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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
Full Text:
##### References:
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