Digamma function
In mathematics, the digamma function is defined as the logarithmic derivative of the gamma function:
It is the first of the polygamma functions.
The digamma function is often denoted as ψ0(x), ψ0(x) or Ϝ (the archaic Greek letter digamma).
The gamma function obeys the equation
Taking the derivative with respect to z gives:
Dividing by Γ(z + 1) or the equivalent zΓ(z) gives:
or:
Since the harmonic numbers are defined as
the digamma function is related to it by:
where Hn is the nth harmonic number, and γ is the Euler–Mascheroni constant. For half-integer values, it may be expressed as
If the real part of x is positive then the digamma function has the following integral representation
This may be written as
which follows from Leonhard Euler's integral formula for the harmonic numbers.
The digamma function can be computed in the complex plane outside negative integers (Abramowitz and Stegun 6.3.16), using
or
This can be utilized to evaluate infinite sums of rational functions, i.e.,
where p(n) and q(n) are polynomials of n.
Performing partial fraction on un in the complex field, in the case when all roots of q(n) are simple roots,
For the series to converge,
otherwise the series will be greater than the harmonic series and thus diverge. Hence
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