Author Archives: Shobhit Bhatnagar
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
Bernoulli numbers and a related integral
Consider the sequence $\{B_r(x)\}_{r=0}^{\infty}$ of polynomials defined using the recursion: $ \begin{aligned} B_0(x) &= 1 \\ B_r^\prime(x) &= r B_{r-1}(x) \quad \forall r\geq 1 \\ \int_0^1 B_r(x) dx &= 0 \quad \forall r\geq 1 \end{aligned} $ The first few Bernoulli … Continue reading
Weyl’s Equidistribution Theorem
In this post, we will prove the Weyl’s Equidistribution theorem. A sequence of real numbers $x_1, x_2, \cdots$ is said to be equidistributed (mod 1) if for every sub-interval $(a,b)\subset [0,1]$, we have $$\lim_{N\to \infty}\frac{|\{1\leq n\leq N:\; \langle x_n \rangle\in … Continue reading