3.4 Harmonic functions

LEARNING OBJECTIVES

1. Complex numbers are introduced in two different ways.

2. Geometric interpretation of the complex numbers as points in a plane is provided and representation in terms of polar coordinates is given.

3. The point at infinity is introduced and explained via stereographic projection.

4. The concepts of neighbourhood and limit points, Jordan curves, bounded sets and domains in the complex plane are introduced. Jordan theorem about closed Jordan curves is enunciated.

5. The concept of limit and continuity and derivative of a function are introduced in the context of complex variables. Cauchy-Riemann equations for the existence of a derivative are derived.

6. It is explained as to how the real and imaginary parts of a differentiable function are harmonic functions.

C o m p l e x  n u m b e r s

1. Introduction

Students of physics are familiar with the concept of complex numbers as an extension of the set of real numbers. Functions of a complex variable, a variable that can take complex values, are of great interest and applied often in the analysis of problems of physical sciences. One area of application is the theory of oscillations where representation of the trigonometric functions in terms of complex variables often simplifies the analysis of the problem considerably. Representation of voltage, current and impedance in circuit analysis by complex numbers is the most familiar example. The theory of functions of a complex variables is fully utilized in the study of integral transforms (Fourier and Laplace transforms for example) which are so important in the study of response of a system to an arbitrary response function and solution of ordinary and partial differential equations.

1.1 Complex Numbers

We can develop a well defined algebra of complex numbers if we suppose that the “number” i obeys all the usual laws of algebra. Let

1.2 An alternative route

There is an alternative, but completely equivalent way of defining complex numbers. Taking cue from rationals, which can be defined as an ordered pair of integers, we now define a complex number as an ordered pair of real numbers (a, b) and make the following definitions:

This is completely equivalent to the algebra of real numbers. Thus without any ambiguity we can identify the pair (a, 0) with the real number a. Next we notice that

1.3 Geometrical interpretation

1.3.1 Polar coordinates

then

Thus the product of two complex numbers is represented by the point whose modulus is the product of the moduli and argument is the sum of arguments of the points.

We have the Euler’sformula in complex analysis, which relates the exponential and the trigonometric functions:

Euler’s formula is ubiquitous in mathematics, physics, and engineering. Feynman called the formula a jewel and the most remarkable formula in mathematics. The formula is actually valid even if x is a complex number, and so some authors refer to the more general complex version as Euler’s formula. Many proofs of this formula are possible. One simple proof depends on the power series expansion of the sine, cosine and exponential functions, which we will deal with in later units.

Using Euler formula we can write

1.4 The point at infinity

1.4.1 Stereographic projection

In geometry, the stereographic projection is a particular mapping that projects a sphere onto a plane. The projection is defined on the entire sphere, except at one point, the projection point. Intuitively stereographic projection is a way of picturing the sphere as the plane, with some inevitable compromises. Stereographic projection appears in many areas of mathematics; it finds use in diverse fields including complex analysis, geography, geology, and photography.

To make a stereographic projection of a sphere onto a plane we take a point on the sphere as the vertex of this projection and its equatorial plane as the plane of projection. Then to any point on the sphere, save for the vertex, there corresponds a unique point in the plane and vice versa.

Let the sphere, called the Riemann sphere, be defined by

There is thus a one-to-one and continuous transformation between the complex numbers and the points on a spherical surface.

so that

2. Preliminaries

2.1 Neighbourhood and limit points

We now prepare ourselves for the study of functions of a complex variable. Here we lay the groundwork by defining concepts like neighbourhood, limit points and domain et cetera. These concepts are similar to those in the case of real variables but with obvious differences due to fact that we are now dealing with points in a plane rather than along a line. To some extent it may seem like we are dealing with a function of two real variables, but differentiability will bring out the basic difference between functions of real and complex variables.

2.2 Jordan curves

The equation

2.3 Bounded set

A set of points is said to be bounded if there exists positive real number R, such that |z| < R for every point in the set. If no such number exists the set is said to be unbounded.

2.4 Domain

A set of points in the Argand plane is said to be a connex if every pair of its points can be connected by a curve consisting of points of the set only. An open connex set of points is called an open domain. Two open domains which have no points in common are said to be separated.

2.5 The Jordan theorem

It is easy to see, for example, that the circle |z| = 1 divides the Argand plane into two separated open domains, viz, |z| < 1 and |z| > 1 with the circle as the common boundary. This result is a particular case of the Jordan curve theorem which states that a simply closed Jordan curve divides the plane into two open domains which have the curve as their common boundary. One of these two open domains is bounded and is called the interior domain and the other is unbounded and is called the exterior domain. The theorem is intuitively clear though a rigorous proof is extremely complicated.

2.6 The Bolzano-Weierstrass theorem

The fundamental property of a bounded set of points is contained in the Bolzano-Weierstrass theorem. The theorem states that if a bounded set contains infinite number of points then it contains at least one limit point.

3. Functions of a complex variable

We now consider functions of a complex variable; i.e., functions in which the domain is a subset of complex numbers. In some sense, these too are familiar to us from elementary calculus—they are simply functions from a subset of the plane into the plane:

The complex perspective generally provides richer and more profitable insight into these functions.

3.1 Continuous functions

Function of a function

We next deal with continuity of a function of a function. The result is given by the following theorem:

Theorem

Proof

3.1.1 Uniform continuity

3.2 Existence of a derivative

The derivative of a function of a complex variable behaves very much like that for plain old real-valued functions. All the “usual” results for real-valued functions also hold for these new complex functions: the derivative of a constant is zero, the derivative of the sum of two functions is the sum of the derivatives, the ”product” and ”quotient” rules for derivatives are valid, the chain rule for the composition of functions holds, etc., etc. Proofs are also similar to the ones for functions of real variables.

3.3 Cauchy-Riemann equations

These equations are called Cauchy-Riemann equations.

3.3.1 Sufficient conditions

Use Cauchy-Riemann conditions to eliminate derivatives with respect to y in favour of those with respect to x, and we obtain

3.4 Harmonic functions

Assume that the functions u and v are twice differentiable with continuous second order partial derivatives. Then the Cauchy-Riemann equations can be differentiated. Differentiate the first equation with respect to x and the second with respect to y, so that

Exactly similarly

So we see that the functions u and v are solutions of the Laplace equation in two dimensions. Functions that satisfy the Laplace equation are called harmonic functions. Thus the real and imaginary parts of an analytic function are harmonic. Not only that, for any real harmonic function u(x, y) in a simply connected domain there is a harmonic function v(x, y) called the harmonic conjugate of u such that f = u + iv is an analytic function. The harmonic conjugate is determined up to a constant.

Example

Let

which is the harmonic conjugate of u.

SUMMARY

• We begin our study of functions of a complex variable by first introducing the concept of complex numbers. This we do in two different ways.
• Next we provide a geometric interpretation of the complex numbers as points in a plane and give their representation in terms of polar coordinates.
• We introduce infinity as a single point in the plane and explain it via stereographic projection.
• Next we introduce the concepts of neighbourhood and limit points, Jordan curves, bounded sets and domains in the complex plane and enunciate the Jordan theorem about closed Jordan curves.
• We introduce limit and continuity and derivative of a function in the context of complex variables. We then derive the Cauchy-Riemann equations as necessary and sufficient conditions for the existence of a derivative.
• Finally we explained as to how the real and imaginary parts of a differentiable function are harmonic functions, that is, they are solutions of Laplace equation in two dimensions.