Polygamma function

In mathematics, the Polygamma functions are a set of special functions, which are defined as the derivatives of the function. This refers to the gamma function, and the natural logarithm.

The first two are called Polygammafunktionen Digammafunktion and Trigammafunktion.

Notation

The Polygammafunktionen be designated by the small Greek letter Psi (). In the first Polygammafunktion, the Digammafunktion, the index is usually omitted or as a set. The second Polygammafunktion, so the Trigammafunktion is then referred to as ( or rarely ); it is the second derivative of. In general, the - te Polygammafunktion or Polygammafunktion the order with or referred to, and as the -th derivative of defined.

Definition and other representations

It is

With the Digammafunktion. Such derivatives are referred to as derivatives of logarithmic.

An integral representation is

For and

Properties

Difference equations

The Polygammafunktionen have the difference equations

Reflection formula

Another important relationship is

Multiplication formula

The multiplication formula is given by

For case, the Digammafunktion, see there.

Series representations

A series representation of Polygammafunktion is

Where and any complex number except the negative integers. The formula is easier to write using the Hurwitz zeta function as

The generalization of the Polygammafunktionen to any non- whole systems is set forth below.

Another series representation is

Where the Kronecker delta denotes that follows from the decomposition of the gamma function by the Weierstrass product theorem.

The Taylor series for is given by

Which converges for. designated it 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 and Dirichlet Catalansche constant β function; e.g.

General shall also apply:

The m-th derivative of the tangent can also be expressed with the Polygammafunktion:

Generalized Polygammafunktion

The generalized Polygammafunktion met and the functional equation

The Euler - Mascheroni constant. because of

For integer indicated further above differential equation is included for natural.

With the aid of the Hurwitz function we obtain the relation

Which satisfies the functional equation.

As a consequence, can the doubling formula

Derived. A generalization of these is

Which is equivalent to Gaussian multiplication formula for the gamma function and contains the multiplication formula as a special case for.

Q- Polygammafunktion

The - Polygammafunktion is defined by

Credentials

• Milton Abramowitz and Irene Stegun, Handbook of Mathematical Functions, (1964 ) Dover Publications, New York. ISBN 978-0-486-61272-0. See § 6.4
• Eric W. Weisstein: Polygamma Function at MathWorld, in functions.wolfram.com in references.worlfram.com.

• Analytical function