Digamma function

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File:Complex Polygamma 0.jpg
Digamma function 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

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

External links

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

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