Digamma function: Definition from Answers.com

Digamma function

ψ(s)

in the complex plane. The color of a point

s

encodes the value of

ψ(s)

. Strong colors denote values close to zero and hue encodes the value's argument.

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 Series formulae
4 Taylor series
5 Newton series
6 Reflection formula
7 Recurrence formula
8 Gaussian sum
9 Gauss's digamma theorem
10 Computation & approximation
11 Special values
12 See also
13 References
14 External links

Relation to harmonic numbers

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

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

Integral representations

It has the integral representation

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

valid if the real part of $x$ is positive. 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.

Series formulae

Digamma can be computed in the complex plane outside negative integers (Abramowitz and Stegun 6.3.16), using

$\psi(z+1)= -\gamma +\sum_{n=1}^\infty \left( \frac{z}{n(n+z)} \right), z \neq -1, -2, -3, ...$

Taylor series

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

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

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.

Newton series

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.

Reflection formula

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

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

Recurrence formula

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 $\gamma\,$ is the Euler-Mascheroni constant.

More generally, one has

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

Gaussian sum

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.

Gauss's digamma theorem

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

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

Computation & approximation

According to J.M. Bernardo AS 103 algorithm you can compute digamma function for x, a real number, with

$\psi(x) = ln(x) -\frac{1}{2x} - \frac{1}{12x^2} + \frac{1}{120x^4} - \frac{1}{252x^6} + O\left(\frac{1}{x^8}\right)$

$\psi(x) = ln(x) - \frac{1}{2x} + \sum_{n=1}^\infty \frac{\zeta(1-2n)}{x^{2n}}$

$\psi(x) = ln(x) - \frac{1}{2x} - \sum_{n=1}^\infty \frac{B(2n)}{2n(x^{2n})}$

n as integer, where B(n) is the nth Bernouilli number for and $ζ(n)$ is the Riemann zeta function.

Special values

The digamma function has the following special values:

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

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

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

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

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

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

References

Abramowitz, M. and Stegun, I. A. (Eds.). "Psi (Digamma) Function." §6.3 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 258-259, 1972. See section §6.4
Weisstein, Eric W., "Digamma function" from MathWorld.

External links

Cephes - C and C++ language special functions math library
[1] - Bernardo Statistical algorithm Psi(digamma function) computation, pp. 1

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