The first polygamma functions in real terms
m = 0 m = 1 m = 2 m = 3 m = 4
In mathematics, the polygamma functions are a series of special functions that are defined as the derivatives of the function . The gamma function and the natural logarithm denote .
The first two polygamma functions are called the digamma function and the trigamma function .
Representation of the first five polygamma functions in the complex plane
|
|
|
|
|
|
|
|
|
|
|
|
notation
The polygamma functions are identified with the small Greek letter Psi . In the case of the first polygamma function, the index is usually left out or defined as 0; it is called the digamma function . The second polygamma function, i.e. the trigamma function , has the symbol (or less often ) and is the second derivative of . In general, the is -th polygamma function or polygamma function of the order with or referred to and than the th derivative of defined.
Definition and further representations
It is
with the digamma function . Such derivatives are also referred to as logarithmic derivatives of .
An integral representation is
for and
properties
Difference equations
The polygamma functions have the difference equations
Reflection formula
Another important relationship is
Multiplication formula
The multiplication formula is given for by
For the case of the digamma function , see there.
Series representations
A series representation of the polygamma function is
where and is any complex number other than negative integers. The formula is easier to write using Hurwitz's zeta function than
The generalization of the polygamma functions to arbitrary, non-whole orders is given below .
Another series representation is
where the Kronecker delta denotes, which follows from the decomposition of the gamma function according to Weierstrass' product theorem.
The Taylor series um is given by
which converges for. referred to the Riemann zeta function .
Special values
The values of the Polygammafunktionen for rational arguments can usually be expressed using constants and functions such as , square root , Clausen function , Riemann ζ function , Catalan's constant and Dirichlet β function ; z. B.
In general, the following also applies:
-
.
The mth derivative of the tangent can also be expressed using the polygamma function:
-
.
In addition, special values of polygamma functions have repeatedly proven useful as universal constants in a closed limit value description of series or integrals, for example applies
Generalized polygamma function
The generalized polygamma function satisfies for and the functional equation
where denotes the Euler-Mascheroni constant . Because of
for integers , the difference equation given above for natural is included.
The relationship is then obtained
with the aid of the Hurwitz function
which satisfies the functional equation.
As a consequence, the doubling formula
derive. A generalization of this is
which is an equivalent to the Gaussian multiplication formula of the gamma function and contains the
multiplication formula as a special case for .
q-polygamma function
The polygamma function is defined by
-
.
literature
credentials
-
↑ Oliver Espinosa and Victor H. Moll:
A Generalized Polygamma Function on arXiv.org e-Print archive 2003.
-
↑ Eric W. Weisstein : q-Polygamma Function . In: MathWorld (English).