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The incomplete gamma functions since Tricomi. (English) Zbl 0965.33001
Tricomi’s ideas and contemporary applied mathematics. Proceedings of the international conference held on the occasion of the 100th anniversary of the birth of Francesco G. Tricomi, Rome, Italy, November 28-29 and Turin, Italy, December 1-2, 1997. Rome: Accademia Nazionale dei Lincei, Atti Convegni Lincei. 147, 203-237 (1998).
This is a very thorough and enjoyable survey on the incomplete gamma functions and their Tricomi-variants $\gamma^\ast(a,x)=x^{-a}\int_0^x e^{-t}t^{a-1} dt/\Gamma(a)$ and $\gamma_1(a,x)=\Gamma(a)x^a \gamma^\ast(a,-x).$ There are 160 references and the author even corrects errors in a few. In addition to what the title promiseses, the paper starts with a brief account of the history of the subject before 1950, followed by an affectionate but not uncritical description of Tricomi’s contributions. The main body (two thirds) of the paper deals with approximation, asymptotics, Stokes’s phenomenon, zeros, inverse functions, inequalities, monotonicity, numerical methods, generalizations, and links to other special functions after Tricomi. For the entire collection see [Zbl 0948.00034].

33-02Research monographs (special functions)
33B20Incomplete beta and gamma functions
33-03Historical (special functions)
33C15Confluent hypergeometric functions, Whittaker functions, ${}_1F_1$
33C45Orthogonal polynomials and functions of hypergeometric type
33F05Numerical approximation and evaluation of special functions
41A30Approximation by other special function classes
41A58Series expansions (e.g. Taylor, Lidstone series, but not Fourier series)
26A48Monotonic functions, generalizations (one real variable)
26D07Inequalities involving other types of real functions