Digamma function

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(Redirected from Gauss's digamma theorem)

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

$\psi(x) =\frac{d}{dx} \ln{\Gamma(x)}= \frac{\Gamma'(x)}{\Gamma(x)}.$

It is the first of the polygamma functions.

1 Relation to harmonic numbers
2 Integral representations
3 Taylor series
4 Newton series
5 Reflection formula
6 Recurrence formula
7 Gaussian sum
8 Gauss's digamma theorem
9 Special values
10 See also
11 References
12 External links

The digamma function, often denoted also as ψ₀(x), ψ⁰(x) or $\digamma$ (after the shape of the obsolete Greek letter Ϝ digamma), is related to the harmonic numbers 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

$\psi\left(n+{\frac{1}{2}}\right) = -\gamma - 2\ln 2 + \sum_{k=1}^n \frac{2}{2k-1}$

It has the integral representation

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

This may be written as

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

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

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

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

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

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

$\psi(s+1)=-\gamma-\sum_{k=1}^\infty \frac{(-1)^k}{k} {s \choose k}$

where $\textstyle{s \choose k}$ is the binomial coefficient.

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

$\psi(1 - x) - \psi(x) = \pi\,\!\cot{ \left ( \pi x \right ) }$

The digamma function satisfies the recurrence relation

$\psi(x + 1) = \psi(x) + \frac{1}{x}$

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

$\Delta [\psi] (x) = \frac{1}{x}$

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

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

where $γ$ is the Euler-Mascheroni constant.

More generally, one has

$\psi(x+1) = -\gamma + \sum_{k=1}^\infty \left( \frac{1}{k}-\frac{1}{x+k-1} \right)$

The digamma has a Gaussian sum of the form

$\frac{-1}{\pi k} \sum_{n=1}^k \sin \left( \frac{2\pi nm}{k}\right) \psi \left(\frac{n}{k}\right) = \zeta\left(0,\frac{m}{k}\right) = -B_1 \left(\frac{m}{k}\right) = \frac{1}{2} - \frac{m}{k}$

for integers $0 < m < k$ . Here, ζ(s,q) is the Hurwitz zeta function and $B n (x)$ is a Bernoulli polynomial. A special case of the multiplication theorem is

$\sum_{n=1}^k \psi \left(\frac{n}{k}\right) =-k(\gamma+\log k),$

and a neat generalization of this is

$\sum_{p=0}^{q-1}\psi(a+p/q)=q(\psi(qa)-\ln(q)),$

in which it is assumed that q is a natural number, and that 1-qa is not.

For positive integers m and k (with m < k), the digamma function may be expressed in terms of elementary functions as

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

The digamma function has the following special values:

$\psi(1) = -\gamma\,\!$

$\psi\left(\frac{1}{2}\right) = -2\ln{2} - \gamma$

$\psi\left(\frac{1}{3}\right) = -\frac{\pi}{2\sqrt{3}} -\frac{3}{2}\ln{3} - \gamma$

$\psi\left(\frac{1}{4}\right) = -\frac{\pi}{2} - 3\ln{2} - \gamma$

$\psi\left(\frac{1}{6}\right) = -\frac{\pi}{2}\sqrt{3} -2\ln{2} -\frac{3}{2}\ln(3) - \gamma$

$\psi\left(\frac{1}{8}\right) = -\frac{\pi}{2} - 4\ln{2} - \frac{1}{\sqrt{2}} \left\{\pi + \ln(2 + \sqrt{2}) - \ln(2 - \sqrt{2})\right\} - \gamma$

Trigamma function

Milton Abramowitz and Irene A. Stegun, Handbook of Mathematical Functions, (1964) Dover Publications, New York. ISBN 0486612724 . See section §6.3
Eric W. Weisstein, Digamma function at MathWorld.

Cephes - C and C++ language special functions math library

Retrieved from "http://en.wikipedia.org/wiki/Digamma_function"

Category: Gamma and related functions

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