# 双伽玛函数

$\psi(x) =\frac{d}{dx} \ln{\Gamma(x)}= \frac{\Gamma'(x)}{\Gamma(x)}.$

## 与调和数的关系

$\psi(n) = H_{n-1}-\gamma\!$

$\psi\left(n+{\frac{1}{2}}\right) = -\gamma - 2\ln 2 + \sum_{k=1}^n \frac{2}{2k-1}$

## 积分表示法

$\psi(x) = \int_0^{\infty}\left(\frac{e^{-t}}{t} - \frac{e^{-xt}}{1 - e^{-t}}\right)\,dt$

$\psi(s+1)= -\gamma + \int_0^1 \frac {1-x^s}{1-x} dx$

## 泰勒级数

$\psi(z+1)= -\gamma -\sum_{k=1}^\infty \zeta (k+1)\;(-z)^k$,

## 牛顿级数

$\psi(s+1)=-\gamma-\sum_{k=1}^\infty \frac{(-1)^k}{k} {s \choose k}$

## 反射公式

$\psi(1 - x) - \psi(x) = \pi\,\!\cot{ \left ( \pi x \right ) }$

## 递推关系

$\psi(x + 1) = \psi(x) + \frac{1}{x}$

## 高斯和

$\frac{-1}{\pi k} \sum_{n=1}^k \sin \left( \frac{2\pi nm}{k}\right) \psi \left(\frac{n}{k}\right) = \zeta\left(0,\frac{m}{k}\right) = -B_1 \left(\frac{m}{k}\right) = \frac{1}{2} - \frac{m}{k}$

$\sum_{n=1}^k \psi \left(\frac{n}{k}\right) =-k(\gamma+\log k),$

$\sum_{p=0}^{q-1}\psi(a+p/q)=q(\psi(qa)-\ln(q)),$

## 高斯双伽玛定理

$\psi\left(\frac{m}{k}\right) = -\gamma -\ln(2k) -\frac{\pi}{2}\cot\left(\frac{m\pi}{k}\right) +2\sum_{n=1}^{\lfloor \frac{k-1}{2}\rfloor} \cos\left(\frac{2\pi nm}{k} \right) \ln \sin\left(\frac{n\pi}{k} \right)$

## 特殊值

$\psi(1) = -\gamma\,\!$
$\psi\left(\frac{1}{2}\right) = -2\ln{2} - \gamma$
$\psi\left(\frac{1}{3}\right) = -\frac{\pi}{2\sqrt{3}} -\frac{3}{2}\ln{3} - \gamma$
$\psi\left(\frac{1}{4}\right) = -\frac{\pi}{2} - 3\ln2 - \gamma$
$\psi\left(\frac{1}{6}\right) = -\frac{\pi}{2}\sqrt{3} -2\ln{2} -\frac{3}{2}\ln3 - \gamma$
$\psi\left(\frac{1}{8}\right) = -\frac{\pi}{2} - 4\ln{2} - \frac{\sqrt2}{2} \left[\pi + \ln(3+2\sqrt{2})\right] - \gamma$
$\psi\left(\frac{3}{4}\right) = \frac{\pi}{2} - 3\ln{2} - \gamma$