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 HarmonicNumber

The best-known properties and formulas for differentiated gamma functions

Real values for real arguments

For real values of the argument and nonnegative integer , the differentiated gamma functions , , , and are real (or infinity). The function is real (or infinity) for real values of argument and integer .

Simple values at zero

The differentiated gamma functions , , , and have simple values for zero arguments:

Values at fixed points

The differentiated gamma functions , , , and with rational arguments can sometimes be evaluated through classical constants and logarithms, for example:

Specific values for specialized variables

The previous relations are particular cases of the following general formulas:

The differentiated gamma functions and with integer parameters and have the following representations:

Analyticity

The digamma function and the harmonic number are defined for all complex values of the variable . The functions and are analytical functions of and over the whole complex ‐ and ‐planes. For fixed , the generalized harmonic number is an entire function of .

Poles and essential singularities

The differentiated gamma functions and have an infinite set of singular points , where for and for . These points are the simple poles with residues . The point is the accumulation point of poles for the functions , , and (with fixed nonnegative integer ), which means that is an essential singular point.

For fixed nonnegative integer , the function has an infinite set of singular points: are the simple poles with residues ; and are the poles of order with residues .

For fixed , the function does not have poles and the function has only one singular point at , which is an essential singular point.

Branch points and branch cuts

The functions and do not have branch points and branch cuts.

For integer , the function does not have branch points and branch cuts.

For fixed noninteger , the function has two singular branch points and , and it is a single‐valued function on the ‐plane cut along the interval , where it is continuous from above:

For fixed , the functions and do not have branch points and branch cuts.

Periodicity

The differentiated gamma functions , , , and do not have periodicity.

Parity and symmetry

The differentiated gamma functions , , , and have mirror symmetry:

Series representations

The differentiated gamma functions , , and have the following series expansions near regular points:

Near singular points, the differentiated gamma functions , , , and can be expanded through the following series:

Here is the Euler gamma constant .

Except for the generalized power series, there are other types of series through which differentiated gamma functions , , , and can be represented, for example:

Asymptotic series expansions

The asymptotic behavior of the differentiated gamma functions , , , and can be described by the following formulas (only the main terms of the asymptotic expansions are given):

Integral representations

The differentiated gamma functions , , , and can also be represented through the following equivalent integrals:

Transformations

The following formulas describe some of the transformations that change the differentiated gamma functions into themselves:

Transformations with arguments that are integer multiples take the following forms:

The following transformations represent summation theorems:

Identities

The differentiated gamma functions , , , and satisfy the following recurrence identities:

The previous formulas can be generalized to the following recurrence identities with a jump of length :

Representations of derivatives

The derivatives of the differentiated gamma functions , , , and have rather simple representations:

The corresponding symbolic -order derivatives of all the differentiated gamma functions , , , and can be expressed by the following formulas: