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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.

Mathematics. --- Functional analysis. --- Functions of complex variables. --- Functions of real variables. --- Special functions. --- Functions of a Complex Variable. --- Functional Analysis. --- Real Functions. --- Special Functions. --- Several Complex Variables and Analytic Spaces. --- Special functions --- Real variables --- Complex variables --- Functional calculus --- Math --- Functions, special. --- Differential equations, partial. --- Elliptic functions --- Functions of real variables --- Calculus of variations --- Functional equations --- Integral equations --- Science --- Partial differential equations --- Mathematical analysis --- Functions, Special. --- Functions of complex variables

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The four contributions collected in this volume deal with several advanced results in analytic number theory. Friedlander's paper contains some recent achievements of sieve theory leading to asymptotic formulae for the number of primes represented by suitable polynomials. Heath-Brown's lecture notes mainly deal with counting integer solutions to Diophantine equations, using among other tools several results from algebraic geometry and from the geometry of numbers. Iwaniec's paper gives a broad picture of the theory of Siegel's zeros and of exceptional characters of L-functions, and gives a new proof of Linnik's theorem on the least prime in an arithmetic progression. Kaczorowski's article presents an up-to-date survey of the axiomatic theory of L-functions introduced by Selberg, with a detailed exposition of several recent results.

Number theory --- Geometry --- landmeetkunde --- getallenleer

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Mathematics

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Mathematics