自己紹介

好きな積分級数
\begin{eqnarray} \int_{0}^{\pi}\frac{\sin^{2n}x} {(1-2r\cos x+r^2)^{n+1}}dx =\frac{\pi}{2^{2n}(1-r^2)}\binom{2n}{n} \end{eqnarray}
\begin{eqnarray} \int_{-\infty}^{\infty}\binom{n}{x}\binom{m}{x}\binom{n+m+x}{x}dx=\sum_{k=0}^{n}\binom{n}{k}\binom{m}{k}\binom{n+m+k}{k}=\binom{n+m}{n}^2 \end{eqnarray}
\begin{eqnarray} \int_{0}^{\infty}\frac{\sin ax}{\sinh\frac{\pi}{2}x}\frac{dx}{1+x^2}dx=e^a\arctan e^{-a}-e^{-a}\arctan e^a \end{eqnarray}
\begin{eqnarray} \sum_{n>0}\frac{\pi}{n^{4m-1}(e^{\pi n}-1)}=\pi\zeta(4m-1)+\sum_{k=0}^{2m}(-1)^k\zeta(2k)\zeta(4m-2k) \end{eqnarray}
\begin{eqnarray} \sum_{n,m>0}\frac{(n-1)!(m-1)!}{(n+m)!}=\zeta(2) \end{eqnarray}
\begin{eqnarray} \sum_{n,m≠(0,0)}\frac{1}{(n^2+m^2)^s}=4\beta(s)\zeta(s) \end{eqnarray}
\begin{eqnarray} \int_{-\infty}^{\infty} \frac{\sin x}{x}\prod_{k=1}^{m} \frac{\sin a_kx}{a_kx}dx =\sum_{n=-\infty}^{\infty} \frac{\sin n}{n}\prod_{k=1}^{m} \frac{\sin a_kn}{a_kn}=\pi \end{eqnarray}