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## Über dieses Buch

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.

## Inhaltsverzeichnis

### Chapter 1. Picard’s Theorems

Abstract
Our main goal in this chapter is to give an elementary proof of Picard’s first and second theorems, which we base upon Schottky’s theorem (Theorem 1.2).
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### Chapter 2. The Weierstrass Factorization Theorem

Abstract
A function f(z) is meromorphic in an open set $$A\subset {\mathbb C}$$ if it is regular in A except for a finite or infinite sequence $$z_1,z_2,\ldots \in A$$ of poles of f(z) (of any multiplicities).
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### Chapter 3. Entire Functions of Finite Order

Abstract
From now on, we shall use Vinogradov’s asymptotic symbol $$\ll$$ which is defined as follows.
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### Chapter 4. Bernoulli Numbers and Polynomials

Abstract
By a standard application of the Weierstrass–Hadamard factorization formula (3.​12).
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### Chapter 5. Summation Formulae

Abstract
For $$n\in {\mathbb N}$$ let.
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### Chapter 6. The Euler Gamma-Function

Abstract
The gamma-function $$\varGamma (z)$$ was introduced by Euler with the purpose of interpolating in a natural way the sequence n! (see (6.9)).
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### Chapter 7. Linear Differential Equations

Abstract
In this chapter we deal with some basic facts concerning ordinary linear differential equations in the analytic domain, culminating in Fuchs’ theory on regular singular points.
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### Chapter 8. Hypergeometric Functions

Abstract
Let the functions $$p_1(z)$$ and $$p_2(z)$$ be one-valued and regular for any sufficiently large |z|, say for $$|z|>R>0$$.
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### Backmatter

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