Author Archives: Shobhit Bhatnagar
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
An alternating Euler Sum
In this post, we will evaluate the famous Euler sum: $$\sum_{n=1}^\infty (-1)^{n+1}\frac{H_n}{n^3}\quad \color{blue}{\cdots (*)} $$ where $H_n = \sum_{k=1}^n \frac{1}{k}=\int_0^1 \frac{1-t^n}{1-t}dt$ is the $n$-th harmonic number. This series resists contour integration techniques which makes it’s computation quite challenging. We will … Continue reading
Ramanujan’s Master Theorem: Part I
Ramanujan’s master theorem (RMT) is powerful tool that provides the Mellin transform of an analytic function using it’s power series around zero. Theorem (RMT): If the complex-valued function $f(z)$ has the following expansion $$ f(z) = \sum_{n=0}^\infty \frac{\varphi(n)}{n!}(-z)^n $$ then … Continue reading