# 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.

## Relation to harmonic numbers

The digamma function, often denoted also as ψ_{0}(*x*), ψ^{0}(*x*) or (after the shape of the obsolete Greek letter Ϝ digamma), is related to the harmonic numbers in that

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

## Integral representations

It has the integral representation

This may be written as

which follows from Euler's integral formula for the harmonic numbers.

## Taylor series

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.

## Newton series

The Newton series for the digamma follows from Euler's integral formula:

where is the binomial coefficient.

## Reflection formula

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

## Recurrence formula

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

## Gaussian sum

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

## Special values

The digamma function has the following special values:

## See also

## References

- Milton Abramowitz and Irene A. Stegun,
*Handbook of Mathematical Functions*, (1964) Dover Publications, New York. ISBN 0486612724 . See section §6.3 - Template:Mathworld

## External links

- Cephes - C and C++ language special functions math library

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