Integrating gamma products and quotients over a vertical line - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T04:40:41Z http://mathoverflow.net/feeds/question/86988 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/86988/integrating-gamma-products-and-quotients-over-a-vertical-line Integrating gamma products and quotients over a vertical line spanferkel 2012-01-29T21:21:44Z 2012-01-29T21:27:26Z <p>The <a href="http://functions.wolfram.com/GammaBetaErf/Gamma/21/01/01/" rel="nofollow">Wolfram functions collection</a> contains a small number of integrals of products and quotients of terms $\Gamma(a_i\pm t)$ over a vertical line, all of which can be evaluated in terms of only gamma functions. </p> <p><strong>Does somebody know references concerning formulae of this type?</strong> If they are obtained by the residue theorem, how?</p> <p>The second one seems to be false. By putting $2b=a-c+1$, we would get from it, after using the doubling formula, $$\int_{\gamma-i\infty}^{\gamma+i\infty}\Gamma(a+t)\Gamma(c-t)dt=2\pi i\cdot2^{a+c}{\Gamma(a+c)},$$ whereas for $a,c\in\mathbb Z$ such that $a+c\ge1$, we have $$\int\limits_{\gamma-i\infty}^{\gamma+i\infty}\Gamma(a+t)\Gamma(c-t)dt=\pi i\int\limits_{-\infty}^{\infty}\frac{\overbrace{(1-a-t)\cdots(c-1-t)}^{(a+c-1)\ {terms}}}{\cosh \pi t}dt=\pi i\int\limits_{-\infty}^{\infty}\frac{\Gamma(c-t)\; dt}{\Gamma(1-a-t)\cosh \pi t}$$ which may be written in umbral form as $(1-a-\frac E2)\cdots(c-1-\frac E2)$, i.e. as a linear combination of terms $\int\limits_{-\infty}^{\infty}\dfrac{t^{k}dt}{\cosh \pi t}=\dfrac{E_k}{2^k}$, where the $E_k$ are the (absolute) Euler numbers, thus it can definitely not be written in terms of $\Gamma(a+c)$ etc.</p> <p>Some others of the Wolfram identities <strong>look like they could be generalized</strong>, e.g. the first and third one. Is it true e.g. that for $n\ge3$, $$\int_{\gamma-i\infty}^{\gamma+i\infty}\prod\limits_{j=1}^n\Bigl(\Gamma(a_j+t)\Gamma(b_j-t)\Bigr)dt=2\pi i\frac{\prod_j\prod_k\Gamma(a_j+b_k)}{\Gamma(\sum a_j+\sum b_j)^{n-1}}$$ where all products and sums run from $1$ to $n$? (Note that again, this does not hold for $n=1$.)</p>