Introduction to Complex Analysis Michael Taylor

1 Introduction to Complex AnalysisMichael Taylor12 ContentsChapter 1. Basic calculus in the Complex domain0. Complex numbers, power series, and exponentials1. Holomorphic functions, derivatives, and path integrals2. Holomorphic functions de ned by power series3. Exponential and trigonometric functions: Euler's formula4. Square roots, logs, and other inverse functionsI. 2is irrationalChapter 2. Going deeper { the Cauchy integral theorem and consequences5. The Cauchy integral theorem and the Cauchy integral formula6. The maximum principle, Liouville' s theorem , and the fundamental theorem of al-gebra7. Harmonic functions on planar regions8. Morera' s theorem , the Schwarz re ection principle, and Goursat's theorem9. In nite products10. Uniqueness and analytic continuation11. Singularities12. Laurent seriesC.}

2 Green's theoremF. The fundamental theorem of algebra (elementary proof)L. Absolutely convergent seriesChapter 3. Fourier Analysis and Complex function theory13. Fourier series and the Poisson integral14. Fourier transforms15. Laplace transforms and Mellin transformsH. Inner product spacesN. The matrix exponentialG. The Weierstrass and Runge approximation theoremsChapter 4. Residue calculus, the argument principle, and two very specialfunctions16. Residue calculus17. The argument principle18. The Gamma function19. The Riemann zeta function and the prime number theoremJ. Euler's constantS. Hadamard's factorization theorem3 Chapter 5. Conformal maps and geometrical aspects of Complex function the-ory20. Conformal maps21. Normal families22. The Riemann sphere (and other Riemann surfaces)23.

3 The Riemann mapping theorem24. Boundary behavior of conformal maps25. Covering maps26. The disk coversCnf0;1g27. Montel's theorem28. Picard's theorems29. Harmonic functions IID. Surfaces and metric tensorsE. Poincar e metricsChapter 6. Elliptic functions and elliptic integrals30. Periodic and doubly periodic functions - in nite series representations31. The Weierstrass}in elliptic function theory32. Theta functions and}33. Elliptic integrals34. The Riemann surface of q( )K. Rapid evaluation of the Weierstrass}-functionChapter 7. Complex Analysis and differential equations35. Bessel functions36. Differential equations on a Complex domainO. From wave equations to Bessel and Legendre equationsAppendicesA. Metric spaces, convergence, and compactnessB. Derivatives and diffeomorphismsP. The Laplace asymptotic method and Stirling's formulaM.

4 The Stieltjes integralR. Abelian theorems and Tauberian theoremsQ. Cubics, quartics, and quintics4 PrefaceThis text is designed for a rst course in Complex Analysis , for beginning graduate stu-dents, or well prepared undergraduates, whose background includes multivariable calculus,linear algebra, and advanced calculus. In this course the student will learn that all the basicfunctions that arise in calculus, rst derived as functions of a real variable, such as powersand fractional powers, exponentials and logs, trigonometric functions and their inverses,and also many new functions that the student will meet, are naturally de ned for complexarguments. Furthermore, this expanded setting reveals a much richer understanding ofsuch is taken to introduce these basic functions rst in real settings.

5 In the openingsection on Complex power series and exponentials, in Chapter 1, the exponential functionis rst introduced for real values of its argument, as the solution to a differential is used to derive its power series, and from there extend it to Complex sintand costare rst given geometrical de nitions, for real angles, and theEuler identity is established based on the geometrical fact thateitis a unit-speed curve onthe unit circle, for realt. Then one sees how to de ne sinzand coszfor central objects in Complex Analysis are functions that are Complex -differentiable( , holomorphic). One goal in the early part of the text is to establish an equivalencebetween being holomorphic and having a convergent power series expansion. Half of thisequivalence, namely the holomorphy of convergent power series, is established in 2 starts with two major theoretical results, the Cauchy integral theorem , andits corollary, the Cauchy integral formula.

6 These theorems have a major impact on theentire rest of the text, including the demonstration that if a functionf(z) is holomorphicon a disk, then it is given by a convergent power series on that disk. A useful variant ofsuch power series is the Laurent series, for a function holomorphic on an text segues from Laurent series to Fourier series, in Chapter 3, and from there to theFourier transform and the Laplace transform. These three topics have many applications inanalysis, such as constructing harmonic functions, and providing other tools for differentialequations. The Laplace transform of a function has the important property of beingholomorphic on a half space. It is convenient to have a treatment of the Laplace transformafter the Fourier transform, since the Fourier inversion formula serves to motivate andprovide a proof of the Laplace inversion on these transforms illuminate the material in Chapter 4.

7 For example, thesetransforms are a major source of important de nite integrals that one cannot evaluate byelementary means, but that are amenable to Analysis by residue calculus, a key applicationof the Cauchy integral theorem . Chapter 4 starts with this, and proceeds to the study oftwo important special functions, the Gamma function and the Riemann zeta Gamma function, which is the rst \higher" transcendental function, is essentiallya Laplace transform. The Riemann zeta function is a basic object of analytic number5theory, arising in the study of prime numbers. One sees in Chapter 4 roles of Fourieranalysis, residue calculus, and the Gamma function in the study of the zeta function. Forexample, a relation between Fourier series and the Fourier transform, known as the Poissonsummation formula, plays an important role in its Chapter 5, the text takes a geometrical turn, viewing holomorphic functions asconformal maps.

8 This notion is pursued not only for maps between planar domains,but also for maps to surfaces inR3. The standard case is the unit sphereS2, and theassociated stereographic projection. The text also considers other surfaces. It constructsconformal maps from planar domains to general surfaces of revolution, deriving for the mapa rst-order differential equation, nonlinear but separable. These surfaces are discussedas examples of Riemann surfaces. The Riemann spherebC=C[f1gis also discussed asa Riemann surface, conformally equivalent toS2. One sees the group of linear fractionaltransformations as a group of conformal automorphisms ofbC, and certain subgroups asgroups of conformal automorphisms of the unit disk and of the upper half also bring in the notion of normal families, to prove the Riemann mapping of this theorem to a special domain, together with a re ection argument, showsthat there is a holomorphic covering ofCnf0;1gby the unit disk.]

9 This leads to key resultsof Picard and Montel, and applications to the behavior of iterations of holomorphic mapsR:bC!bC, and the Julia sets that treatment of Riemann surfaces includes some differential geometric material. Inan appendix to Chapter 5, we introduce the concept of a metric tensor, and show how itis associated to a surface in Euclidean space, and how the metric tensor behaves undersmooth mappings, and in particular how this behavior characterizes conformal discuss the notion of metric tensors beyond the setting of metrics induced on surfacesin Euclidean space. In particular, we introduce a special metric on the unit disk, calledthe Poincar e metric, which has the property of being invariant under all conformal auto-morphisms of the disk. We show how the geometry of the Poincar e metric leads to anotherproof of Picard' s theorem , and also provides a different perspective on the proof of theRiemann mapping text next examines elliptic functions, in Chapter 6.

10 These are doubly periodicfunctions onC, holomorphic except at poles (that is, meromorphic). Such a functioncan be regarded as a meromorphic function on the torusT =C= , where Cis alattice. A prime example is the Weierstrass function} (z), de ned by a double shows that} (z)2is a cubic polynomial in} (z), so the Weierstrass functioninverts an elliptic integral. Elliptic integrals arise in many situations in geometry andmechanics, including arclengths of ellipses and pendulum problems, to mention two basiccases. The Analysis of general elliptic integrals leads to the problem of nding the latticewhose associated elliptic functions are related to these integrals. This is the Abel inversionproblem. Section 34 of the text tackles this problem by constructing the Riemann surfaceassociated to p(z), wherep(z) is a cubic or quartic in this text, the exponential function was de ned by a differential equation andgiven a power series solution, and these two characterizations were used to develop itsproperties.