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The book discusses major topics in complex analysis with applications to number theory. This book is intended as a text for graduate students of mathematics and undergraduate students of engineering, as well as to researchers in complex analysis and number theory. This theory is a prerequisite for the study of many areas of mathematics, including the theory of several finitely and infinitely many complex variables, hyperbolic geometry, two and three manifolds and number theory. In additional to solved examples and problems, the book covers most of the topics of current interest, such as Cauchy theorems, Picard’s theorems, Riemann–Zeta function, Dirichlet theorem, gamma function and harmonic functions.
Introduction and Simply Connected Regions
The Cauchy Theorems and Their Applications
Conformal Mappings and the Riemann Mapping Theorem
Harmonic Functions
The Picard Theorems
The Weierstrass Factorisation Theorem, Hadamard’s Factorisation Theorem and the Gamma Function
The Riemann Zeta Function and the Prime Number Theorem
The Prime Number Theorem with an Error Term
The Dirichlet Series and the Dirichlet Theorem on Primes in Arithmetic Progressions
The Baker Theorem