Trigamma function explained

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

\psi _{1(z)} = (d ^{2})/(dz ^{2}) ln \Gamma(z) .

It follows from this definition that

\psi _{1(z)} = (d)/(dz) \psi(z)

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

\psi _{1(z)} = \sum ^{inf ty} _{n =0} (1)/((z + n) ^{2}) ,

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 ^{x (x ^{z-1})/(y( 1- x))} _{0} dydx using the formula for the sum of a geometric series. Integration over yields:\psi _{1(z)} = - \int ^{1 (x ^{z-1} ln {x)/()} _{0} }\,dx

An asymptotic expansion as a Laurent series can be obtained via the derivative of the asymptotic expansion of the digamma function:

\begin{align} \psi _{1(z) &\sim} {\operatorname{d}\over\operatorname{d}z} \left( ln z - \sum ^{inf ty} _{n =1} (B _{n})/(nz ^{n}) \right) \\ & = (1)/(z) + \sum ^{inf ty} _{n =1} (B _{n})/(z ^{n+1}) = \sum ^{inf ty} _{n =0} (B _{n})/(z ^{n+1}) \\ & = (1)/(z) + (1)/(2 z ^{2}) + (1)/(6 z ^{3}) - (1)/(30 z ^{5}) + (1)/(42 z ^{7}) - (1)/(30 z ^{9}) + (5)/(66 z ^{11}) - (691)/(2730 z ^{13}) + (7)/(6 z ^{15}) … \end{align} where is the th Bernoulli number and we choose .

Recurrence and reflection formulae

The trigamma function satisfies the recurrence relation

\psi _{1(z} + 1 ) = \psi _{1(z)} - (1)/(z ^{2})

and the reflection formula

\psi _{1(1} - z) + \psi _{1(z)} = (\pi ^{2})/(\sin ^{2} \pi z)

which immediately gives the value for z :

\psi ^{2 }{ 2 }} _{1 (\tfrac{ 1 }{ 2 }) = \tfrac{\pi} .

Special values

At positive half integer values we have that

\psi _{1 \left(n + (12 \right) = (\pi ^{2})/(2))/(-4 \sum)} ^{n (1)/(( 2 k -1 ) ^{2})} ^{k =1} .

Moreover, the trigamma function has the following special values:

\begin{align} \psi _{1\left(\tfrac14\right)} & = \pi ^{2} + 8 G & \psi _{1\left(\tfrac12\right)} & = (\pi ^{2})/(2) & \psi _{1(1)} & = (\pi ^{2})/(6) \\[ 6 px] \psi _{1\left(\tfrac32\right)} & = (\pi ^{2})/(2) - 4 & \psi _{1(2)} & = (\pi ^{2})/(6) - 1 \\ \psi _{1(n)} & = (\pi ^{2})/(6) - \sum ^{n -1} _{k =1} (1)/(k ^{2}) \end{align}

where represents Catalan's constant and is a positive integer.

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,^[1]

\psi _{1 \left( (p)/(q)} \right) = (\pi ^{2})/(2 \sin ^{2(\pi} p/q)) +2 q\sum ^{(q -1 )/ 2} _{m =1} \sin\left( (2 \pi mp)/(q) \right) rm{Cl} _{2 \left( (2 \pi m)/(q)} \right).

Appearance

The trigamma function appears in this sum formula:

\sum ^{inf ty (n ^{2- (12)/()})/(\left(n ^{2+ (12 \right))/(^{2})})} _{n =1} \left(\psi _{1 ( n - (i)/(\sqrt{ 2)} }\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

References

• Milton Abramowitz and Irene A. Stegun, Handbook of Mathematical Functions, (1964) Dover Publications, New York. . See section §6.4
• Eric W. Weisstein. Trigamma Function -- from MathWorld--A Wolfram Web Resource

Notes and References

1. Book: Structural properties of polylogarithms. Lewin. L. . American Mathematical Society. 1991. 978-0821816349.