Difference between revisions of "Digamma"
(Created page with "The digamma function $\psi$ is defined by $$\psi(z) = \dfrac{d}{dz} \log \Gamma(z) = \dfrac{\Gamma'(z)}{\Gamma(z)}.$$") |
|||
| Line 1: | Line 1: | ||
The digamma function $\psi$ is defined by | The digamma function $\psi$ is defined by | ||
$$\psi(z) = \dfrac{d}{dz} \log \Gamma(z) = \dfrac{\Gamma'(z)}{\Gamma(z)}.$$ | $$\psi(z) = \dfrac{d}{dz} \log \Gamma(z) = \dfrac{\Gamma'(z)}{\Gamma(z)}.$$ | ||
| + | |||
| + | =Properties= | ||
| + | <div class="toccolours mw-collapsible mw-collapsed"> | ||
| + | <strong>Theorem:</strong> $\psi(1)=-\gamma$ and for integers $n\geq 2$, | ||
| + | $$\psi(n)=-\gamma + \displaystyle\sum_{k=1}^{n-1} \dfrac{1}{k}$$ | ||
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> █ | ||
| + | </div> | ||
| + | </div> | ||
| + | |||
| + | <div class="toccolours mw-collapsible mw-collapsed"> | ||
| + | <strong>Theorem:</strong> $\psi\left(\dfrac{1}{2}\right)=-\gamma-2\log(2)$ and for integers $n \geq 1$, | ||
| + | $$\psi \left( n + \dfrac{1}{2} \right) = -\gamma - 2 \log(2) + 2 \left( 1 + \dfrac{1}{3} + \ldots + \dfrac{1}{2n-1} \right).$$ | ||
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> █ | ||
| + | </div> | ||
| + | </div> | ||
| + | |||
| + | <div class="toccolours mw-collapsible mw-collapsed"> | ||
| + | <strong>Theorem:</strong> $\psi(z+1) = \psi(z) + \dfrac{1}{z}$ | ||
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> █ | ||
| + | </div> | ||
| + | </div> | ||
| + | |||
| + | <div class="toccolours mw-collapsible mw-collapsed"> | ||
| + | <strong>Theorem:</strong> $\psi(z+n)=\dfrac{1}{(n-1)+z} + \dfrac{1}{(n-2)+z} + \ldots + \dfrac{1}{2+z} + \dfrac{1}{1+z} + \psi(1+z)$ | ||
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> █ | ||
| + | </div> | ||
| + | </div> | ||
| + | |||
| + | <div class="toccolours mw-collapsible mw-collapsed"> | ||
| + | <strong>Theorem:</strong> $\psi(1-z)=\psi(z) + \pi \cot(\pi z)$ | ||
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> █ | ||
| + | </div> | ||
| + | </div> | ||
| + | |||
| + | <div class="toccolours mw-collapsible mw-collapsed"> | ||
| + | <strong>Theorem:</strong> $\psi(2z)=\dfrac{1}{2}\psi(z) + \dfrac{1}{2} \psi \left( z + \dfrac{1}{2} \right) + \log(2)$ | ||
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> █ | ||
| + | </div> | ||
| + | </div> | ||
| + | |||
| + | <div class="toccolours mw-collapsible mw-collapsed"> | ||
| + | <strong>Theorem:</strong> $\psi(\overline{z})=\overline{\psi(z)}$ | ||
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> █ | ||
| + | </div> | ||
| + | </div> | ||
Revision as of 06:50, 31 October 2014
The digamma function $\psi$ is defined by $$\psi(z) = \dfrac{d}{dz} \log \Gamma(z) = \dfrac{\Gamma'(z)}{\Gamma(z)}.$$
Properties
Theorem: $\psi(1)=-\gamma$ and for integers $n\geq 2$, $$\psi(n)=-\gamma + \displaystyle\sum_{k=1}^{n-1} \dfrac{1}{k}$$
Proof: █
Theorem: $\psi\left(\dfrac{1}{2}\right)=-\gamma-2\log(2)$ and for integers $n \geq 1$, $$\psi \left( n + \dfrac{1}{2} \right) = -\gamma - 2 \log(2) + 2 \left( 1 + \dfrac{1}{3} + \ldots + \dfrac{1}{2n-1} \right).$$
Proof: █
Theorem: $\psi(z+1) = \psi(z) + \dfrac{1}{z}$
Proof: █
Theorem: $\psi(z+n)=\dfrac{1}{(n-1)+z} + \dfrac{1}{(n-2)+z} + \ldots + \dfrac{1}{2+z} + \dfrac{1}{1+z} + \psi(1+z)$
Proof: █
Theorem: $\psi(1-z)=\psi(z) + \pi \cot(\pi z)$
Proof: █
Theorem: $\psi(2z)=\dfrac{1}{2}\psi(z) + \dfrac{1}{2} \psi \left( z + \dfrac{1}{2} \right) + \log(2)$
Proof: █
Theorem: $\psi(\overline{z})=\overline{\psi(z)}$
Proof: █
