\pi=\left(\int_{-\infty}^\infty e^{-x^2}dx\right)^2 \\
\pi=\left(\int_{-\infty}^\infty e^{-x^2}dx\right)^2 \\
\pi=\left((-1)^{m+1}\frac{(2m)!}{2^{2m-1}B_{2m}}\sum_{n=1}^\infty\frac{1}{n^{2m}}\right)^{\frac{1}{2m}} \\
\pi=\left((-1)^{m+1}\frac{(2m)!}{2^{2m-1}B_{2m}}\sum_{n=1}^\infty\frac{1}{n^{2m}}\right)^{\frac{1}{2m}} \\
\pi=2\prod_{k=1}^{\infty} \frac{4k^2}{4k^2 – 1} \\
\pi=2\prod_{k=1}^{\infty} \frac{4k^2}{4k^2 – 1} \\
\gamma :=\lim _{n\rightarrow \infty }\left(\sum _{k=1}^{n}{\frac {1}{k}}-\ln(n)\right)=\int _{1}^{\infty }\left({1 \over \lfloor x\rfloor }-{1 \over x}\right)\,dx \\
\gamma :=\lim _{n\rightarrow \infty }\left(\sum _{k=1}^{n}{\frac {1}{k}}-\ln(n)\right)=\int _{1}^{\infty }\left({1 \over \lfloor x\rfloor }-{1 \over x}\right)\,dx \\
\Gamma (z)=\lim _{n\to \infty }{\frac {n^{z}n!}{\displaystyle \prod _{k=0}^{n}{(z+k)}}} \\
\Gamma (z)=\lim _{n\to \infty }{\frac {n^{z}n!}{\displaystyle \prod _{k=0}^{n}{(z+k)}}} \\
\Psi (z)={\frac {\Gamma '(z)}{\Gamma (z)}}=\lim _{n\to \infty }\left(\log {n}-\sum _{k=0}^{n}{\frac {1}{z+k}}\right)\ \\
\Psi (z)={\frac {\Gamma '(z)}{\Gamma (z)}}=\lim _{n\to \infty }\left(\log {n}-\sum _{k=0}^{n}{\frac {1}{z+k}}\right)\ \\
{\begin{aligned}\left|\int _{\delta }^{e^{\delta }-1}{\frac {1}{u(u+1)}}du\right|&\leq \left|\int _{\delta }^{e^{\delta }-1}{\frac {1}{\delta }}du\right|=O(\delta )\\\end{aligned}} \\
{\begin{aligned}\left|\int _{\delta }^{e^{\delta }-1}{\frac {1}{u(u+1)}}du\right|&\leq \left|\int _{\delta }^{e^{\delta }-1}{\frac {1}{\delta }}du\right|=O(\delta )\\\end{aligned}} \\
\xi (s)=\pi ^{\!-s/2}\,\,\Gamma \!\left({\frac {s}{2}}\right)\,\zeta (s) \\
\xi (s)=\pi ^{\!-s/2}\,\,\Gamma \!\left({\frac {s}{2}}\right)\,\zeta (s) \\
\pi (x)=\sum _{n=1}^{\infty }{\frac {\mu (n)}{n}}\,\Pi (x^{1/n}) \\
\pi (x)=\sum _{n=1}^{\infty }{\frac {\mu (n)}{n}}\,\Pi (x^{1/n}) \\
{\begin{aligned}&\Gamma (z)={\frac {1}{e^{2{\pi }iz}-1}}\int _{C}s^{z-1}e^{-s}ds\qquad (z\in \mathbb {C} \setminus \mathbb {Z} )\\\end{aligned}} \\
{\begin{aligned}&\Gamma (z)={\frac {1}{e^{2{\pi }iz}-1}}\int _{C}s^{z-1}e^{-s}ds\qquad (z\in \mathbb {C} \setminus \mathbb {Z} )\\\end{aligned}} \\
{\begin{aligned}\lim _{\Re {z}\to +\infty }f(z)&=\lim _{\Re {z}\to +\infty }{\frac {n^{nz-1/2}\left[\prod _{k=0}^{n-1}{{\sqrt {\frac {2{\pi }}{z+k/n}}}\left({\frac {z+k/n}{e}}\right)^{z+k/n}}\right]}{(2\pi )^{(n-1)/2}{\sqrt {\frac {2{\pi }}{nz}}}\left({\frac {nz}{e}}\right)^{nz}}}\end{aligned}} \\
{\begin{aligned}\lim _{\Re {z}\to +\infty }f(z)&=\lim _{\Re {z}\to +\infty }{\frac {n^{nz-1/2}\left[\prod _{k=0}^{n-1}{{\sqrt {\frac {2{\pi }}{z+k/n}}}\left({\frac {z+k/n}{e}}\right)^{z+k/n}}\right]}{(2\pi )^{(n-1)/2}{\sqrt {\frac {2{\pi }}{nz}}}\left({\frac {nz}{e}}\right)^{nz}}}\end{aligned}} \\
\sin z=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n+1)!}}z^{2n+1}\quad {\text{for all}}\ z \\
\sin z=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n+1)!}}z^{2n+1}\quad {\text{for all}}\ z \\
\cos z=\sum _{n=0}^{\infty }{\frac {\left(-1\right)^{n}}{\left(2n\right)!}}z^{2n}\quad {\text{for all}}\ z \\
\cos z=\sum _{n=0}^{\infty }{\frac {\left(-1\right)^{n}}{\left(2n\right)!}}z^{2n}\quad {\text{for all}}\ z \\
\tan z=\sum _{n=1}^{\infty }{\frac {\left(-1\right)^{n}2^{2n}\left(1-2^{2n}\right)B_{2n}}{\left(2n\right)!}}z^{2n-1}\quad {\text{for}}\ |z|<{\frac {\pi }{2}} \\
\tan z=\sum _{n=1}^{\infty }{\frac {\left(-1\right)^{n}2^{2n}\left(1-2^{2n}\right)B_{2n}}{\left(2n\right)!}}z^{2n-1}\quad {\text{for}}\ |z|<{\frac {\pi }{2}} \\
\cot z=\sum _{n=0}^{\infty }{\frac {\left(-1\right)^{n}2^{2n}B_{2n}}{\left(2n\right)!}}z^{2n-1}\quad {\text{for}}\ 0<|z|<\pi \\
\cot z=\sum _{n=0}^{\infty }{\frac {\left(-1\right)^{n}2^{2n}B_{2n}}{\left(2n\right)!}}z^{2n-1}\quad {\text{for}}\ 0<|z|<\pi \\
\sec z=\sum _{n=0}^{\infty }{\frac {\left(-1\right)^{n}E_{2n}}{\left(2n\right)!}}z^{2n}\quad {\text{for}}\ |z|<{\frac {\pi }{2}} \\
\sec z=\sum _{n=0}^{\infty }{\frac {\left(-1\right)^{n}E_{2n}}{\left(2n\right)!}}z^{2n}\quad {\text{for}}\ |z|<{\frac {\pi }{2}} \\
\csc z=\sum _{n=0}^{\infty }{\frac {\left(-1\right)^{n}\left(2-2^{2n}\right)B_{2n}}{\left(2n\right)!}}z^{2n-1}\quad {\text{for}}\ 0<|z|<\pi
\csc z=\sum _{n=0}^{\infty }{\frac {\left(-1\right)^{n}\left(2-2^{2n}\right)B_{2n}}{\left(2n\right)!}}z^{2n-1}\quad {\text{for}}\ 0<|z|<\pi