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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
Euler Sums involving square of Harmonic numbers
In my previous post on Euler sums, we evaluated sums containing $H_n$ and $H_n^{(2)}$. In this post, we’ll derive some further results using the integral $\int_0^x \frac{\log^3(1-t)}{t}dt$. Our starting point is the following generating function identity: $$ \sum_{n=1}^\infty (H_n)^2 x^n … Continue reading
The Contour Integration approach to Infinite Series
In this post, we will evaluate the series $\sum_{n=0}^\infty \frac{\cot\left(\frac{2n+1}{2}\pi\sqrt{2} \right)}{(2n+1)^3}$ using contour integration. Let’s define $f:\mathbb{C}\to \mathbb{C}$ as $$ f(z) = \frac{\pi \tan(\pi z)\tan(\pi z \theta)}{z^3} $$ where the parameter $\theta$ is a positive irrational number. Let $C_N$ denote … Continue reading