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This book is based on courses of lectures given to undergraduates in the Universities of Edinburgh and St. Andrews, and is intended to provide an easy introduction to the methods of the theory of functions of a complex variable. The reader is assumed to have a knowledge of the elements of the theory of functions of a real variable, such as is contained, for example, in Hardy's Course of Pure Mathematics an acquaintance with the easier parts of Bromwich's Infinite Series would prove advantageous, but is not essential.
The first six chapters contain an exposition, based on Cauchy's Theorem, of the properties of one-valued differentiable functions of a complex variable. In the rest of the book the problem of conformal representation, the elements of the theory of integral functions and the behaviour of some of the special functions of analysis are discussed by the methods developed in the earlier part. The book concludes with the classical proof of Picard's Theorem.
No attempt has been made to give the book an encyclopaedic character. My object has been to interest the reader and to encourage him to study further some of the more advanced parts of the subject suggestions for further reading have been made at the end of each chapter
Complex Numbers
The Convergence of Infinite Series
Functions of a Complex Variable
Cauchy's Theorem
Uniform Convergence
The Calculus of Residues
Integral Functions
Conformal Representation
The Gamma Function
The Hypergeometric Functions
Legendre Functions
Bessel Functions
The Elliptic Functions of Weierstrass
Jacobi's Elliptic Functions
Elliptic Modular Functions and Picard's Theorem