#### Vol. 8, No. 4, 2015

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A type of multiple integral with log-gamma function

### Duokui Yan, Rongchang Liu and Geng-zhe Chang

Vol. 8 (2015), No. 4, 593–614
##### Abstract

In this paper, we give a general formula for the multiple integral

$I={\int }_{0}^{1}{\int }_{0}^{1}\dots {\int }_{0}^{1}f\left({x}_{1}+{x}_{2}+\cdots +{x}_{n}\right)\phantom{\rule{0.3em}{0ex}}d{x}_{1}\phantom{\rule{0.3em}{0ex}}d{x}_{2}\dots \phantom{\rule{0.3em}{0ex}}d{x}_{n}.$

As an application, the integral $I$ with $f\left(x\right)=log\Gamma \left(x\right)$ is evaluated for all $n\in ℕ$. The subsidiary computational challenges are interesting in their own right.

##### Keywords
multiple integral, log-gamma function
##### Mathematical Subject Classification 2010
Primary: 05A19, 54C30