Gleichmässige asymptotische Entwicklungen für unvollständige Integrale

Abstract

A type of integral is considered in which the integrand exp $(—g(s,t))$ has an only maximum in $t=y(s),a(s)<y(s)<b(s)$ for all (real) $s$ near $s_0$, the upper limit is $b$ and the lower limit can vary in $[a,-b]$. For integrals of this type an asymptotic expansion is derived holding for $s\to s_0$ and uniformly with respect to all $x\in [a,b]$. The results are applied to the incomplete gamma function and incomplete beta function.