Difference between revisions of "Digamma"

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The digamma function $\psi$ is defined by
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The digamma function $\psi \colon \mathbb{C} \setminus \{0,-1,-2,\ldots\} \rightarrow \mathbb{C}$ is defined by
$$\psi(z) = \dfrac{d}{dz} \log \Gamma(z) = \dfrac{\Gamma'(z)}{\Gamma(z)}.$$
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$$\psi(z) = \dfrac{\mathrm{d}}{\mathrm{d}z} \log \Gamma(z) = \dfrac{\Gamma'(z)}{\Gamma(z)}.$$
  
 
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<gallery>
File:Plot digamma.png|Graph of $\psi$ on $[-5,5]$.
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File:Plot digamma.png|Graph of $\psi$.
File:Domain coloring digamma.png|[[Domain coloring]] of [[analytic continuation]] of $\psi(z)$.
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File:Complexdigammaplot.png|[[Domain coloring]] of $\psi(z)$.
 
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File:Domain coloring digamma.png
 
  
 
=Properties=
 
=Properties=
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[[Partial derivative of beta function]]<br />
<strong>Theorem:</strong> $\psi(1)=-\gamma$ and for integers $n\geq 2$,
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[[Digamma at 1]]<br />
$$\psi(n)=-\gamma + \displaystyle\sum_{k=1}^{n-1} \dfrac{1}{k}$$
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[[Digamma functional equation]]<br />
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[[Digamma at n+1]]<br />
<strong>Proof:</strong>
 
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=See Also=
<strong>Theorem:</strong> $\psi\left(\dfrac{1}{2}\right)=-\gamma-2\log(2)$ and for integers $n \geq 1$,
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[[Gamma]] <br />
$$\psi \left( n + \dfrac{1}{2} \right) = -\gamma - 2 \log(2) + 2 \left( 1 + \dfrac{1}{3} + \ldots + \dfrac{1}{2n-1} \right).$$
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[[Polygamma]]<br />
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[[Trigamma]] <br />
<strong>Proof:</strong>  █
 
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<strong>Theorem:</strong> $\psi(z+1) = \psi(z) + \dfrac{1}{z}$
 
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<strong>Proof:</strong>  █
 
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<strong>Theorem:</strong> $\psi(z+n)=\dfrac{1}{(n-1)+z} + \dfrac{1}{(n-2)+z} + \ldots + \dfrac{1}{2+z} + \dfrac{1}{1+z} + \psi(1+z)$
 
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<strong>Proof:</strong>  █
 
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<strong>Theorem:</strong> $\psi(1-z)=\psi(z) + \pi \cot(\pi z)$
 
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<strong>Proof:</strong>
 
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=References=
<strong>Theorem:</strong> $\psi(2z)=\dfrac{1}{2}\psi(z) + \dfrac{1}{2} \psi \left( z + \dfrac{1}{2} \right) + \log(2)$
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* {{BookReference|Higher Transcendental Functions Volume I|1953|Arthur Erdélyi|author2=Wilhelm Magnus|author3=Fritz Oberhettinger|author4=Francesco G. Tricomi|prev=findme|next=findme}}: $\S 1.7 (1)$
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* {{BookReference|Handbook of mathematical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev=Beta is symmetric|next=Digamma at 1}}: $6.3.1$
<strong>Proof:</strong>  █
 
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[[Category:SpecialFunction]]
<strong>Theorem:</strong> $\psi(\overline{z})=\overline{\psi(z)}$
 
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<strong>Proof:</strong>  █
 
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{{:Partial derivative of beta function}}
 
 
 
=See Also=
 
[[Gamma function]] <br />
 
[[Polygamma function]]<br />
 
[[Trigamma function]] <br />
 

Latest revision as of 23:21, 3 March 2018

The digamma function $\psi \colon \mathbb{C} \setminus \{0,-1,-2,\ldots\} \rightarrow \mathbb{C}$ is defined by $$\psi(z) = \dfrac{\mathrm{d}}{\mathrm{d}z} \log \Gamma(z) = \dfrac{\Gamma'(z)}{\Gamma(z)}.$$

Properties

Partial derivative of beta function
Digamma at 1
Digamma functional equation
Digamma at n+1

See Also

Gamma
Polygamma
Trigamma

References