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Complex Analysis Lecture Notes

Dan Romik

About this document.These notes were created for use as primary reading material for the graduate courseMath 205A: Complex Analysisat UC Davis. The current 2020 revision (dated June 15, 2021) updates my earlier version of the notes from 2018. With some exceptions, the exposition follows the textbookComplex Analysisby E. M. Stein and R. Shakarchi (Princeton Uni- versity Press, 2003).

The notes are typeset in the Bera Serif font.

Acknowledgements.I am grateful to Christopher Alexander, Jennifer Brown, Brynn Caddel, Keith Conrad, Bo Long, Anthony Nguyen, Jianping Pan, and Brad Velasquez for comments that helped me improve the notes. Figure 5 on page 27 was created by Jennifer Brown and is used with her permission. An anonymous contributor added an index and suggested the Bera Serif font and a few other improvements to the document design. You too can help me continue to improve these notes by emailing me at romik@math.ucdavis.eduwith any comments or corrections you have.

Complex Analysis Lecture Notes

Document version: June 15, 2021

Copyright © 2020 by Dan Romik

Cover figure: a heat map plot of the entire functionz7!z(z1)z=2(z=2)(z). Created with Mathematica using code by Simon Woods, available at

Contents

1 Introduction: why study complex analysis? 1

2 The fundamental theorem of algebra 3

3 Analyticity 7

4 Power series 13

5 Contour integrals 16

6 Cauchy"s theorem 21

7 Consequences of Cauchy"s theorem 26

8 Zeros, poles, and the residue theorem 35

9 Meromorphic functions and the Riemann sphere 38

10The argument principle 41

11Applications of Rouché"s theorem 45

12Simply-connected regions and Cauchy"s theorem 46

13The logarithm function 50

14The Euler gamma function 52

15The Riemann zeta function 59

16The prime number theorem 71

17Introduction to asymptotic analysis 79

Problems 92

Suggested topics for course projects 119

References 121

Index122

1 1 INTRODUCTION: WHY STUDY COMPLEX ANALYSIS?

1 Introduction: why study complex analysis?

These notes are aboutcomplex analysis, the area of mathematics that studies analytic functions of a complex variable and their properties. While this may sound a bit specialized, there are (at least) two excellent reasons why all mathematicians should learn about complex analysis. First, it is, in my humble opinion, one of the most beautiful areas of mathematics. One way of putting it that has occurred to me is that complex analysis has a very high ratio of theorems to definitions (i.e., a very low "entropy"): you get a lot more as "output" than you put in as "input." The second reason is complex analysis has a large number of applications (in both the pure math and applied math senses of the word) to things that seem like they ought to have little to do with complex numbers. For example: Solving polynomial equations: historically ,this was the motivation for introducing complex numbers by Cardano, who published the famous formula for solving cubic equations in 1543, after learning of the solu- tion found earlier by Scipione del Ferro. An important point to keep in mind is that Cardano"s formula sometimes requires taking operations in the complex plane as an intermediate step to get to the final answer, even when the cubic equation being solved has only real roots. Example 1.Using Cardano"s formula, it can be found that the solutions to the cubic equation z

3+ 6z2+ 9z+ 3 = 0

are z

1= 2cos(2=9)2;

z

2= 2cos(8=9)2;

z

3= 2sin(=18)2:

Proving

Stirling"s formula

:n!p2n(n=e)n. Here,anbnis the stan- dard "asymptotic to" relation, defined to meanlimn!1an=bn= 1.

Proving the

prime number theorem :(n)nlogn, where(n)denotes the number of primes less than or equal ton(the prime-counting func- tion).

2 1 INTRODUCTION: WHY STUDY COMPLEX ANALYSIS?

Proving many other asymptotic formulas in number theory and combi- natorics, e.g. (to name one other of my favorite examples), the Hardy-

Ramanujan formula

p(n)14 p3nep2n=3; wherep(n)is the number ofinteger partitions of n. Evaluation of complicated definite integrals, for example Z 1 0 sin(t2)dt=12 r 2 (This application is strongly emphasized in older textbooks, and has been known to result in a mild case of post-traumatic stress disorder.) Solving physics problems in hydrodynamics, heat conduction, electro- statics and more. Analyzing alternating current electrical networks by extending Ohm"s law to electrical impedance . Complex analysis also has many other important applications in electrical engineering, signals processing and control theory. Probability and combinatorics, e.g., the Cardy-Smirnov formula in per- colation theory and the connective constant for self -avoidingwalks on the hexagonal lattice It was proved in 2016 that the optimal densities for sphere packing in

8 and 24 dimensions are4=384and12=12!, respectively. The proofs

make spectacular use of complex analysis (and more specifically, a part of complex analysis that studies certain special functions known as modular forms). field theory. This is not a mere mathematical convenience or sleight-of- hand, but in fact appears to be a built-in feature of the very equations describing our physical universe. Why? No one knows. Conformal maps, which come up in purely geometric applications where the algebraic or analytic structure of complex numbers seems irrele- vant, are in fact deeply tied to complex analysis. Conformal maps were used by the Dutch artist M.C. Escher (though he had no mathematical training) to create amazing art, and used by others to better under- stand, and even to improve on, Escher"s work. See Fig. 1, and see [10] for more on the connection of Escher"s work to mathematics.

3 2 THE FUNDAMENTAL THEOREM OF ALGEBRA

Figure 1:Print Gallery, a lithograph by M.C. Escher which was discovered to be based on a mathematical structure related to a complex functionz7!z for a certain complex number, although it was constructed by Escher purely using geometric intuition. See the paper [8] and this website , which has animated versions of Escher"s lithograph brought to life using the math- ematics of complex analysis. Complex dynamics, e.g., the iconic Mandelbrot set. See Fig. 2. There are many other applications and beautiful connections of complex analysis to other areas of mathematics. (If you run across some interesting ones, please let me know!) In the next section I will begin our journey into the subject by illustrating a few beautiful ideas and along the way begin to review the concepts from undergraduate complex analysis.

2 The fundamental theorem of algebra

One of the most famous theorems in complex analysis is the not-very-aptly named Fundamental Theorem of Algebra. This seems like a fitting place to start our journey into the theory. Theorem 1(The Fundamental Theorem of Algebra.).Every nonconstant polynomialp(z)over the complex numbers has a root.

4 2 THE FUNDAMENTAL THEOREM OF ALGEBRA

Figure 2: The Mandelbrot set. [Source:

W ikipedia

The fundamental theorem of algebra is a subtle result that has many beautiful proofs. I will show you three of them. Let me know if you see any "algebra"...

First proof: analytic proof.Let

p(z) =anzn+an1zn1+:::+a0 be a polynomial of degreen1, and consider wherejp(z)jattains its infi- mum.

First, note that it can"t happen asjzj ! 1, since

jp(z)j=jzjn(jan+an1z1+an2z2+:::+a0znj); and in particularlimjzj!1jp(z)jjzjn=janj, so for largejzjit is guaranteed that jp(z)j jp(0)j=ja0j. Fixing some radiusR >0for whichjzj> Rimplies jp(z)j ja0j, we therefore have that m

0:= infz2Cjp(z)j= infjzjRjp(z)j= minjzjRjp(z)j=jp(z0)j

wherez0= argmin jzjRjp(z)j, and the minimum exists becausep(z)is a continu- ous function on the discDR(0). Denotew0=p(z0), so thatm0=jw0j. We now claim thatm0= 0. As- sume by contradiction that it doesn"t, and examine the local behavior ofp(z) aroundz0; more precisely, expandingp(z)in powers ofzz0we can write p(z) =w0+nX j=1c j(zz0)j=w0+ck(zz0)k+:::+cn(zz0)n;

5 2 THE FUNDAMENTAL THEOREM OF ALGEBRA

wherekis the minimal positive index for whichcj6= 0. (Exercise: why can we expandp(z)in this way?) Now imagine starting withz=z0and traveling away fromz0in some directionei. What happens top(z)? Well, the expansion gives p(z0+rei) =w0+ckrkeik+ck+1rk+1ei(k+1)+:::+cnrnein: Whenris very small, the powerrkdominates the other termsrjwithk < j n, i.e., p(z0+rei) =w0+rk(ckeik+ck+1rei(k+1)+:::+cnrnkein) =w0+ckrkeik(1 +g(r;)); wherelimr!0jg(r;)j= 0. To reach a contradiction, it is now enough to chooseso that the vectorckrkeik"points in the opposite direction" from w

0, that is, such that

c krkeikw

02(1;0):

Obviously this is possible: take=1k

(argw0arg(ck)+). It follows that, for rsmall enough, jw0+ckrkeikja contradiction. This completes the proof.Exercise 1.Complete the last details of the proof (for whichrare the in-

equalities valid, and why?) Note that "complex analysis" is part of "analysis" - you need to develop facility with such estimates until they become second nature. Second proof: topological proof.Letw0=p(0). Ifw0= 0, we are done. Otherwise consider the image underpof the circlejzj=r. Specifically: 1. F orrvery small the image is contained in a neighborhood ofw0, so it cannot "go around" the origin. 2.

F orrvery large we have

p(rei) =anrnein

1 +an1a

nr1ei+:::+a0a nrnein =anrnein(1 +h(r;))

6 2 THE FUNDAMENTAL THEOREM OF ALGEBRA

wherelimr!1h(r;) = 0(uniformly in). Asgoes from0to2, this is a closed curve that goes around the originntimes (approximately in a circular path, that becomes closer and closer to a circle asr! 1). As we gradually increaserfrom0to a very large number, in order to tran- sition from a curve that doesn"t go around the origin to a curve that goes around the originntimes, there has to be a value ofrfor which the curve crosses0. That means the circlejzj=rcontains a point such thatp(z) = 0, which was the claim.Remark 1.The argument presented in the topological proof is imprecise. It can be made rigorous in a couple of ways - one way we will see a bit later is using Rouché"s theorem and the argument principle. This already gives a hint as to the importance of subtle topological arguments in complex analysis. Remark 2.The topological proof should be compared to the standard calcu- lus proof that any odd-degree polynomial over the reals has a real root. That argument is also "topological" (based on the mean value theorem), although much more trivial. Third proof: standard textbook proof (or: "hocus-pocus" proof).Recall: Theorem 2(Liouville"s theorem.).A bounded entire function is constant. Assuming this result, ifp(z)is a polynomial with no root, then1=p(z) is an entire function. Moreover, it is bounded, since as we noted before lim jzj!1jp(z)jjzjn=janj, solimjzj!11=p(z) = 0. It follows that1=p(z)is a constant,

which then has to be0, which is a contradiction.To summarize this section, we saw three proofs of the fundamental the-

orem of algebra. They are all beautiful - the "hocus-pocus" proof certainly packs a punch, which is why it is a favorite of complex analysis textbooks - but personally I like the first one best since it is elementary and doesn"t use Cauchy"s theorem or any of its consequences, or subtle topological con- cepts. Moreover, it is a "local" argument that is based on understanding how a polynomial behaves locally, where by contrast the other two proofs can be characterized as "global." It is a general philosophical principle in analysis (that has analogies in other areas, such as number theory and graph theory) that local arguments are easier than global ones.

7 3 ANALYTICITY

3 Analyticity, conformality and the Cauchy-Riemann

equations In this section we begin to build the theory by laying the most basic corner- stone of the theory, the definition of analyticity, along with some of the useful ways to think about this fundamental concept.

3.1 Definition and basic meanings of analyticity

Definition 1(analyticity).A functionf(z)of a complex variable isholomor- phic(a.k.a.complex-differentiable,analytic1) atzif the limit f

0(z) := limh!0f(z+h)f(z)h

exists. In this case we callf0(z)thederivative offatz. In the case whenf0(z)6= 0, the existence of the derivative has a geometric meaning: if we write the polar decompositionf0(z) =reiof the derivative, then for pointswthat are close toz, we will have the approximate equality f(w)f(z)wzf0(z) =rei; or equivalently f(w)f(z) +rei(wz) +[lower order terms]; where "lower order terms" refers to a quantity that is much smaller in mag- nitude thatjwzj. Geometrically, this means that to computef(w), we start fromf(z), and move by a vector that results by taking the displacement vec- torwz, rotating it by an angle of, and then scaling it by a factor ofr (which corresponds to a magnification ifr >1, a shrinking if0< r <1, or doing nothing ifr= 1). This idea can be summarized by the slogan: "Analytic functions behave locally as a rotation composed with a scaling."1 Note: some people use "analytic" and "holomorphic" with two a priori different defini- tions that are then proved to be equivalent; I find this needlessly confusing so I may use these two terms interchangeably.

8 3 ANALYTICITY

The local behavior of analytic functions in the casef0(z) = 0is more subtle; we will consider that a bit later. A further interpretation of the meaning of analyticity is that analytic func- tions areconformal mappingswhere their derivatives don"t vanish. More precisely, if 1,

2are two differentiable planar curves such that

1(0) =

2(0) =z,fis differentiable atzandf0(z)6= 0, then, denotingv1=

01(0),

v 2=

02(0),w1= (f

1)0(0),w2= (f

2)0(0), we can write the inner prod-

ucts (in the ordinary sense of vector geometry) between the complex number pairsv1;v2andw1;w2as hv1;v2i= Re(v1v 2); hw1;w2i=h(f0( 1(0))

01(0));(f0(

2(0))

02(0))i

=f0(z)f

0(z)hv1;v2i=jf0(z)j2hv1;v2i:

If we denote by(resp.') the angle betweenv1;v2(resp.w1;w2), it then follows that cos'=hw1;w2ijw1jjw2j=jf0(z)j2hv1;v2ijf0(z)v1jjf0(z)v2j=hv1;v2ijv1jjv2j= cos: That is, the functionfmaps two curves meeting at an angleatzto two curves that meet at the same angle atf(z). A function with this property is said to beconformalatz. Conversely, iffis conformal in a neighborhood ofzthen (under some additional mild assumptions) it is analytic - we will prove this below after discussing the Cauchy-Riemann equations. Thus the theory of analytic func- tions contains the theory of planar conformal maps as a special (and largely equivalent) case, although this is by no means obvious from the purely geo- metric definition of conformality. Let us briefly review some properties of derivatives. Lemma 1.Under appropriate assumptions, we have the relations (f+g)0(z) =f0(z) +g0(z); (fg)(z) =f0(z)g(z) +f(z)g0(z);1fquotesdbs_dbs21.pdfusesText_27

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