Digamma function

In mathematics, the digamma function is defined by

[itex]\psi(x) =\frac{d}{dx} \ln{\Gamma(x)}= \frac{\Gamma'(x)}{\Gamma(x)}.[itex]

It is the first of the polygamma functions.

 Contents

Calculation

The digamma function, often denoted also ψ0(x) or even ψ0(x), is related to the harmonic numbers in that

[itex]\psi(n) = H_{n-1}-\gamma[itex]

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

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

Recurrence formulae

The digamma function satisfies a reflection formula similar to that of the Gamma function,

[itex]\psi(1 - x) - \psi(x) = \pi\,\!\cot{ \pi x}[itex]

which cannot be used to compute ψ(1/2), which is given below. The digamma function satisfies the recurrence relation [itex]\psi(x + 1) = \psi(x) + \frac{1}{x}[itex]

Note that this satisfies the recurrence relation of a partial sum of the harmonic series, thus implying the formula [itex] \psi(x) = H_{n-1} - \gamma[itex]

Special values

The digamma function has the following special values:

[itex] \psi(1) = -\gamma\,\![itex]
[itex] \psi\left(\frac{1}{4}\right) = -\frac{\pi}{2} - 3\ln{2} - \gamma[itex]
[itex] \psi\left(\frac{1}{3}\right) = -\frac{\pi\sqrt{3} + 9\ln{3}}{6} - \gamma[itex]
[itex] \psi\left(\frac{1}{2}\right) = -2\ln{2} - \gamma[itex]

Also see

Digamma function -- from MathWorld (http://mathworld.wolfram.com/DigammaFunction.html|)es:Función digamma it:Funzione digamma

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