Trigamma function

In mathematics, the Trigamma function is the second Polygammafunktion; the first Polygammafunktion is the Digammafunktion. The Trigammafunktion is therefore a specific function and is commonly referred to and defined as the second derivative of the function, wherein the gamma function called.

Definition and other representations

The definition is:

It follows from the connection with the Digammafunktion that

Is the derivative of the Trigammafunktion Digammafunktion.

From the sum representation

Follows, that is a special case of Trigammafunktion Hurwitz function.

A representation as a double integral

Also, applies

Calculation and properties

The asymptotic calculation includes the Bernoulli numbers:

Although the series is not convergent with, however, this formula does not set too large a very good approximation dar. is greater, the greater can be selected.

The recursion formula of Trigammafunktion is:

The functional equation of the form of a reflection Trigammafunktion equation and is given by:

Here is the cosecant.

Special values

The following is a listing of some special values ​​of the Trigammafunktion, the Catalansche constant, the Riemann zeta function and the Clausen function called.

Credentials

• Milton Abramowitz and Irene A. Stegun, Handbook of Mathematical Functions, (1964 ) Dover Publications, New York. ISBN 0-486-61272-4. Section § 6.4
• Eric W. Weisstein: Trigamma Function. In: MathWorld (English).

• Analytical function