Asymptotics and closed form of a generalized incomplete gamma function. (English) Zbl 0853.33003
Summary: We consider the asymptotic behavior of the function
\[
\Gamma (\alpha, x; b) = \int^\infty_x t^{\alpha - 1} e^{- t - b/t} dt, \quad x > 0,\;\alpha > 0,\;b \geq 0,
\]
as \(x\) tends to infinity. We give several expansions for the case that \(\alpha\) and \(b\) are fixed and we give a uniform expansion in which \(\alpha\) and \(b\) may range through unbounded intervals. We give a closed form for half-integer values of \(\alpha\).
MSC:
| 33B20 | Incomplete beta and gamma functions (error functions, probability integral, Fresnel integrals) |
| 33C10 | Bessel and Airy functions, cylinder functions, \({}_0F_1\) |
| 41A60 | Asymptotic approximations, asymptotic expansions (steepest descent, etc.) |
Digital Library of Mathematical Functions:
§8.16 Generalizations ‣ Incomplete Gamma Functions ‣ Chapter 8 Incomplete Gamma and Related FunctionsReferences:
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