Some definite integrals
Proceedings of the London Mathematical Society, 2, XVII, 1918,
Records for 17 Jan. 1918

Typical formulæ are:

$$\int\limits^\infty_{-\infty} \frac{e^{nix}dx}{\Gamma(\alpha+x)\Gamma(\beta-x)} = \frac{(2\cos \half n)^{\alpha+\beta-2}}{\Gamma(\alpha+\beta-1)} e^{\frac{1}{2}n(\beta-\alpha)i}\:\: ({\rm or}\:0),$$
$$\int\limits^\infty_{-\infty} \frac{\Gamma(\alpha+x)}{\Gamma(\beta+x)} e^{nix} dx = \pm \frac{2 \pi i(2\sin \half N)^{\beta-\alpha-1}}{\Gamma(\beta-\alpha)} e^{-n \alpha i + \frac{1}{2}(\pi-N)(\beta-\alpha-1)i} \:\:({\rm or}\: 0),$$
\begin{eqnarray} && \int\limits^\infty_{-\infty} \Gamma(\alpha+x) \Gamma(\beta-x) e^{n i x} dx \nonumber \\ = \frac{2\pi i \Gamma(\alpha+\beta)}{(2\sin \half N)^{\alpha+\beta}} e^{\frac{1}{2}n(\beta-\alpha)i} && \left[\epsilon_n (\beta) e^{k \pi(\alpha+\beta)i} - \epsilon_n(-\alpha) e^{-k\pi(\alpha+\beta)i}\right]. \end{eqnarray}
Here $n$ is real, $n=2k\pi+N (0 \leq N \lt 2 \pi)$ in (2), and $n = (2k-1)\pi+N (0\leq N \lt 2\pi)$ in (3). In (1) the zero value is to be taken if $|n| \geq \pi$, the non-zero value otherwise. In (2) $\alpha$ must be complex: the zero value is to be taken if $n$ and ${\mathfrak I} (\alpha)$ have the same sign, the positive sign if $n \geq 0$ and ${\mathfrak I}(\alpha) \lt 0$, and the negative sign if $n \leq 0$ and ${\mathfrak I} (\alpha) \gt 0$. In (3) $\alpha$ and $\beta$ must both be complex; and $\epsilon_n (\zeta)$ is 0,1, or $-1$ according as (i) $\pi-n$ and ${\mathfrak J} (\zeta)$ have the same sign, (ii) $n \leq \pi$ and ${\mathfrak I} (\zeta) \lt 0$, (iii) $n \geq \pi$ and ${\mathfrak I}(\zeta) \gt 0$. The convergence conditions are, in general,

(1) ${\mathfrak R}(\alpha+\beta) \gt 1$, (2) ${\mathfrak R}(\alpha-\beta) \lt 0$, (3)${\mathfrak R}(\alpha+\beta)\lt 1$.

But there are certain special cases in which a more stringent condition is required. A formula of a different character, deduced from (1), is

\begin{eqnarray*} \int\limits^\infty_{-\infty} \frac{J_{\alpha+x}(\lambda)} {\lambda^{\alpha + x}} \frac{J_{\beta -x}(\mu)}{\mu^{\beta-x}} e^{nix} dx = \left(\frac{2 \cos \half n}{\Omega}\right)^{\frac{1}{2}(\alpha+\beta)} \\ e^{\frac{1}{2}n(\beta-\alpha)i} J_{\alpha+\beta} \{\sqrt{(2 \Omega \cos \half n)}\} \:({\rm or} \: 0). \end{eqnarray*} Here $$\Omega = \lambda^2 e^{\frac{1}{2} ni} + \mu^2 e^{-\frac{1}{2} n i};$$ the zero value is to be taken if $|n|+ \geq \pi$, the non-zero value otherwise; and the condition of convergence is, in general, that $${\mathfrak R} (\alpha+ \beta) \gt - 1.$$ The formulæ include a large number of interesting special cases, such as \begin{eqnarray*} \int\limits^\infty_{-\infty} \frac{dx}{\Gamma(\alpha+x)\Gamma(\beta-x)} &=& \frac{2^{\alpha+\beta-2}}{\Gamma(\alpha+\beta-1)}, \\ \int\limits^\infty_0 \frac{\sin \pi x dx}{x(x^2-1^2)(x^2-2^2) \cdots(x^2-k^2)} &=& (-1)^k \frac{2^{2k-1}\pi}{(2k)!}, \\ \int\limits^\infty_{-\infty} J_{\alpha+x} (\lambda) J_{\beta-x} (\lambda) dx &=& J_{\alpha+\beta} (2 \lambda). \end{eqnarray*} The formula \begin{eqnarray*} && \int\limits^\infty_{-\infty} \frac{dx}{\Gamma(\alpha+x)\Gamma(\beta-x)\Gamma(\gamma+x)\Gamma(\delta-x)} \\ && = \frac{\Gamma(\alpha+\beta+\gamma+\delta-3)} {\Gamma(\alpha+\beta-1)\Gamma(\beta+\gamma-1)\Gamma(\gamma+\delta-1) \Gamma(\delta+\alpha-1)}, \end{eqnarray*} may also be mentioned : it holds, in general, if $${\mathfrak R} (\alpha+ \beta+\gamma+\delta) \gt 3.$$ A fuller account of these formulæ will be published in the Quarterly Journal of Mathematics1

Endnotes

1[See No.27 of this volume]