Trigamma Function

In mathematics, the trigamma function, denoted or , is the second of the polygamma functions, and is defined by

: $\psi_1(z) = \frac \ln\Gamma(z)$.

It follows from this definition that

: $\psi_1(z) = \frac \psi(z)$

where is the digamma function. It may also be defined as the sum of the series

: $\psi_1(z) = \sum_^\frac,$

making it a special case of the Hurwitz zeta function

: $\psi_1(z) = \zeta(2,z).$

Note that the last two formulas are valid when 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:

: $\psi_1(z) = \int_0^1\!\!\int_0^x\frac\,dy\,dx$

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

: $\psi_1(z) = -\int_0^1\frac\,dx$

An asymptotic expansion as a Laurent series is

: $\psi_1(z) = \frac + \frac + \sum_^\frac = \sum_^\frac$

if we have chosen , i.e. the Bernoulli numbers of the second kind.

Recurrence and reflection formulae

The trigamma function satisfies the recurrence relation

: $\psi_1(z + 1) = \psi_1(z) - \frac$

and the reflection formula

: $\psi_1(1 - z) + \psi_1(z) = \frac \,$

which immediately gives the value for ''z'' : $\psi_1(\tfrac)=\tfrac$.

Special values

At positive half integer values we have that
:$\psi_1\left(n+\frac12\right)=\frac-4\sum_^n\frac.$

Moreover, the trigamma function has the following special values:

: \begin
\psi_1\left(\tfrac14\right) &= \pi^2 + 8G \quad &
\psi_1\left(\tfrac12\right) &= \frac &
\psi_1(1) &= \frac \\ px\psi_1\left(\tfrac32\right) &= \frac - 4 &
\psi_1(2) &= \frac - 1 \quad
\end

where represents Catalan's constant.

There are no roots on the real axis of , but there exist infinitely many pairs of roots for . Each such pair of roots approaches quickly and their imaginary part increases slowly logarithmic with . For example, and are the first two roots with .

Relation to the Clausen function

The digamma function at rational arguments can be expressed in terms of trigonometric functions and logarithm by the digamma theorem. A similar result holds for the trigamma function but the circular functions are replaced by Clausen's function. Namely,
:$\psi_1\left(\frac\right)=\frac+2q\sum_^\sin\left(\frac\right)\textrm_2\left(\frac\right).$

Computation and approximation

An easy method to approximate the trigamma function is to take the derivative of the asymptotic expansion of the digamma function.

:$\psi_1(x) \approx \frac + \frac + \frac - \frac + \frac - \frac + \frac - \frac + \frac$

Appearance

The trigamma function appears in this surprising sum formula:

: $\sum_^\infty\frac\left(\psi_1\bigg(n-\frac\bigg)+\psi_1\bigg(n+\frac\bigg)\right)= -1+\frac\pi\coth\frac-\frac+\frac\left(5+\cosh\pi\sqrt\right).$

See also

* Gamma function
* Digamma function
* Polygamma function
* Catalan's constant

Notes

References

* Milton Abramowitz and Irene A. Stegun, '' Handbook of Mathematical Functions'', (1964) Dover Publications, New York. {{ISBN, 0-486-61272-4. See sectio
§6.4
* Eric W. Weisstein


Gamma and related functions