In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, the trigamma function, denoted ψ₁(z), is the second of the polygamma functions, and is defined by

.

It follows from this definition that

where ψ(z) is the digamma function. It may also be defined as the sum of the series

Series (mathematics)

A series is the sum of the terms of a sequence. Finite sequences and series have defined first and last terms, whereas infinite sequences and series continue indefinitely....

making it a special case of the Hurwitz zeta function

Note that the last two formulæ are valid when 1-z is not a natural number.

Calculation

A double integral representation, as an alternative to the ones given above, may be derived from the series representation:

using the formula for the sum of a geometric series. Integration by parts yields:

An asymptotic expansion in terms of the Bernoulli number

Bernoulli number

In mathematics, the Bernoulli numbers Bn are a sequence of rational numbers with deep connections to number theory. They are closely related to the values of the Riemann zeta function at negative integers....

s is

.

Recurrence and reflection formulae

The trigamma function satisfies the recurrence relation

Recurrence relation

In mathematics, a recurrence relation is an equation that recursively defines a sequence, once one or more initial terms are given: each further term of the sequence is defined as a function of the preceding terms....

:

and the reflection formula

Reflection formula

In mathematics, a reflection formula or reflection relation for a function f is a relationship between f and f...

:

Special values

The trigamma function has the following special values:

where K represents Catalan's constant.

See also

• Gamma function
  Gamma function
  In mathematics, the gamma function is an extension of the factorial function, with its argument shifted down by 1, to real and complex numbers...
  
  
• Digamma function
• Polygamma function
• Catalan's constant