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Complex Analysis Complex Analysis Xue-Mei Li May 16, 2016 Contents 1 Complex Differentiation 6 1.1 The Complex Plane . .6 1.2 Complex Functions of One Variables . .6 1.3 Complex Linear Functions . 10 1.4 Complex Differentiation . 10 1.5 The @ and @¯ operator . 13 1.6 Harmonic Functions . 13 1.7 Holomorphic Functions . 14 1.8 Rules of Differentiation . 14 2 The Riemann Sphere and Mobius¨ Transforms 16 2.1 Conformal Mappings . 16 2.2 Mobius¨ Transforms (Lecture 5) . 17 2.2.1 The Extended Complex Plane . 18 2.2.2 Properties of Mobius¨ Transforms . 18 2.3 The Riemann Sphere and Stereographic Projection (lecture 7) . 22 3 Power Series 25 3.1 Power series is holomorphic in its disc of convergence . 25 3.2 Analytic Continuation (Lecture 8) . 27 3.3 The Exponential and Trigonometric Functions . 29 3.4 The Logarithmic Function and Power Function (Lecture 9) . 30 4 Complex Integration 31 4.1 Complex Integration . 31 4.2 Integration Along a Curve (lecture 9) . 32 4.3 Existence of Primitives (Lecture 9) . 34 4.4 Goursat’s Lemma . 37 4.5 Cauchy’s Theorem: simply connected domains (Lecture 12) . 39 4.6 Supplementary . 41 5 Cauchy’s Integral Formula 42 5.1 Keyhole Operation and Other Techniques (Lecture 12) . 42 5.2 Cauchy’s Integral Formula . 44 5.3 Taylor Expansion, Cauchy’s Derivative Formulas . 45 5.4 Estimates, Liouville’s Thm and Morera’s Thm . 46 5.4.1 Supplementary . 48 5.5 Locally Uniform Convergent Sequence of Holomorphic Functions . 48 1 5.6 Schwartz Reflection Principle (Lecture 15) . 49 5.7 The Fundamental Theorem of Algebra . 51 5.8 Zero’s of Analytic Functions . 52 5.9 Uniqueness of Analytic Continuation (Lecture 17) . 55 5.10 Supplementary . 55 6 Laurent Series and Singularities 56 6.1 Laurent Series Development (Lecture 17-18) . 56 6.2 Classification of Isolated Singularities (Lecture 19) . 61 6.2.1 Poles . 62 6.2.2 Essential Singularities . 64 6.3 Meromorphic Function . 64 6.3.1 Supplementary . 64 7 Winding Numbers and the Residue Theorem 66 7.1 The Index of a Closed Curve (Lecture 22) . 67 7.1.1 Supplementary . 69 7.2 The Residue Theorem (Lecture 23) . 71 7.3 Compute Real Integrals . 72 8 Fundamental Theorems 74 8.1 The Argument Principle (Lecture 24) . 74 8.2 Rouche’s´ Theorem . 75 8.2.1 Supplementary . 76 8.3 The Open mapping Theorem (Lecture 25) . 76 8.3.1 Supplementary . 78 8.4 Bi-holomorphic Maps on the Disc (lecture 25-26) . 79 9 The Riemann Mapping Theorem 81 9.1 Hurwitz’s Theorem (lecture 26) . 81 9.2 Family of Holomorphic Functions (Lecture 28) . 82 9.3 The Riemann Mapping Theorem (Lecture 28-29) . 84 9.4 Supplementary . 86 10 Special Functions 87 10.1 Constructing Holomorphic Functions by Integration (Lecture 30) . 87 10.2 The Gamma function . 87 10.3 The zeta function(Lecture 30) . 89 2 Prologue This is the lecture notes for the third year undergraduate module: MA3B8. If you need not be motivated, skip this section. Complex Analysis is concerned with the study of complex number valued functions with complex number as domain. Let f : C ! C be such a function. What can we say about it? Where do we use suchp an analysis? The complex number i = −1 appears in Fourier Transform, an important tool in analysis and engineering, and in the Schrodinger¨ equation, @ 2 @2 i = − ~ + V (x) (t; x); ~ @t 2m @2x a fundamental equation of physics, that describes how a wave function of a physical system evolves. Complex Differentiation is a very important concept, this is allured to by the fact that a number of terminologies are associated with ‘complex differentiable’. A function, complex differentiable on its domain, has two other names: a holomor- phic map and an analytic function, reflecting the original approach. The first meant the function is complex differentiable at every point, and the latter refers to functions with a power series expansion at every point. The beauty is that the two concepts are equivalent. A complex valued function defined on the whole complex domain is an entire function. Quotients of entire functions are Meromorphic functions on the whole plane. A map is conformal at a point if it preserves the angle between two tangent vec- tors at that point. A complex differentiable function is conformal at any point where its derivative does not vanish. Bi-holomorphic functions, a bi-jective holomorphic function between two regions, are conformal in the sense they preserve angles. Of- ten by conformal maps people mean bi-holomorphic maps. Conformal maps are the building blocks in Conformal Field Theory. It is conjectured that 2D statistical mod- els at criticality are conformal invariant. An exciting development is SLE, evolved from the Loewner differential equation describing evolutions of conformal maps. The Schramann-Loewner Evolution (also known as Stochastic Loewner Evolution, abbre- viated as SLE ) has been identified to describe the limits of a number of lattice models in statistical mechanics. Two mathematicians, W. Werner and S. Smirnov, have been awarded the Fields medals for their works on and related to SLE. Complex valued functions are built into the definition for Fourier transforms. For f : R ! R, Z 1 1 −ikx f^(k) = p e dx; k 2 R: 2π 1 Fourier transform extends the concept of Fourier series for period functions, is an im- portant tool in analysis and in image and sound processing, and is widely used in elec- trical engineering. 3 A well known function in number theory is the Riemann zeta-function, 1 X 1 ζ(s) = : ns n=1 The interests in the Riemann-zeta function began with Euler who discovered that the Riemann zeta function can be related to the study of prime numbers. 1 ζ(s) = Π : 1 − p−s The product on the right hand side is over all prime numbers: 1 1 1 1 1 1 1 Π = · · · · ::: :::: 1 − p−s 1 − 2−s 1 − 3−s 1 − 5−s 1 − 7−s 1 − 11−s 1 − p−s The Riemann-zeta function is clearly well defined for s > 1 and extends to all complex numbers except s = 1, a procedure known as the analytic /meromorphic continuation of a real analytic function. Riemann was interested in the following question: how many prime number are below a given number x? Denote this number π(x). Riemann found an explicit formula for π(x) in his 1859 paper in terms of a sum over the zeros of ζ. The Riemann hypothesis states that all non-trivial zeros of the Riemann zeta 1 function lie on the critical line s = 2 . The Clay institute in Canada has offered a prize of 1 million dollars for solving this problem. In symplectic geometry, symplectic manifolds are often studied together with a complex structure. The space C is a role model for symplectic manifold. A 2-dimensional symplectic manifold is a space that looks locally like a piece of R2 and has a symplectic form, which we do not define here. We may impose in addition a complex structure Jx 2 at each point of x 2 M. The complex structure Jx is essentially a matrix s.t. −Jx is the identity and defines a complex structure and leads to the concept of Khaler¨ manifolds. Finally we should mention that complex analysis is an important tool in combina- torial enumeration problems: analysis of analytic or meromorphic generating functions provides means for estimating the coefficients of its series expansions and estimates for the size of discrete structures. Topics Holomorphic Functions, meromorphic functions, poles, zeros, winding numbers (rota- tion number/index) of a closed curve, closed curves homologous to zero, closed curves homotopic to zero, classification of isolated singularities, analytical continuation, Con- formal mappings, Riemann spheres, special functions and maps. Main Theorems: Goursat’s theorem, Cauchy’s theorem, Cauchy’s derivative formulas, Cauchy’s inte- gral formula for curves homologous to zero, Weirerstrass Theorem, The Argument principle, Rouche’s´ theorem, Open Mapping Theorem, Maximum modulus principle, Schwartz’s lemma, Mantel’s Theorem, Hurwitz’s¨ theorem, and the Riemann Mapping Theorem. References • L. V. Ahlfors, Complex Analysis, Third Edition, Mc Graw-Hill, Inc. (1979) 4 • J. B. Conway. Functions of one complex variables. • T. Gamelin. Complex Analysis, Springer. (2001) • E. Hairer, G. Wanner, Analyse Complexe et Series´ de Fourier. http://www.unige.ch/hairer/poly_complexe/complexe.pdf • G. J. O. Jameson. A First course on complex functions. Chapman and Hall, (1970). • E. M. Stein and R. Shakarchi. Complex Analysis. Princeton University Press. (2003) Acknowledgement. I would like to thank E. Hairer and G. Wanner for the figures in this note. 5 Chapter 1 Complex Differentiation 1.1 The Complex Plane The complex plane C = fx+iy : x; y 2 Rg is a field with addition and multiplication, on which is also defined the complex conjugation x + iy = x − iy and modulus (also p p called absolute value) jzj = zz¯ = x2 + y2. It is a vector space over R and over C with the norm jz1 − z2j. We will frequently treat C as a metric space, with distance d(z1; z2) = jz1 − z2j, and so we understand that a sequence of complex numbers zn converges to a complex number z is meant by that the distance jzn − zj converges to zero. The space C with the above mentioned distance is a complete metric space and so a sequence converges if and only if it is a Cauchy sequence. Since 2 2 2 jzn − zj = jRe(zn) − Re(z)j + jIm(zn) − Im(z)j ; zn converges to z if and only if the real parts of (zn) converge to the real part of z and the imaginary parts of (zn) converge to the imaginary part of z.
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