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