At the beginning of his paper Ramanujan makes the following statement:
It is well known that, $$ \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \left( \cos x \right)^m e^{inx} dx = \frac{\pi}{2^m} \frac{\Gamma(1+m)}{\Gamma\left(1+\frac{m+n}{2}\right)\Gamma\left(1+\frac{m-n}{2}\right)} $$ if $\Re[m]>-1$.
Unfortunately I do not know that. Can someone explain how the l.h.s. produces the r.h.s. (or give a reference that explains this)?
Then Ramanujan makes another statement:
It follows from this and Fourier's theorem that, if $n$ is any real number except $\pm \pi$ and $\Re[\alpha + \beta]>1$, or if $n=\pm \pi$ and $\Re[\alpha+\beta]>2$ then $$ \int_{-\infty}^{\infty} \frac{e^{inx}}{\Gamma(\alpha+x)\Gamma(\beta-x)} = \frac{\left(2 \cos\left(\frac{n}{2}\right)\right)^{\alpha+\beta-2}}{\Gamma(\alpha+\beta-1)} e^{\frac{in(\beta-\alpha)}{2}}\quad \mathrm{or} \quad 0, $$ according as $\vert n \vert < \pi$ or $\vert n \vert \geq \pi$.
This is another statement that is not trivial for me. It would be great if someone could provide proof or a reference.
