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Gamma, Beta, Erf > HarmonicNumber[z] > Introduction to the differentiated gamma functions

Definitions of the differentiated gamma functions

The digamma function , polygamma function , harmonic number , and generalized harmonic number are defined by the following formulas (the first formula is a general definition for complex arguments and the second formula is for positive integer arguments):

Here is the Euler gamma constant:

Remark: This formula presents the (not unique) continuation of the classical definition of from positive integer values of to its arbitrary complex values.

For positive integer and arbitrary complex , , the following definitions are commonly used:

The previous definition for uses the Mathematica definition for the Hurwitz zeta function . Branch cuts and related properties are thus inherited from .

The previous functions are interconnected and belong to the differentiated gamma functions group. These functions are widely used in the coefficients of series expansions for many mathematical functions (especially the so‐called logarithmic cases).