$${\psi(z) = \frac{d}{dz} \left( \ln \Gamma(z) \right) = \frac{\Gamma'(z)}{\Gamma(z)}}$$
$${\psi(z) = -\gamma + \sum_{n=0}^{\infty} \left( \frac{1}{n+1} - \frac{1}{n+z} \right) = -\gamma + \sum_{n=0}^{\infty} \frac{z-1}{(n+1)(n+z)} }$$
$${\psi(z+1) = -\gamma + \sum_{n=1}^{\infty} \left( \frac{1}{n} - \frac{1}{n+z} \right) = -\gamma + \sum_{n=1}^{\infty} \frac{z}{n(n+z)} }$$
$${\psi(z) - \psi(s) = \sum_{n=0}^{\infty} \left( \frac{1}{n+s} - \frac{1}{n+z} \right) }$$
$${\sum_{n = 0}^{\infty} \frac{(-1)^n}{n + c} = \frac{\psi\left( \frac{c+1}{2} \right) - \psi\left( \frac{c}{2} \right)}{2} }$$
$${\psi(z+1) = - \gamma + \int_{0}^{1} \frac{1-x^z}{1-x} \ dx }$$
$${\psi(z+1) - \psi(s+1) = \int_{0}^{1} \frac{x^s-x^z}{1-x} \ dx }$$
$${\psi(z) = \frac{1}{\Gamma(z)} \int_{0}^{\infty} x^{z-1} \ln(x) e^{-x} \ dx }$$
$${\int_{0}^{1} \frac{x^{c-1}}{x + 1} \ dx = \frac{\psi\left( \frac{c+1}{2} \right) - \psi\left( \frac{c}{2} \right)}{2} }$$
$${\int_{0}^{1} \frac{x^{s-1}-x^{z-1}}{1-x^a} \ dx = \frac{\psi\left( \frac{z}{a} \right) - \psi\left( \frac{s}{a} \right)}{a} }$$
$${\int_{1}^{\infty} \frac{x^{s-1}-x^{z-1}}{1-x^a} \ dx = \frac{\psi\left(1 - \frac{s}{a} \right) - \psi\left(1 - \frac{z}{a} \right)}{a} }$$
$${\int_{0}^{\infty} \frac{x^{s-1}-x^{z-1}}{1-x^a} \ dx = \frac{ \left( \psi\left(1 - \frac{s}{a} \right) - \psi\left( \frac{s}{a} \right) \right) - \left( \psi\left(1 - \frac{z}{a} \right) - \psi\left( \frac{z}{a} \right)\right)}{a} }$$
$${\int_{0}^{\infty} \frac{x^{s-1}-x^{z-1}}{1-x^a} \ dx = \frac{\pi \left( \cot \left( \frac{\pi s}{a} \right) - \cot \left( \frac{\pi z}{a} \right) \right) }{a} }$$
$${\psi(z+1) = \lim_\limits{n \to \infty} \left(\ln(n) - \sum_{k=1}^{n} \frac{1}{k+z} \right)}$$
$${\psi(z+1) = \psi(z) + \frac{1}{z}}$$
$${\psi(z) = \psi(z+1) - \frac{1}{z}}$$
$${\psi(n) = H_{n-1} - \gamma, \ n \in \mathbb{Z}^+}$$
$${\psi(n) - \psi(m) = H_{n-1} - H_{m-1}, \ n, m \in \mathbb{Z}^+}$$
Where H is a Harmonic number: $${H_{n} = \sum_{k = 1}^{n} \frac{1}{k}}$$
$${\psi \left(k + \frac{1}{2} \right) = - \gamma - 2 \ln(2) + 2 \sum_{n=1}^{k} \frac{1}{2n-1}, \ n \in \mathbb{Z}^+}$$
Reflection Formula:$${\psi(1-z) - \psi(z) = \pi \cot(\pi z) }$$
$${\psi\left(- \frac{1}{n} \right) - \psi\left(\frac{1}{n} \right) = n + \pi \cot \left(\frac{\pi}{n} \right), \ n \in \mathbb{Z}^+ }$$
$${\psi\left(- a \right) - \psi\left( a \right) = \frac{1}{a} + \pi \cot \left( \pi a \right), \ n \notin \mathbb{Z} }$$
Multiplication Theorem:$${\psi(nz) = \frac{1}{n} \sum_{k = 0}^{n - 1} \psi \left(z + \frac{k}{n} \right) + \ln n }$$
$${\sum_{k = 1}^{n - 1} \psi \left(\frac{k}{n} \right) = (1-n) \gamma - n \ln n , \ n \in \mathbb{Z}^+ }$$
Gauss's Digamma Theorem$${}$$
$${r,m \in \mathbb{Z} \ \text{and} \ r < m }$$
$${\psi\left( \frac{r}{m} \right) = - \gamma - \ln(2m) - \frac{\pi}{2} \cot \left( \frac{r \pi}{m} \right) + 2 \sum_{n = 1}^{\left \lfloor \frac{m-1}{2} \right \rfloor} \cos \left( \frac{2 \pi n r}{m} \right) \ln \sin \left( \frac{\pi n}{m} \right) }$$
$${\psi(1) = -\gamma}$$
$${\psi\left( \frac{1}{2} \right) = -\gamma - 2 \ln 2}$$
$${\psi\left(- \frac{1}{2} \right) = -\gamma - 2 \ln 2 + 2}$$
$${\psi\left( \frac{1}{3} \right) = -\gamma - \frac{3}{2} \ln 3 - \frac{\pi}{2\sqrt{3}} }$$
$${\psi\left( \frac{2}{3} \right) = -\gamma - \frac{3}{2} \ln 3 + \frac{\pi}{2\sqrt{3}} }$$
$${\psi\left(- \frac{1}{3} \right) = -\gamma - \frac{3}{2} \ln 3 + \frac{\pi}{2\sqrt{3}} + 3 }$$
$${\psi\left( \frac{1}{4} \right) = -\gamma - 3 \ln 2 - \frac{\pi}{2} }$$
$${\psi\left( \frac{3}{4} \right) = -\gamma - 3 \ln 2 + \frac{\pi}{2} }$$
$${\psi\left(- \frac{1}{4} \right) = -\gamma - 3 \ln 2 + \frac{\pi}{2} + 4 }$$
$${\psi\left( \frac{1}{6} \right) = -\gamma - \frac{3}{2} \ln 3 - 2 \ln 2 - \frac{\sqrt{3}}{2} \pi }$$
$${\psi\left( \frac{5}{6} \right) = -\gamma - \frac{3}{2} \ln 3 - 2 \ln 2 + \frac{\sqrt{3}}{2} \pi }$$
$${\psi\left(- \frac{1}{6} \right) = -\gamma - \frac{3}{2} \ln 3 - 2 \ln 2 + \frac{\sqrt{3}}{2} \pi + 6 }$$