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Complex analysis. (English) Zbl 1020.30001
Princeton Lectures in Analysis. 2. Princeton, NJ: Princeton University Press. xvii, 379 p. (2003).
Chapter 1 is devoted to preliminaries (complex numbers, power series) to complex analysis. Chapter 2 focusses attention on Cauchy’s theorem and its applications. Cauchy integral formulae, Morera’s theorem, Schwarz’s reflection principle and Runge’s approximation theorem are delineated. Chapter 3 deals with meromorphic functions and the applications are presented here. Chapter 4 gives an account of the Fourier transforms. Chapter 5 explores the notion of entire functions. Jensens’s formulae and Hadamard’s factorization theorem are formulated and proved. The gamma and zeta functions are studied in Chapter 6. Analytic continuation is also a topic in this chapter. Chapter 7 is on the zeta function and prime number theorem. Chapter 8 elucidates a detailed treatment of conformal mappings. The Schwarz lemma and the Riemann mapping theorem are included in this chapter. Chapter 9 is an introduction to elliptic functions. Applications of theta functions are given in Chapter 10. Appendix A contains asymptotics. Simple connectivity and the Jordan curve theorem form contents of Appendix B. Some exercises are quoted below in order to assess the standard of the text.
Consider the function \(f\) defined on \(\mathbb{R}\) by \[ f(x) =\begin{cases} 0\quad &\text{if }x\leq 0\\ e^{-1/x^2}\quad &\text{if }x>0.\end{cases} \] Prove that \(f\) is indefinitely differentiable.
2. Evaluate the integral \(\int^\infty_{-\infty} {dx\over 1+x^4}\).
3. Find the order of growth of the following entire functions: (a) \(p(z)\) where \(p\) is a polynomial. (b) \(e^{\zeta^z}\).
4. Prove that the function \(\zeta\) has many zeros in the critical strip.
5. Does there exist a holomorphic surjection from the unit disc to \(\mathbb{C}\).
6. Prove that \(\wp''\) is a quadratic polynomial in \(\wp\).
To help the reader, bibliography, symbol glossary and index are furnished. The get-up and printing are attractive. This book will serve as a text at undergraduate and graduate levels for students of mathematics, engineering and financial management courses.

30-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to functions of a complex variable