Digamma function
Appearance
In mathematics, the digamma function is defined by
It is the first of the polygamma functions and is denoted by
Calculation
The digamma function, often denoted also ψ0(x) or even ψ0(x), is related to the harmonic numbers in that
where Hn−1 is the (n−1)th harmonic number, and γ is the well-known Euler-Mascheroni constant.
and may be calculated with the integral
Recurrence formulae
The digamma function satisfies a reflection formula similar to that of the Gamma function,
which may not be used to compute ψ(1/2), which is given below. The digamma function satisfies the recurrence relation
Note that this satisfies the recurrence relation of a partial sum of the harmonic series, thus implying the formula
Special values
The digamma function has the following special values:
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