Bemerkungen über das Integral \(\int _0^{\infty } z^s e^{-e^z} dz\).
type: Item
instance of: scholarly article
title: Bemerkungen über das Integral \(\int _0^{\infty } z^s e^{-e^z} dz\).
zbMATH Open document ID: 58.0326.01
DOI: 10.1007/BF01699066
author: W. Wirtinger
published in: Monatshefte für Mathematik und Physik
publication date: 1932-01-01T00:00:00Z
review text: Es sei \[ e^{-e^z} = e^{-1} \left ( c_0 + c_1z + \frac {c_2}{2!}z^2 + \dots \right ) \] (die \(c_{\nu }\) sind ganze rationale Zahlen: \(c_0 =1\), \(c_1 = -1\), \(c_2 =0\), \(c_3 = +1\), \(\dots \)). In \[ G(s) = \int _0^{\infty } z^s e^{-e^z} dz = \int _0^1 + \int _1^{\infty } = G_1 (s) +G_2(s) \] hat \(G_1(s)\) Pole in allen und nur denjenigen Punkten \(s = - \nu -1\) \((\nu =0,1,2, \dots )\), für die \(c_{\nu } \neq 0\) (also z. B. in \(s = -1, -2\), aber nicht in \(s=-3\)). Weiter ist \(G_2(s)\) eine ganze Funktion und in \[ G_2 (s) = \sum _{n=0}^{\infty } K(n) \frac {(s+1)^n}{n!} \] ist (wie \textit{K.Frey} berechnete) \[ K(n) = \nu ^n e^{-e^{\nu }} \left ( e^{-\frac {e^{\nu }}{2}} \sqrt {\frac {2 \nu \pi }{e^{\nu } (e^{\nu } + \nu +1)}} + 72 \nu ^3 e^{2\nu } e^{-e^{\nu }} \theta _{\nu } \right ) \,. \] wo \[ n = \nu e^{e^{\nu }} e^{\nu }, \quad -1 \leq \theta _{\nu } \leq +1. \]
zbMATH DE Number: 2549937
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/bf01699066
OpenAlex ID: W2088316447
description: scientific article; zbMATH DE number 2549937