Digamma function
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In mathematics, the digamma function is defined by
- <math>\psi(x) =\frac{d}{dx} \ln{\Gamma(x)}= \frac{\Gamma'(x)}{\Gamma(x)}.<math>
It is the first of the polygamma functions.
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Calculation
The digamma function, often denoted also ψ0(x) or even ψ0(x), is related to the harmonic numbers in that
- <math>\psi(n) = H_{n-1}-\gamma<math>
where Hn−1 is the (n−1)th harmonic number, and γ is the well-known Euler-Mascheroni constant.
and may be calculated with the integral
<math>\psi(x) = \int_0^{\infty}\left(\frac{e^{-t}}{t} - \frac{e^{-xt}}{1 - e^{-t}}\right)\,dt<math>
Recurrence formulae
The digamma function satisfies a reflection formula similar to that of the Gamma function,
<math>\psi(1 - x) - \psi(x) = \pi\,\!\cot{ \pi x}<math>
which cannot be used to compute ψ(1/2), which is given below. The digamma function satisfies the recurrence relation <math>\psi(x + 1) = \psi(x) + \frac{1}{x}<math>
Note that this satisfies the recurrence relation of a partial sum of the harmonic series, thus implying the formula <math> \psi(x) = H_{n-1} - \gamma<math>
Special values
The digamma function has the following special values:
- <math> \psi(1) = -\gamma\,\!<math>
- <math> \psi\left(\frac{1}{4}\right) = -\frac{\pi}{2} - 3\ln{2} - \gamma<math>
- <math> \psi\left(\frac{1}{3}\right) = -\frac{\pi\sqrt{3} + 9\ln{3}}{6} - \gamma<math>
- <math> \psi\left(\frac{1}{2}\right) = -2\ln{2} - \gamma<math>
References
- Milton Abramowitz and Irene A. Stegun, Handbook of Mathematical Functions, (1964) Dover Publications, New York. ISBN 486-61272-4 . See section §6.3 (http://www.math.sfu.ca/~cbm/aands/page_258.htm)
- Wolfram Research's MathWorld by Eric Weisstein Digamma function -- from MathWorld (http://mathworld.wolfram.com/DigammaFunction.html|)
Also see
External links
Digamma function -- from MathWorld (http://mathworld.wolfram.com/DigammaFunction.html|)es:Función digamma it:Funzione digamma