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useful because these functions are the eigenfunctions of the translation operator. However,

this also points to a limitation of the Fourier transform. Consider a linear lter that is

not time-invariant, that is a lter where the output does not depend only on the di er-

ence between the input time and the output time t. Such a lter satis es the general

equation (11.55) rather than the convolution integral (11.57). The action of a lter that

is not time-invariant can in general not be written as a combination of the following op-

erations: multiplication, translation and integration. This means that for such a lter the

functions exp (;i!t) that form the basis of the Fourier transform are not the appropriate

eigenfunctions. The upshot of this is that in practice the Fourier transform is only useful

for systems that are time-dependent, or in general that are translationally invariant in the

coordinate that is used.

CHAPTER 11. FOURIER ANALYSIS

152

Chapter 12

Analytic functions

In this section we will consider complex functions in the complex plane. The reason

for doing this is that the requirement that the function \behaves well" (this is de ned

later) imposes remarkable constraints on such complex functions. Since these constraints

coincide with some of the laws of physics, the theory of complex functions has a number of

important applications in mathematical physics. In this chapter complex functions h(z)

are treated that are decomposed in a real and imaginary parts:

h(z) = f(z) + ig(z) (12.1)

hence the functions f(z) and g(z) are assumed to be real. The complex number z will

frequently be written as z = x + iy, so that x = and y = where and denote

<(z) =(z) < =

the real and imaginary part respectively.

12.1 The theorem of Cauchy-Riemann

Let us rst consider a real function F(x) of a real variable x. The derivative of such a

function is de ned by the rule

dF = lim F(x + x) F(x) : (12.2)

;

dx x

x!0

In general there are two ways in which x can approach zero from above and from below.

For a function that is di erentiable it does not matter whether x approaches zero from

above or from below. If the limits x 0 and x 0 do give a di erent result it is

# "

a sign that the function does not behave well, it has a kink and the derivative is not

unambiguously de nes, see gure (12.1).

For complex functions the derivative is de ned in the same way as in equation (12.1)

for real functions:

dh = lim h(z + z) h(z) : (12.3)

;

dz z

z!0

For real functions, x could approach zero in two ways, from below and from above.

However, the limit z 0 in (12.3) can be taken in in nitely many ways. As an example

!

see gure (12.2) where several paths are sketched that one can use to let z approach

zero. This does not always give the same result.

153

CHAPTER 12. ANALYTIC FUNCTIONS

154

F(x)

âˆ†x < 0 âˆ†x> 0

x

Figure 12.1: A function F(x) that is not di erentiable.

Figure 12.2: Examples of paths along which the limit can be taken.

12.1. THE THEOREM OF CAUCHY-RIEMANN 155

Problem a: Consider the function h(z) = exp(1=z). Using the de nition (12.3) compute

dh=dz at the point z = 0 when z approaches zero (i) from the positive real axis,

(ii) from the negative real axis, (iii) from the positive imaginary axis and (iv) from

the negative imaginary axis.

You have discovered that for some functions the result of the limit z depends critically

on the path that one uses in the limit process. The derivative of such a function is not

de ned unambiguously. However, for many functions the value of the derivative does not

depend on the way that z approaches zero. When these functions and their derivative

are also nite, they are called analytic functions. The requirement that the derivative

does not depend on the way in which z approaches zero imposes a strong constraint on

the real and imaginary part of the complex function. To see this we will let z approach

zero along the real axis and along the imaginary axis.

Problem b: Consider a complex function of the form (12.1) and compute the derivative

dh=dz by setting z = x with x a real number. (Hence z approaches zero

along the real axis). Show that the derivative is given by dh=dz = @f=@x + i@g=@x.

Problem c: Compute the derivative dh=dz also by setting z = i y with y a real

number. (Hence z approaches zero along the imaginary axis.) Show that the

derivative is given by dh=dz = @g=@y i@f=@y.

;

Problem d: When h(z) is analytic these two expressions for the derivative are by de -

nition equal. Show that this implies that:

@f = @g (12.4)

@x @y

@g = @f : (12.5)

@x @y

;

These are puzzling expressions since the conditions (12.4) and (12.5) imply that the real

and imaginary part of an analytic complex functions are not independent of each other,

they are coupled by the constraints imposed by the equations above. The expressions

(12.4) and (12.5) are called the Cauchy-Riemann relations.

Problem e: Use these relations to show that both f(x y) and g(x y) are harmonic func-

tions. These are functions for which the Laplacian vanishes:

2 f = r2 g = 0 : (12.6)

r

Hence we have found not only that f and g are coupled to each other in addition the

functions f and g must be harmonic functions. This is exactly the reason why this theory is

so useful in mathematical physics because harmonic functions arise in several applications,

see the examples of the coming sections. However, we have not found all the properties of

harmonic functions yet.

Problem f: Show that:

=0:

(rf (12.7)

rg)

CHAPTER 12. ANALYTIC FUNCTIONS

156

Since the gradient of a function is perpendicular to the lines where the function is

constant this implies that the curves where f is constant and where g is constant intersect

each other at a xed angle.

Problem g: Determine this angle.

Problem h: Verify the properties (12.4) through (12.7) explicitly for the function h(z) =

z 2 . Also sketch the lines in the complex plane where f = and g = are

<(h) =(h)

constant.

Still we haveHnot fully explored all the properties of analytic functions. Let us consider a

line integral C h(z)dz along a closed contour C in the complex plane.

Problem i: Use the property dz = dx + idy to deduce that:

I I I

v dr + i C w dr

h(z)dz = (12.8)

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