$  \exp(x) = e^x := \sum_{k=0}^{\infty} \frac{ x^k }{ k! } = \frac x 1 + \frac{ x^2 }{ 2! } + \frac{ x^3 }{ 3! } + \dots  $ 

$  \exp(x) = \lim_{k \rightarrow \infty} \left( 1 + \frac x k \right)^k  $ 

$  \sin(x) := \frac 1 {2i} \left( e^{ix} - e^{-ix} \right) = \sum_{k=0}^{\infty} (-1)^k \frac{ x^{2k+1} }{ (2k + 1)! } = \frac x 1 - \frac{ x^3 }{ 3! } + \frac{ x^5 }{ 5! } - \dots  $ 

$  \cos(x) := \frac 1 2 \left( e^{ix} + e^{-ix} \right) = \sum_{k=0}^{\infty} (-1)^k \frac{ x^{2k} }{ (2k)! } = \frac{ x^0 }{ 0! } - \frac{ x^2 }{ 2! } + \frac{ x^4 }{ 4! } - \dots  $ 

$  \Gamma(x) := \int_0^\infty t^{x-1} e^{-t} \mathrm{d}t  $ 

$  \Gamma(n + 1) = n!  $ 

$  \Gamma(x + 1) = x \Gamma(x)  $ 

$  {}_pF_q(a_1,\dots,a_p;b_1,\dots,b_q;z)=\sum_{k=0}^\infty\prod_{i=1}^p\frac{\Gamma(k+a_i)}{\Gamma(a_i)}\prod_{j=1}^q\frac{\Gamma(b_j)}{\Gamma(k+b_j)}\frac{z^k}{k!};\quad p,q\in \mathbb{N}_0  $ 

$  \zeta(s) := \sum_{k=1}^\infty \frac 1 {k^s} = \prod_{p \mathrm{prim}} \frac{ 1 }{ 1 - \frac 1 {p^s} }  $ 

$  \zeta(s) = \frac{ 1 }{\Gamma(s)} \int_0^\infty \frac{ x^{s-1} }{ e^x - 1 } \mathrm{d}x  $