An Introduction to Special 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.

Prof. Carlo Viola began his scientific activity in 1970. He is interested in various aspects of analytic as well as of geometric number theory, and especially in diophantine approximation. In the decade 1970-1980 he worked on several problems in analytic number theory, with contributions to the theory of diophantine equations, sieve methods, mean values of Dirichlet L-functions. In the 1980’s he was mainly interested in effective diophantine approximation to algebraic numbers. In this field he introduced new multiplicity estimates based on local-to-global analysis of singularities of highly reducible algebraic curves. More recently, he worked on polynomial-exponential diophantine equations, on factorization of lacunary polynomials, on the vanishing multiplicity of linear recurrence sequences, and on diophantine approximation to values of hypergeometric functions and to logarithms and dilogarithms of algebraic numbers. In 1996 he introduced, jointly with G. Rhin, a newalgebraic method in the study of arithmetical properties of values of the Riemann zeta-function, consisting in the action of groups of birational transformations and of permutation groups on multiple Euler-type integrals. In 1997 Carlo Viola was the editor of a volume on the arithmetic of elliptic curves with papers by J. Coates, R. Greenberg, K.A. Ribet and K. Rubin, and in 2002 he was co-editor with A. Perelli of a volume on analytic number theory with papers by J.B. Friedlander, D.R. Heath-Brown, H. Iwaniec and J. Kaczorowski, both published in the C.I.M.E. subseries of the Springer Lecture Notes in Mathematics. He was a member of the Institute for Advanced Study at Princeton in the academic year 1980-1981, and a visiting professor at Columbia University in New York in 1987-1988. Since 2002, he is a fellow of the Academy of Sciences in Turin. He received many invitations for research periods and for conferences and workshops in France, USA, Germany, Great Britain, Poland, Sweden, Hungary, Japan, Russia, etc.

Born in Rome, Italy, April 11, 1943.

Assistant professor at the University of Pisa since November 1970.

Full professor at the University of L’Aquila since November 1981.

Full professor at the University of Rome II since November 1983.

Full professor at the University of Pisa since November 1988.