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Asymptotic expansions for the incomplete gamma function in the transition regions. (English) Zbl 1412.33006
The authors give a historical overview of the various types of asymptotic expansions existing in the literature for the incomplete gamma function \(\Gamma(a,z)\) for large \(a\) and \(z\). The principal new result of this paper is the asymptotic expansion for the normalised incomplete gamma function \(Q(a,a+\tau a^{1/2})\) for large \(a\) and bounded \(\tau\), where \(Q(a,z)=\Gamma(a,z)/\Gamma(a)\). The statement of this result is given in Theorem 1.1. This is basically a re-organised version of Temme’s well-known expansion for \(Q(a,z)\) for large \(a\), which covered the transition region but which has the disadvantage of possessing coefficients that present a removable singularity at \(\tau=0\). The authors’ expansion has coefficients \(C_n(\tau)\) that are polynomials in \(\tau\) and so do not possess this awkward inconvenience.
An expansion for the solution \(x=x(a,q)\) of the inverse problem \(Q(a,x)=q\), with \(0<q<1\), is derived also involving polynomial coefficients \(d_n(\tau_0)\), where \(\tau_0\) is the unique real root of \(q=\frac{1}{2} \text{erfc} (\tau_0/\surd 2)\). Finally, an asymptotic expansion as \(a\to -\infty\) for the unique negative zero \(x_-(a)\) of the regularised incomplete gamma function \(\gamma^*(a,x)\) is given, extending previous known asymptotic approximations. This last expansion also involves the polynomial coefficients \(d_n\) of imaginary argument.

MSC:
33B20 Incomplete beta and gamma functions (error functions, probability integral, Fresnel integrals)
41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
Software:
Algorithm 955; DLMF
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