Author Archives: Shobhit Bhatnagar
A proof of GR 3.255
In this post, we will prove the following monstrous looking identity from Gradshteyn and Ryzhik (3.255): $$ \int _0^1 \frac{x^{\mu+\frac{1}{2}} (1-x)^{\mu-\frac{1}{2}}}{(c+2bx-ax^2)^{\mu+1}}dx = \frac{\sqrt{\pi}}{\left\{a + \left(\sqrt{c+2b-a} + \sqrt{c}\right)^2\right\}^{\mu+\frac{1}{2}}\sqrt{c+2b-a}} \frac{\Gamma \left(\mu + \frac{1}{2}\right)}{\Gamma\left(\mu+1\right)}$$ where $c+2b-a>0$, $a + \left(\sqrt{c+2b-a} + \sqrt{c}\right)^2 > 0$ … Continue reading
Integral representations of the reciprocal beta function
In this post, we’ll prove a very interesting identity: $$ \int_0^\pi \left(\sin(\theta) \right)^{\alpha-1} e^{i \beta \theta} \; d\theta = \frac{\pi e^{\frac{i \pi}{2} \beta}}{\alpha 2^{\alpha-1} B \left(\frac{\alpha+\beta+1}{2}, \frac{\alpha-\beta+1}{2} \right)} $$ where $\beta + 1> \alpha > 0$ and $B(x,y) = \frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}$ … Continue reading
Introduction to Theta Functions II
Infinite product representations of theta functions Let $f(z) = \prod_{n=1}^{\infty} (1-q^{2n-1}e^{2iz})(1-q^{2n-1}e^{-2iz})$. Each of these two products converge absolutely and uniformly in any bounded domain of values of $z$. Hence $f(z)$ is analytic throughout the finite part of the $z$ plane. … Continue reading