Difference between revisions of "Digamma"
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| − | The digamma function $\psi$ is defined by | + | The digamma function $\psi \colon \mathbb{C} \setminus \{0,-1,-2,\ldots\} \rightarrow \mathbb{C}$ is defined by |
$$\psi(z) = \dfrac{\mathrm{d}}{\mathrm{d}z} \log \Gamma(z) = \dfrac{\Gamma'(z)}{\Gamma(z)}.$$ | $$\psi(z) = \dfrac{\mathrm{d}}{\mathrm{d}z} \log \Gamma(z) = \dfrac{\Gamma'(z)}{\Gamma(z)}.$$ | ||
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<gallery> | <gallery> | ||
File:Plot digamma.png|Graph of $\psi$. | File:Plot digamma.png|Graph of $\psi$. | ||
| − | File: | + | File:Complexdigammaplot.png|[[Domain coloring]] of $\psi(z)$. |
</gallery> | </gallery> | ||
</div> | </div> | ||
=Properties= | =Properties= | ||
| − | < | + | [[Partial derivative of beta function]]<br /> |
| − | + | [[Digamma at 1]]<br /> | |
| − | + | [[Digamma functional equation]]<br /> | |
| − | + | [[Digamma at n+1]]<br /> | |
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| − | </ | ||
| − | </ | ||
| − | + | =See Also= | |
| − | + | [[Gamma]] <br /> | |
| − | + | [[Polygamma]]<br /> | |
| − | + | [[Trigamma]] <br /> | |
| − | < | ||
| − | </ | ||
| − | </ | ||
| − | + | =References= | |
| − | + | * {{BookReference|Higher Transcendental Functions Volume I|1953|Arthur Erdélyi|author2=Wilhelm Magnus|author3=Fritz Oberhettinger|author4=Francesco G. Tricomi|prev=findme|next=findme}}: $\S 1.7 (1)$ | |
| − | + | * {{BookReference|Handbook of mathematical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev=Beta is symmetric|next=Digamma at 1}}: $6.3.1$ | |
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| − | + | [[Category:SpecialFunction]] | |
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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 $$\psi(z) = \dfrac{\mathrm{d}}{\mathrm{d}z} \log \Gamma(z) = \dfrac{\Gamma'(z)}{\Gamma(z)}.$$
Graph of $\psi$.
Domain coloring of $\psi(z)$.
Properties
Partial derivative of beta function
Digamma at 1
Digamma functional equation
Digamma at n+1
See Also
References
- 1953: Arthur Erdélyi, Wilhelm Magnus, Fritz Oberhettinger and Francesco G. Tricomi: Higher Transcendental Functions Volume I ... (previous) ... (next): $\S 1.7 (1)$
- 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): $6.3.1$
