An Introduction to Special Functions

Authors: Carlo Viola



Presents material which connects the elementary theory of analytic functions of one complex variable with the theory of some of the most important special functions

Focuses especially on the Euler gamma function, the Gauss hypergeometric function, and the Kummer confluent hypergeometric function

Assumes only limited prior knowledge, to the level gained from a basic course on analytic functions

The subjects treated in this book have been especially chosen to represent a bridge connecting the content of a first course on the elementary theory of analytic functions with a rigorous treatment of some of the most important special functions: the Euler gamma function, the Gauss hypergeometric function, and the Kummer confluent hypergeometric function. Such special functions are indispensable tools in "higher calculus" and are frequently encountered in almost all branches of pure and applied mathematics. The only knowledge assumed on the part of the reader is an understanding of basic concepts to the level of an elementary course covering the residue theorem, Cauchy's integral formula, the Taylor and Laurent series expansions, poles and essential singularities, branch points, etc. The book addresses the needs of advanced undergraduate and graduate students in mathematics or physics.

Table of contents

Front Matter

Picard’s Theorems

The Weierstrass Factorization Theorem

Entire Functions of Finite Order

Bernoulli Numbers and Polynomials

Summation Formulae

The Euler Gamma-Function

Linear Differential Equations

Hypergeometric Functions

Back Matter

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