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Generalized incomplete gamma functions with applications. (English) Zbl 0833.33002

The authors introduce the following generalization of the incomplete gamma function: \[ \int^\infty_x e^{-t} t^{\alpha - 1} e^{- t - b/t} dt, \quad \text{Re} (\alpha),\;b > 0, \] and its complement. These have been found useful in their researches in heat conduction, probability theory and in the study of Fourier and Laplace transforms. They derive their main properties, which include their asymptotic behaviour, Laplace transform, decomposition formula, integral representations, convolutions, recurrence relations and differentiation formula. Some applications are given to the evaluation of certain inverse Laplace transforms, to definite integrals and to infinite series of exponential functions. The paper contains graphs and tables.

MSC:

33B10 Exponential and trigonometric functions

Software:

Algorithm 597
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