Digamma function

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In mathematics, the digamma function is defined by

<math>\psi(x) =\frac{d}{dx} \ln{\Gamma(x)}= \frac{\Gamma'(x)}{\Gamma(x)}.</math>

It is the first of the polygamma functions.

Contents 1 Calculation 1.1 Recurrence formulae 1.2 Special values 2 See also 3 References

Calculation

The digamma function, often denoted also 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 H_n−1 is the (n−1)th harmonic number, and γ is the well-known Euler-Mascheroni constant.

and may be calculated with the integral

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

Recurrence formulae

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

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

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

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

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>

See also

• Gamma function
• Trigamma function
• Polygamma function
• Harmonic number
• Euler-Mascheroni constant

References

• Milton Abramowitz and Irene A. Stegun, Handbook of Mathematical Functions, (1964) Dover Publications, New York. ISBN 486-61272-4 . See section §6.3
• Wolfram Research's MathWorld by Eric Weisstein Digamma function -- from MathWorld