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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
Fourier Series of the Log-Gamma Function and Vardi’s Integral
In this post, we will compute the Fourier series expansion of the Log-Gamma function and use it to prove the beautiful Vardi’s integral: $$\int_0^1 \frac{\log\log \left(\frac{1}{x}\right)}{1+x^2}dx = \frac{\pi}{2}\log\left(\sqrt{2\pi}\frac{\Gamma\left(\frac{3}{4}\right)}{\Gamma\left(\frac{1}{4}\right)} \right)$$ Fourier Series of the Log-Gamma Function: For $s\in (0,1)$, we have … Continue reading