Chapter 2 Complex Analysis

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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 ¯rst 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 in¯nite 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 di®erentiability 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 basicleitmotiffor this part of the course.

2.1.1 The complex plane

We already discussed complex numbers brie°y in Section 1.3.5 . The emphasis in that section was on the algebraic properties of complex numbers, and 73
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?of complex numbers is naturally identi¯ed with the plane?2.

This is often called theArgand plane.

Given a complex numberz=x+iy, its real and imag-

6 z=x+iyy x 7 inary parts de¯ne an element (x;y) of?2, as shown in the ¯gure. In fact this identi¯cation is one of real vec- tor spaces, in the sense that adding complex numbers and multiplying them with real scalars mimic the simi- lar operations one can do in?2. Indeed, if®2?is real, then to®z= (®x) +i(®y) there corresponds the pair (®x;®y) =®(x;y). Similarly, ifz1=x1+iy1andz2=x2+iy2are com- plex numbers, thenz1+z2= (x1+x2) +i(y1+y2), whose associated pair is (x1+x2;y1+y2) = (x1;y1) + (x2;y2). In fact, the identi¯cation is even one of euclidean spaces. Given a complex numberz=x+iy, its modulus x

2+y2which is precisely the norm

k(x;y)kof the pair (x;y). Similarly, ifz1=x1+iy1andz2=x2+iy2, and (x2;y2). In particular, it follows from these remarks and the triangle inequality for the norm in?2, that complex numbers obey a version of the triangle inequality: jz1+z2j · jz1j+jz2j: (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. Let (x;y) be a point in the plane. If we de¯ner= r 7 z=reiµp x

2+y2andµbyµ= arctan(y=x), then we can write

(x;y) = (rcosµ;rsinµ) =r(cosµ;sinµ). The complex numberz=x+iycan then be written asz=r(cosµ+ isinµ). The real numberr, as we have seen, is the modulus jzjofz, and the complex number cosµ+isinµhas unit modulus. Comparing the Taylor series for the cosine and sine functions and the exponential functions we notice that cosµ+isinµ=eiµ. The angleµis called theargumentofzand is written arg(z). Therefore we 74
have the followingpolar formfor a complex numberz: z=jzjeiarg(z): (2.2) Being an angle, the argument of a complex number is only de¯ned up to the

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 ¯rst 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 in¯nite 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 di®erentiability 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 basicleitmotiffor this part of the course.

2.1.1 The complex plane

We already discussed complex numbers brie°y in Section 1.3.5 . The emphasis in that section was on the algebraic properties of complex numbers, and 73
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?of complex numbers is naturally identi¯ed with the plane?2.

This is often called theArgand plane.

Given a complex numberz=x+iy, its real and imag-

6 z=x+iyy x 7 inary parts de¯ne an element (x;y) of?2, as shown in the ¯gure. In fact this identi¯cation is one of real vec- tor spaces, in the sense that adding complex numbers and multiplying them with real scalars mimic the simi- lar operations one can do in?2. Indeed, if®2?is real, then to®z= (®x) +i(®y) there corresponds the pair (®x;®y) =®(x;y). Similarly, ifz1=x1+iy1andz2=x2+iy2are com- plex numbers, thenz1+z2= (x1+x2) +i(y1+y2), whose associated pair is (x1+x2;y1+y2) = (x1;y1) + (x2;y2). In fact, the identi¯cation is even one of euclidean spaces. Given a complex numberz=x+iy, its modulus x

2+y2which is precisely the norm

k(x;y)kof the pair (x;y). Similarly, ifz1=x1+iy1andz2=x2+iy2, and (x2;y2). In particular, it follows from these remarks and the triangle inequality for the norm in?2, that complex numbers obey a version of the triangle inequality: jz1+z2j · jz1j+jz2j: (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. Let (x;y) be a point in the plane. If we de¯ner= r 7 z=reiµp x

2+y2andµbyµ= arctan(y=x), then we can write

(x;y) = (rcosµ;rsinµ) =r(cosµ;sinµ). The complex numberz=x+iycan then be written asz=r(cosµ+ isinµ). The real numberr, as we have seen, is the modulus jzjofz, and the complex number cosµ+isinµhas unit modulus. Comparing the Taylor series for the cosine and sine functions and the exponential functions we notice that cosµ+isinµ=eiµ. The angleµis called theargumentofzand is written arg(z). Therefore we 74
have the followingpolar formfor a complex numberz: z=jzjeiarg(z): (2.2) Being an angle, the argument of a complex number is only de¯ned up to the