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Complex Analysis Complex Analysis Joachim Wehler Ludwig-Maximilians-Universitat¨ Munchen¨ Department of Mathematics Summer Term 2019 - Lecture Notes DRAFT, Release 2.6 July 24, 2019 2 c Joachim Wehler, 2019 I prepared these notes for the students of my lecture. The present lecture took place during the summer term 2019 at the mathematical department of LMU (Ludwig-Maximilians-Universitat)¨ at Munich. The first input are notes taken from former lectures by Otto Forster. Compared to the oral lecture in class these written notes contain some additional material. In particular, at the end of several chapters I added an outlook to some more advanced topics from complex analysis of several variables and from the theory of complex spaces. Please report any errors or typos to [email protected] Release notes: • Release 2.6: Chapter 8: Minor revisions. • Release 2.5: Minor revisions. • Release 2.4: Chapter 8 completed. List of results added. • Release 2.3: Chapter 8, Section 8.2 added. • Release 2.2: Definition 7.6 corrected. • Release 2.1: Chapter 8, Section 8.1 added. References to problem sheet 12 removed • Release 2.0: Chapter 7 completed. Chapter 5, references to problem sheet 11 removed. • Release 1.9: Chapter 6 completed. References to problem sheet 12 to be removed later. • Release 1.8: Minor revisions. • Release 1.7: Chapter 6, Section 6.1 added. • Release 1.6: Chapter 5 completed. References to problem sheet 11 to be removed later. • Release 1.5: Chapter 5, Section 5.2 added. • Release 1.4: Proposition 5.7: typo corrected. • Release 1.3: Chapter 5, Section 5.1 added. • Release 1.2: Chapter 4, Section 4.3 added. • Release 1.1: Definition 3.10: typo corrected. • Release 1.0: Remark 3.28 added, Chapter 4 completed. • Release 0.9: Chapter 4, Section 4.1 created. • Release 0.8: Section 2.2: Jacobi matrix replaced by transposed matrix, typo C-R differential equation corrected. • Release 0.7: Chapter 3 completed. • Release 0.6: Chapter 3, Section 3.2 created. • Release 0.5: Chapter 3, Section 3.1 created. • Release 0.4: Chapter 2 created, Section 1.4 updated. 3 • Release 0.3: Chapter 1 completed. • Release 0.2: Section 1.2 created. • Release 0.1: Chapter 1, Section 1.1 created. Contents Introduction ....................................................... 1 1 Analytic Functions ............................................. 3 1.1 Calculus of convergent power series . .3 1.2 Fundamental properties of analytic functions. 19 1.3 Exponential map and related analytic functions . 32 1.4 Outlook . 44 2 Differentiable Functions and Cauchy-Riemann Differential Equations 45 2.1 Differentiability . 45 2.2 Cauchy-Riemann differential equations . 48 3 Cauchy’s Integral Theorem for disk and annulus . 55 3.1 Cauchy kernel and Cauchy integration . 55 3.2 The concept of holomorphy . 63 3.3 Principal theorems about holomorphic functions . 69 3.4 Outlook . 78 4 Isolated Singularities of Holomorphic Functions ................... 81 4.1 Laurent series and types of singularities . 81 4.2 Meromorphic functions and essential singularities . 88 4.3 The generating function of the Bernoulli numbers . 92 4.4 Outlook . 96 5 Mittag-Leffler Theorem and Weierstrass Product Formula . 97 5.1 Meromorphic functions with prescribed principal parts . 97 5.2 Infinite products of holomorphic functions. 107 5.3 Weierstrass product theorem for holomorphic functions . 111 5.4 Outlook . 130 v vi Contents 6 Integral Theorems of Complex Analysis . 131 6.1 Cauchy’s integral theorem and the residue theorem . 131 6.2 Computation of integrals using the residue theorem . 139 6.3 Applications of the residue theorem in complex analysis . 148 7 Homotopy .....................................................155 7.1 Integration along homotopic paths . 155 7.2 Simply connectedness . 160 7.3 Outlook . 166 8 Holomorphic Maps .............................................167 8.1 Montel’s theorem. 167 8.2 Riemann Mapping Theorem. 172 8.3 Projective space and fractional linear transformation . 186 8.4 Outlook . 198 List of results ......................................................199 References .........................................................203 Index .............................................................205 Introduction 1 + eip = 0 Euler’s identity is considered an example of beauty in mathematics. Some people even call it “the most famous formula in all mathematics”. Euler’s identity relates the natural numbers 0 and 1 and the transcendental num- bers e and p to the complex number i. The latter is the imaginary unit satisfying i2 = −1: The formula above exemplifies the way mathematics reveals some unexpected struc- ture, when one considers its concepts in the context of complex numbers. Complex numbers form the context of complex analysis, the subject of the present lecture notes. Complex analysis investigates analytic functions. Locally, analytic functions are convergent power series. The well-known exponential series extends to the complex plane and evaluates at the non-real argument ip from Euler’s identity as ¥ (ix)n ¥ (ix)2n ¥ (ix)2n+1 eix = ∑ = ∑ + ∑ = n=0 n! n=0 (2n)! n=0 (2n + 1)! ¥ x2n ¥ x2n+1 ∑ (−1)n · + i · ∑ (−1)n · = cos x + i · sin x: n=0 (2n)! n=0 (2n + 1)! The result eip = cos p + i · sin p explains the key value eip = −1; hereby proving the Euler identiy. Note. Not every formula has already revealed its deeper meaning. A famous riddle for contem- porary physicists is the formula 1 2 Introduction s hG¯ L = Planck c3 This formula comprises Planck’s constant h¯, Newton’s constant of gravitation G, and the velocity of light c. The formula relates quantum theory, gravitation and the velocity of light to the Planck lenght LPlanck, the scale where spacetime becomes quantized. Chapter 1 Analytic Functions It has been written that the shortest and best way between two truths of the real domain often passes through the imaginary one. Jacques Hadamard, 1945 The concept of holomorphy is accessible from different directions: • Section 1.1 starts from convergent power series. They give reasons for the def- inition of analytic functions in Definition 1.8 (Access due to Weierstrass). All power series from calculus extend from intervals of the real line into the domain of complex numbers, see Section 1.3. • Section 2.1 in Chapter 2 introduces a second path to holomorphy. It starts from differentiability in Definition 2.1 (Access due to Cauchy). Theorem 2.3 shows: Any analytic function in an open set U ⊂ C is differentiable. The derivative is again an analytic function. • A third path starts from the Cauchy-Riemann differential equations for the par- tial derivatives in the real sense, see Theorem 2.6 (Access due to Riemann). The theorem also proves the equivalence of path two and path three. Deeper is the fact that any differentiable function in an open set U ⊂ C is analytic. This result follows from a simple variant of Cauchy’s integral formula. It will be demonstrated not until Chapter 3. The proof will be the final step in establishing the equivalence of the three paths to holomorphy. Cauchy, Weierstrass and Riemann are the three protagonists of complex analysis in the 19th century. 1.1 Calculus of convergent power series Analytic functions are those functions which expand locally into a convergent power series. Therefore the present section investigates the calculus of convergent power series. 3 4 1 Analytic Functions Definition 1.1 (Compact convergence). A sequence ( fn )n2N of functions fn : U −! C; n 2 N; on an open subset U ⊂ C is compact convergent to a function f : U −! C if for every compact subset K ⊂ U the sequence of restrictions ( fn jK)n2N is uni- formly convergent to f jK. Remark 1.2 (Convergence). Consider fn : U −! C; n 2 N; and f : U −! C. For f = lim fn n!¥ the convergence is • pointwise: 8x 2 U 8e > 0 9N 2 N 8n ≥ N : j f (x) − fn(x)j < e • uniform: 8e > 0 9N 2 N 8x 2 U 8n ≥ N : j f (x) − fn(x)j < e • compact: 8K ⊂ U compact 8e > 0 9N 2 N 8x 2 K 8n ≥ N : j f (x) − fn(x)j < e Theorem 1.3 (Convergence of power series). Consider a fixed point a 2 C and a power series ¥ n ∑ cn · (z − a) n=0 with center a and complex coefficients cn 2 C; n 2 N: 1. If the series is convergent for at least one z0 2 C; z0 6= a; then it is absolutely and compactly convergent in the whole open disk Dr(a) := fz 2 C : jz − aj < rg; r := jz0 − aj: The function 1.1 Calculus of convergent power series 5 ¥ n f : Dr(a) −! C; z 7! ∑ cn · (z − a) ; n=0 is continuous. Fig. 1.1 Compact convergence in Dr(a) 2. There exists a radius 0 ≤ R ≤ ¥; the radius of convergence, which discriminates between the following two alter- natives: • The series is convergent for all z 2 C with jz − aj < R and • divergent for all z 2 C with jz − aj > R. The disk DR(a) is the named the disk of convergence. If R > 0 then the power series is named a convergent power series. Proof. A statement about a power series with center a reduces by the linear substi- tution of variables w := z − a to a statement about a power series with center = 0. Hence, w.l.o.g. we assume a = 0. 1. Assume that the series is convergent for z0 2 C. Then n (cn · z0)n2N is a null-sequence, in particular bounded: Exists M > 0 such that for all n 2 N n jcn · z0j ≤ M: For any z 2 C with jzj < jz0j we have 6 1 Analytic Functions n n ¥ ¥ z ¥ z n n ∑ jcn · z j = ∑ jcn · z0j · ≤ M · ∑ n=0 n=0 z0 n=0 z0 and the dominating series is a convergent geometric series.
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