Complex Analysis - School of Mathematics | School of Mathematics

Chapter 2

Complex Analysis

In this part of the course we will study some basic complex analysis. This is

an extremely useful and beautiful part of mathematics and forms the basis

of many techniques employed in many branches of mathematics and physics.

We will extend the notions of derivatives and integrals, familiar from calculus,

to the case of complex functions of a complex variable. In so doing we will

come across analytic functions, which form the centerpiece of this part of the

course. In fact, to a large extent complex analysis is the study of analytic

functions. After a brief review of complex numbers as points in the complex

plane, we will first discuss analyticity and give plenty of examples of analytic

functions. We will then discuss complex integration, culminating with the

generalised Cauchy Integral Formula, and some of its applications. We then

go on to discuss the power series representations of analytic functions and

the residue calculus, which will allow us to compute many real integrals and

infinite sums very easily via complex integration.

2.1

Analytic functions

In this section we will study complex functions of a complex variable. We

will see that differentiability of such a function is a non-trivial property,

giving rise to the concept of an analytic function. We will then study many

examples of analytic functions. In fact, the construction of analytic functions

will form a basic leitmotif for this part of the course.

2.1.1

The complex plane

We already discussed complex numbers briefly in Section 1.3.5. The emphasis

in that section was on the algebraic properties of complex numbers, and

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although these properties are of course important here as well and will be

used all the time, we are now also interested in more geometric properties of

the complex numbers.

The set C of complex numbers is naturally identified with the plane R2 .

This is often called the Argand plane.

Given a complex number z = x+i y, its real and imag6

inary parts define an element (x, y) of R2 , as shown in

z = x + iy

y

the figure. In fact this identification is one of real vec7

tor spaces, in the sense that adding complex numbers

and multiplying them with real scalars mimic the simix

lar operations one can do in R2 . Indeed, if ¦Á ¡Ê R is real,

then to ¦Á z = (¦Á x) + i (¦Á y) there corresponds the pair

(¦Á x, ¦Á y) = ¦Á (x, y). Similarly, if z1 = x1 + i y1 and z2 = x2 + i y2 are complex numbers, then z1 + z2 = (x1 + x2 ) + i (y1 + y2 ), whose associated pair

is (x1 + x2 , y1 + y2 ) = (x1 , y1 ) + (x2 , y2 ). In fact, the identification is even

one of euclidean spaces. Given a complex

p number z = x + i y, its modulus

2

?

|z|, defined by |z| = zz , is given by x2 + y 2 which is precisely the norm

k(x, y)k of the pair (x, y). Similarly, if z1 = x1 + i y1 and z2 = x2 + i y2 ,

then Re(z1? z2 ) = x1 x2 + y1 y2 which is the dot product of the pairs (x1 , y1 )

and (x2 , y2 ). In particular, it follows from these remarks and the triangle

inequality for the norm in R2 , that complex numbers obey a version of the

triangle inequality:

|z1 + z2 | ¡Ü |z1 | + |z2 | .

(2.1)

Polar form and the argument function

Points in the plane can also be represented using polar coordinates, and

this representation in turn translates into a representation of the complex

numbers.

z = rei¦È p Let (x, y) be a point in the plane. If we define r =

x2 + y 2 and ¦È by ¦È = arctan(y/x), then we can write

r 7

¦È

(x, y) = (r cos ¦È, r sin ¦È) = r (cos ¦È, sin ¦È). The complex

number z = x + i y can then be written as z = r (cos ¦È +

i sin ¦È). The real number r, as we have seen, is the modulus

|z| of z, and the complex number cos ¦È + i sin ¦È has unit

modulus. Comparing the Taylor series for the cosine and

sine functions and the exponential functions we notice that cos ¦È+i sin ¦È = ei¦È .

The angle ¦È is called the argument of z and is written arg(z). Therefore we

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have the following polar form for a complex number z:

z = |z| ei arg(z) .

(2.2)

Being an angle, the argument of a complex number is only defined up to the

addition of integer multiples of 2¦Ð. In other words, it is a multiple-valued

function. This ambiguity can be resolved by defining the principal value

Arg of the arg function to take values in the interval (?¦Ð, ¦Ð]; that is, for any

complex number z, one has

?¦Ð < Arg(z) ¡Ü ¦Ð .

(2.3)

Notice, however, that Arg is not a continuous function: it has a discontinuity

along the negative real axis. Approaching a point on the negative real axis

from the upper half-plane, the principal value of its argument approaches ¦Ð,

whereas if we approach it from the lower half-plane, the principal value of

its argument approaches ?¦Ð. Notice finally that whereas the modulus is a

multiplicative function: |zw| = |z||w|, the argument is additive: arg(z1 z2 ) =

arg(z1 ) + arg(z2 ), provided that we understand the equation to hold up to

integer multiples of 2¦Ð. Also notice that whereas the modulus is invariant

under conjugation |z ? | = |z|, the argument changes sign arg(z ? ) = ? arg(z),

again up to integer multiples of 2¦Ð.

Some important subsets of the complex plane

We end this section with a brief discussion of some very important subsets

of the complex plane. Let z0 be any complex number, and consider all those

complex numbers z which are a distance at most ¦Å away from z0 . These

points form a disk of radius ¦Å centred at z0 . More precisely, let us define the

open ¦Å-disk around z0 to be the subset D¦Å (z0 ) of the complex plane defined

by

D¦Å (z0 ) = {z ¡Ê C | |z ? z0 | < ¦Å} .

(2.4)

Similarly one defines the closed ¦Å-disk around z0 to be the subset

D?¦Å (z0 ) = {z ¡Ê C | |z ? z0 | ¡Ü ¦Å} ,

(2.5)

which consists of the open ¦Å-disk and the circle |z ? z0 | = ¦Å which forms its

boundary. More generally a subset U ? C of the complex plane is said to be

open if given any z ¡Ê U , there exists some positive real number ¦Å > 0 (which

can depend on z) such that the open ¦Å-disk around z also belongs to U . A set

C is said to be closed if its complement C c = {z ¡Ê C | z 6¡Ê C}¡ªthat is, all

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those points not in C¡ªis open. One should keep in mind that generic subsets

of the complex plane are neither closed nor open. By a neighbourhood of a

point z0 in the complex plane, we will mean any open set containing z0 . For

example, any open ¦Å-disk around z0 is a neighbourhood of z0 .



Let us see that the open and closed ¦Å-disks are indeed open and closed, respectively. Let

z ¡Ê D¦Å (z0 ). This means that |z ? z0 | = ¦Ä < ¦Å. Consider the disk D¦Å?¦Ä (z). We claim that

this disk is contained in D¦Å (z0 ). Indeed, if |w ? z| < ¦Å ? ¦Ä then,

|w ? z0 | = |(w ? z) + (z ? z0 )|

¡Ü |w ? z| + |z ? z0 |

(adding and subtracting z)

(by the triangle inequality (2.1))

¦Å. We will show that it

is an open set. Let z be such that |z ? z0 | = ¦Ç > ¦Å. Then consider the open disk D¦Ç?¦Å (z),

and let w be a point in it. Then

|z ? z0 | = |(z ? w) + (w ? z0 )|

¡Ü |z ? w| + |w ? z0 | .

(adding and subtracting w)

(by the triangle inequality (2.1))

We can rewrite this as

|w ? z0 | ¡Ý |z ? z0 | ? |z ? w|

> ¦Ç ? (¦Ç ? ¦Å)

(since |z ? w| = |w ? z| < ¦Ç ? ¦Å)

=¦Å.

Therefore the complement of D?¦Å (z0 ) is open, whence D?¦Å (z0 ) is closed.

We should remark that the closed disk D?¦Å (z0 ) is not open, since any open disk around a

point z at the boundary of D?¦Å (z0 )¡ªthat is, for which |z ? z0 | = ¦Å¡ªcontains points which

are not included in D¦Å (z0 ).

Notice that it follows from this definition that every open set is made out of the union of

(a possibly uncountable number of) open disks.

2.1.2

Complex-valued functions

In this section we will discuss complex-valued functions.

We start with a rather trivial case of a complex-valued function. Suppose

that f is a complex-valued function of a real variable. That means that if x is

a real number, f (x) is a complex number, which can be decomposed into its

real and imaginary parts: f (x) = u(x) + i v(x), where u and v are real-valued

functions of a real variable; that is, the objects you are familiar with from

calculus. We say that f is continuous at x0 if u and v are continuous at x0 .



Let us recall the definition of continuity. Let f be a real-valued function of a real variable.

We say that f is continuous at x0 , if for every ¦Å > 0, there is a ¦Ä > 0 such that |f (x) ?

f (x0 )| < ¦Å whenever |x ? x0 | < ¦Ä. A function is said to be continuous if it is continuous

at all points where it is defined.

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Now consider a complex-valued function f of a complex variable z. We

say that f is continuous at z0 if given any ¦Å > 0, there exists a ¦Ä > 0 such

that |f (z) ? f (z0 )| < ¦Å whenever |z ? z0 | < ¦Ä. Heuristically, another way of

saying that f is continuous at z0 is that f (z) tends to f (z0 ) as z approaches

z0 . This is equivalent to the continuity of the real and imaginary parts of f

thought of as real-valued functions on the complex plane. Explicitly, if we

write f = u + i v and z = x + i y, u(x, y) and v(x, y) are real-valued functions

on the complex plane. Then the continuity of f at z0 = x0 + i y0 is equivalent

to the continuity of u and v at the point (x0 , y0 ).

¡°Graphing¡± complex-valued functions

Complex-valued functions of a complex variable are harder to visualise than

their real analogues. To visualise a real function f : R ¡ú R, one simply

graphs the function: its graph being the curve y = f (x) in the (x, y)-plane.

A complex-valued function of a complex variable f : C ¡ú C maps complex

numbers to complex numbers, or equivalently points in the (x, y)-plane to

points in the (u, v) plane. Hence its graph defines a surface u = u(x, y) and

v = v(x, y) in the four-dimensional space with coordinates (x, y, u, v), which

is not so easy to visualise. Instead one resorts to investigating what the

function does to regions in the complex plane. Traditionally one considers

two planes: the z-plane whose points have coordinates (x, y) corresponding

to the real and imaginary parts of z = x + i y, and the w-plane whose points

have coordinates (u, v) corresponding to w = u + i v. Any complex-valued

function f of the complex variable z maps points in the z-plane to points

in the w-plane via w = f (z). A lot can be learned from a complex function

by analysing the image in the w-plane of certain sets in the z-plane. We

will have plenty of opportunities to use this throughout the course of these

lectures.



With the picture of the z- and w-planes in mind, one can restate the continuity of a

function very simply in terms of open sets. In fact, this was the historical reason why the

notion of open sets was introduced in mathematics. As we saw, a complex-valued function

f of a complex variable z defines a mapping from the complex z-plane to the complex

w-plane. The function f is continuous at z0 if for every neighbourhood U of w0 = f (z0 )

in the w-plane, the set

f ?1 (U ) = {z | f (z) ¡Ê U }

is open in the z-plane. Checking that both definitions of continuity agree is left as an

exercise.

2.1.3

Differentiability and analyticity

Let us now discuss differentiation of complex-valued functions. Again, if f =

u + i v is a complex-valued function of a real variable x, then the derivative

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