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Digamma Function
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:\psi(x) = rac{d}{dx} \ln{\Gamma(x)}= rac{\Gamma'(x)}{\Gamma(x)}.

It is the first of the Polygamma Function s.


RELATION TO HARMONIC NUMBERS


The digamma function, often denoted also as ψ0(''x''), ψ0(''x'') or F (after the shape of the obsolete Greek letter Ϝ Digamma ), is related to the Harmonic Number s in that

:\psi(n) = H_{n-1}-\gamma\!

where ''H''''n'' is the ''n'' 'th harmonic number, and γ is the Euler-Mascheroni Constant . For half-integer values, it may be expressed as

:\psi\left(n+{ rac{1}{2}} ight) = -\gamma - 2\ln 2 +
\sum_{k=1}^n rac{2}{2k-1}


INTEGRAL REPRESENTATIONS

It has the Integral representation

:\psi(x) = \int_0^{\infty}\left( rac{e^{-t}}{t} - rac{e^{-xt}}{1 - e^{-t}} ight)\,dt

This may be written as

:\psi(s+1)= -\gamma + \int_0^1 rac {1-x^s}{1-x} dx

which follows from Euler's integral formula for the harmonic numbers.


TAYLOR SERIES

The digamma has a Rational Zeta Series , given by the Taylor Series at ''z''=1. This is

:\psi(z+1)= -\gamma -\sum_{k=1}^\infty \zeta (k+1)\;(-z)^k,



: \psi\left( rac{1}{2} ight) = -2\ln{2} - \gamma


SEE ALSO



REFERENCES