On a class of incomplete gamma functions with applications (Q2735628)

The book gives a general introduction to functions related to the gamma function, but the central idea is the generalisation of these functions by including extra terms or parameters in the standard integral representation of the functions. The incomplete gamma functions are generalised into the form \(\int_0^x t^{\alpha-1}\exp(-t-b/t) dt\), which for \(b=0\) is the standard form, and which for \(x=\infty\) is considered as the generalised gamma function (which in fact becomes a \(K-\)Bessel function). The beta integral, the Riemann zeta function and Fresnel integrals are generalised in a similar way. Many properties and relations are discussed for these new functions, and a further generalisation of the incomplete gamma function is introduced by replacing \(\exp(-b/t)\) by a \(K-\)Bessel function. If possible, relations with generalised hypergeometric functions are given. For example, the Kampé de Fériet function of two variables and the Fox \(H-\)function are used for this purpose. Several introductions are motivated by considering problems from mathematical physics and probability theory (generalised Gaussian distributions); examples from heat conduction problems are discussed in several chapters of the book. Many tables of computed values and graphs of the generalised functions and of quantities arising in problems from physic are given. In an appendix a table of Laplace transforms and a number of theorems on integrals dependent on parameters are included (differentiation, interchanging the order of integration, and so on). Without any doubt, the standard special functions can be generalised or extended in various ways to new classes of functions. A nice point of this book is that several generalisations considered here play a role in interesting problems from mathematical physics.