MISCELLANEOUS FORMULAS

Zeta function and Euler product:
\[\zeta(s)=\sum_{n=1}^\infty\frac{1}{n^s}=\prod_{p\:prime}\frac{1}{1-p^{-s}}\]
Cotangent series:
\[\pi\cot(\pi x)=\lim_{N \to \infty }\sum_{n=-N}^N\frac{1}{x+n}=\frac{1}{x}+\sum_{n=1}^\infty\frac{2x}{x^2-n^2}\]
Basel problem:\(\;\zeta(2)=\pi^2/6\)

Euler's constant:
\[\gamma = \lim_{n \to \infty } \left(\sum_{k=1}^n \frac{1}{k} - \log \left(n+{\scriptsize\frac{1}{2}}\right) \right)\] \[\gamma=\frac{\zeta(2)}{2}-\frac{\zeta(3)}{3}+\frac{\zeta(4)}{4}-\frac{\zeta(5)}{5}+\cdots\]
Gamma function:
\[\Gamma (x) = \int_0^\infty t^{x-1}e^{-t}dt\]
Weierstrass product:
\[\frac{1}{\Gamma(x)}=x e^{\gamma x}\prod_{n=1}^\infty \left( 1+\frac{x}{n}\right)e^{-\frac{x}{n}}\]
Exponential integral:
\[{\rm Ei}(x)=\int_{-\infty}^x\frac{e^t}{t} dt=\gamma+\log |x| + \sum_{n=1}^{\infty} \frac{x^n}{n!\,n}\]
Logarithmic integral:
\[{\rm li}(x)=\int_0^x\frac{dt}{\log t}=\gamma-1+\frac{x-1}{\log x}+\log|\log x|+\sum_{n=1}^\infty \frac{\log^n x}{(n+1)!\,n}\] \[{\rm li}(x)={\rm Ei}(\log x)\]
Harmonic numbers:
\[H_n= 1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}\] \[H_x = \int_0^1 \frac{\,\,\, 1 - t^x}{1 - t}\,dt=x \sum_{k=1}^\infty \frac{1}{k(x+k)}\] \[\int_0^x H_{t}\,dt =\int_0^1 \frac{1}{1-t} \left(x+\frac{1-t^x}{\log t}\right) dt = \gamma x+\log{\Gamma(x+1)}\]