File:Complex Polygamma 0.jpg
in the complex plane
. The color of a point
encodes the value of
. 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:
It is the first of the polygamma functions.
Relation to harmonic numbers
The digamma function, often denoted also as ψ0(x), ψ0(x) or (after the shape of the obsolete Greek letter Ϝ digamma), is related to the harmonic numbers in that
where Hn is the n 'th harmonic number, and γ is the Euler-Mascheroni constant. For half-integer values, it may be expressed as
It has the integral representation
This may be written as
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
which converges for |z|<1. Here, 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:
where is the binomial coefficient.
The digamma function satisfies a reflection formula similar to that of the Gamma function,
The digamma function satisfies the recurrence relation
Thus, it can be said to "telescope" 1/x, for one has
where Δ is the forward difference operator. This satisfies the recurrence relation of a partial sum of the harmonic series, thus implying the formula
where is the Euler-Mascheroni constant.
More generally, one has
The digamma has a Gaussian sum of the form
for integers . Here, ζ(s,q) is the Hurwitz zeta function and is a Bernoulli polynomial. A special case of the multiplication theorem is
and a neat generalization of this is
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
The digamma function has the following special values:
- Cephes - C and C++ language special functions math library