Digamma function

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In mathematics, the digamma function is defined as the logarithmic derivative of the gamma function:

<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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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 numbers in that

<math>\psi(n) = H_{n-1}-\gamma\!</math>

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

<math>\psi\left(n+{\frac{1}{2}}\right) = -\gamma - 2\ln 2 +

\sum_{k=1}^n \frac{2}{2k-1}</math>

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Integral representations

It has the integral representation

<math>\psi(x) = \int_0^{\infty}\left(\frac{e^{-t}}{t} - \frac{e^{-xt}}{1 - e^{-t}}\right)\,dt</math>

This may be written as

<math>\psi(s+1)= -\gamma + \int_0^1 \frac {1-x^s}{1-x} dx</math>

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

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Taylor series

The digamma has a rational zeta series, given by the Taylor series at z=1. This is

<math>\psi(z+1)= -\gamma -\sum_{k=1}^\infty \zeta (k+1)\;(-z)^k</math>,

which converges for |z|<1. Here, <math>\zeta(n)</math> is the Riemann zeta function. This series is easily derived from the corresponding Taylor's series for the Hurwitz zeta function.

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Newton series

The Newton series for the digamma follows from Euler's integral formula:

<math>\psi(s+1)=-\gamma-\sum_{k=1}^\infty \frac{(-1)^k}{k} {s \choose k}</math>

where

<math>{s \choose k}</math>

is the binomial coefficient.

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Reflection formula

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

<math>\psi(1 - x) - \psi(x) = \pi\,\!\cot{ \left ( \pi x \right ) }</math>
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Recurrence formula

The digamma function satisfies the recurrence relation

<math>\psi(x + 1) = \psi(x) + \frac{1}{x}</math>

Thus, it can be said to "telescope" 1/x, for one has

<math>\Delta [\psi] (x) = \frac{1}{x}</math>

where Δ is the forward difference operator. This satisfies the recurrence relation of a partial sum of the harmonic series, thus implying the formula

<math> \psi(n)\ =\ H_{n-1} - \gamma.</math>

More generally, one has

<math>\psi(x) = -\gamma + \sum_{k=1}^\infty

\left( \frac{1}{k}-\frac{1}{x+k-1} \right)</math>

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Gaussian sum

The digamma has a Gaussian sum of the form

<math>\frac{-1}{\pi k} \sum_{n=1}^k

\sin \left( \frac{2\pi nm}{k}\right) \psi \left(\frac{n}{k}\right) = \zeta\left(0,\frac{m}{k}\right) = -B_1 \left(\frac{m}{k}\right) = \frac{1}{2} - \frac{m}{k}</math>

for integers <math>0<m<k</math>. Here, ζ(s,q) is the Hurwitz zeta function and <math>B_n(x)</math> is a Bernoulli polynomial.

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Gauss's digamma theorem

For integers m and k, the digamma may be expressed in terms of elementary functions as

<math>\psi\left(\frac{m}{k}\right) = -\gamma -\ln(2k)

-\frac{\pi}{2}\cot\left(\frac{m\pi}{k}\right) +\sum_{n=1}^{\lfloor k/2\rfloor-1} \cos\left(\frac{2\pi nm}{k} \right) \ln\left(\sin\left(\frac{n\pi}{k}\right)\right) </math>

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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>
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See also

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References

es:Función digamma fr:Fonction digamma it:Funzione digamma fi:Digammafunktio

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