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Complex Analysis Lecture Notes Complex Analysis Lecture Notes Dan Romik About this document. These notes were created for use as primary reading material for the graduate course Math 205A: Complex Analysis at 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 textbook Complex Analysis by 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 [email protected] with 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 function z 7! z(z − 1)π−z=2Γ(z=2)ζ(z). Created with Mathematica using code by Simon Woods, available at http://mathematica.stackexchange.com/questions/7275/how-can-i-generate-this-domain-coloring-plot 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 Index 122 1 1 INTRODUCTION: WHY STUDY COMPLEX ANALYSIS? 1 Introduction: why study complex analysis? These notes are about complex 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 z3 + 6z2 + 9z + 3 = 0 are z1 = 2 cos(2π=9) − 2; z2 = 2 cos(8π=9) − 2; z3 = 2 sin(π=18) − 2: p n • Proving Stirling’s formula: n! ∼ 2πn(n=e) . Here, an ∼ bn is the stan- dard “asymptotic to” relation, defined to mean limn!1 an=bn = 1. n • Proving the prime number theorem: π(n) ∼ log n , where π(n) denotes the number of primes less than or equal to n (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 1 p p(n) ∼ p eπ 2n=3; 4 3n where p(n) is the number of integer partitions of n. • Evaluation of complicated definite integrals, for example Z 1 1rπ sin(t2) dt = : 0 2 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-avoiding walks on the hexagonal lattice. • It was proved in 2016 that the optimal densities for sphere packing in 8 and 24 dimensions are π4=384 and π12=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). • Nature uses complex numbers in Schrödinger’s equation and quantum 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 function z 7! zα for a certain complex number α, 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 polynomial p(z) over the complex numbers has a root. 4 2 THE FUNDAMENTAL THEOREM OF ALGEBRA Figure 2: The Mandelbrot set. [Source: Wikipedia] 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 n n−1 p(z) = anz + an−1z + ::: + a0 be a polynomial of degree n ≥ 1, and consider where jp(z)j attains its infi- mum. First, note that it can’t happen as jzj ! 1, since n −1 −2 −n jp(z)j = jzj · (jan + an−1z + an−2z + ::: + a0z j); jp(z)j and in particular limjzj!1 jzjn = janj, so for large jzj it is guaranteed that jp(z)j ≥ jp(0)j = ja0j. Fixing some radius R > 0 for which jzj > R implies jp(z)j ≥ ja0j, we therefore have that m0 := inf jp(z)j = inf jp(z)j = min jp(z)j = jp(z0)j z2C jz|≤R jz|≤R where z0 = arg min jp(z)j, and the minimum exists because p(z) is a continu- jz|≤R ous function on the disc DR(0). Denote w0 = p(z0), so that m0 = jw0j. We now claim that m0 = 0. As- sume by contradiction that it doesn’t, and examine the local behavior of p(z) around z0; more precisely, expanding p(z) in powers of z − z0 we can write n X j k n p(z) = w0 + cj(z − z0) = w0 + ck(z − z0) + ::: + cn(z − z0) ; j=1 5 2 THE FUNDAMENTAL THEOREM OF ALGEBRA where k is the minimal positive index for which cj 6= 0. (Exercise: why can we expand p(z) in this way?) Now imagine starting with z = z0 and iθ traveling away from z0 in some direction e . What happens to p(z)? Well, the expansion gives iθ k ikθ k+1 i(k+1)θ n inθ p(z0 + re ) = w0 + ckr e + ck+1r e + ::: + cnr e : When r is very small, the power rk dominates the other terms rj with k < j ≤ n, i.e., iθ k ikθ i(k+1)θ n−k inθ p(z0 + re ) = w0 + r (cke + ck+1re + ::: + cnr e ) k ikθ = w0 + ckr e (1 + g(r; θ)); where limr!0 jg(r; θ)j = 0. To reach a contradiction, it is now enough to k ikθ choose θ so that the vector ckr e “points in the opposite direction” from w0, that is, such that c rkeikθ k 2 (−∞; 0): w0 1 Obviously this is possible: take θ = k (arg w0 − arg(ck) + π).
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