# Greek Functions

It’s interesting that each letter of the Greek alphabet has a function so I just decided to list them here for fun! It’s interesting to see how many are actually are related. Note: these aren’t functions that are just denoted by these symbols. They are functions which are named after these symbols

Alpha function
$\alpha_n(z)&space;\equiv&space;\int_1^{\infty}t^ne^{-zt}\textup{d}t$
Beta function
$\beta(p,q)&space;\equiv&space;\int_0^1t^{p-1}(1-t)^{q-1}\textup{d}t=\frac{\Gamma(p)\Gamma(q)}{\Gamma(p+q)}$
Gamma function
$\Gamma(x)\equiv&\int_0^{\infty}t^{x-1}e^{-t}\textup{d}t=(x-1)!$
Delta function
$\delta(x)=\left&space;\{&space;\begin{matrix}&space;0&space;&&space;x\neq&space;0\\&space;\infty&space;&&space;x=0&space;\end{matrix}&space;\right&space;\}$
Zeta function1
$\zeta(n)&space;\equiv&space;\sum_{k=1}^\infty\frac{1}{k^n}=\frac{\eta(n)}{1-2^{1-n}}$
Eta function2
$\eta(s)\equiv\sum_{k=1}^\infty\frac{(-1)^{k-1}}{k^s}=(1-2^{1-s})\zeta(s)$
Theta function
$\theta(z;&space;T)\equiv\sum_{-\infty}^{\infty}e^{\pi&space;in^2T+2\pi&space;inz}$
Iota function3
$\textup{I}(n)\equiv&space;\textrm{Largest&space;possible&space;value&space;of&space;composing&space;all&space;functions&space;in&space;Iota&space;function&space;set&space;at&space;}n$
Lambda function
$\lambda(x)\equiv\sum_{k=0}^\infty\frac{1}{{(2k+1)}^x}=\frac{\zeta(x)+\eta(x)}{2}$
Mu function
$\mu(x,&space;B,&space;a)\equiv\int_0^{\infty}\frac{x^{a+t}t^B\textup{d}t}{\Gamma(B+1)\Gamma(a+t+1)}$
Nu function
$\nu(x,&space;a)\equiv\int_0^{\infty}\frac{x^{a+t}\textup{d}t}{\Gamma(a+t+1)}=\mu(x,&space;0,&space;a)$
Xi function
$\xi(t)\equiv\frac{(t-1)\Gamma(\frac{t}{2}+1)\zeta(t)}{\sqrt{\pi^t}}=\Xi&space;\left(&space;\frac{i}{2}-it&space;\right)$
Pi function
$\Pi(t)\equiv&space;\left\{\begin{matrix}&space;0&space;&&space;|t|>\frac{1}{2}\\&space;\frac{1}{2}&space;&&space;|t|=\frac{1}{2}\\&space;1&space;&&space;|t|<\frac{1}{2}&space;\end{matrix}\right.$
Sigma function4
$\sigma_x(n)\equiv\sum_{d|n}d^x$
Tau function
$\Delta(\tau)\equiv(2\pi)^{12}\sum_{n=1}^\infty&space;\tau(n)e^{2\pi&space;int}$
Phi function
$\Phi(n)&space;\equiv&space;\textup{&space;Number&space;of&space;positive&space;integers}\leq&space;n\textup{&space;that&space;are&space;relativitely&space;prime&space;to&space;}n$
Chi function5
$\chi&space;(z)\equiv\gamma+\textup{ln}(z)+\int_0^z\frac{\textup{cosh}(t)-1}{t}\textup{d}t$
Psi function
$\psi(n)\equiv n\prod&space;_{d|n}\left&space;(&space;1+\frac{1}{d}&space;\right&space;)$
Omega function
$\omega(xe^x)=x$

1 Technically is any function that can be defined by infinite sum of powers but this is most common one (Riemann)
2 Another type of eta function also exists (Dedekind) but this was chosen due to its close relation to Zeta
3 Iota function set is set of all functions ever described
4 Technically called divisor function, sigma function assumes $\inline&space;x=1$
5 There also exists a Legendre Chi function

If you want to learn more about any of these functions or see where I learned them from, refer to Wolfram Mathworld