Trigamma function

The trigamma function in the complex number plane . ${\displaystyle \psi _{1}(z)}$

In mathematics, the trigamma function is the second polygamma function ; the first polygamma function is the digamma function . The trigamma function is therefore a special function and is usually referred to as and defined as the second derivative of the function , where the gamma function is referred to. ${\displaystyle \psi }$ ${\displaystyle \psi _{1}}$ ${\displaystyle \ln }$ ${\displaystyle (\Gamma (x))}$ ${\displaystyle \Gamma }$

Definition and further representations

The definition is:

${\displaystyle \psi _{1}(z)={\frac {\mathrm {d} ^{2}}{\mathrm {d} z^{2}}}\ln \Gamma (z).}$

The connection with the digamma function follows from this that ${\displaystyle \psi (z)}$

${\displaystyle \psi _{1}(z)={\frac {\mathrm {d} }{\mathrm {d} z}}\psi (z)}$

the trigamma function is the derivative of the digamma function.

From the totals display

${\displaystyle \psi _{1}(z)=\sum _{n=0}^{\infty }{\frac {1}{(z+n)^{2}}}=\zeta (2,z)}$

it follows that the trigamma function is a special case of the Hurwitz function ${\displaystyle \zeta }$ .

A representation as a double integral is

${\displaystyle \psi _{1}(z)=\int \limits _{0}^{1}{\frac {\mathrm {d} y}{y}}\int \limits _{0}^{y}{\frac {x^{z-1}\,\mathrm {d} x}{1-x}}.}$

Also applies

${\displaystyle \psi _{1}(z)=-\int \limits _{0}^{1}{\frac {x^{z-1}\ln {x}}{1-x}}\,\mathrm {d} x.}$

Calculation and properties

The asymptotic calculation includes the Bernoulli numbers : ${\displaystyle B_{2k}}$

${\displaystyle \psi _{1}(z)\sim {\frac {1}{z}}+{\frac {1}{2z^{2}}}+\sum _{k=1}^{N}{\frac {B_{2k}}{z^{2k+1}}}}$.

Although the series does not converge with any , this formula represents a very good approximation for those that are not selected too large . The larger is, the larger the selection. ${\displaystyle z}$ ${\displaystyle N\to \infty }$ ${\displaystyle N}$ ${\displaystyle |z|}$ ${\displaystyle N}$

The recursion formula of the trigamma function is:

${\displaystyle \psi _{1}(z+1)=\psi _{1}(z)-{\frac {1}{z^{2}}}}$

The functional equation of the trigamma function has the form of a reflection equation and is given by:

${\displaystyle \psi _{1}(1-z)+\psi _{1}(z)=\pi ^{2}\csc ^{2}(\pi z).\,}$

Here is the cosecan . ${\displaystyle \csc }$

Special values

The following is a listing of some special values of Trigammafunktion, with the Catalan's constant , the Riemann zeta function and the Clausen function called. ${\displaystyle G}$ ${\displaystyle \zeta (x)}$ ${\displaystyle {\rm {{Cl}_{2}}}}$

${\displaystyle {\begin{aligned}&\psi _{1}\left({\tfrac {1}{4}}\right)&={}&\pi ^{2}+8G\\&\psi _{1}\left({\tfrac {1}{3}}\right)&={}&{\tfrac {2}{3}}\pi ^{2}+3{\sqrt {3}}\cdot {\rm {{Cl}_{2}({\tfrac {2}{3}}\pi )}}\\&\psi _{1}\left({\tfrac {1}{2}}\right)&={}&{\tfrac {1}{2}}\pi ^{2}\\&\psi _{1}\left({\tfrac {2}{3}}\right)&={}&{\tfrac {2}{3}}\pi ^{2}-3{\sqrt {3}}\cdot {\rm {{Cl}_{2}({\tfrac {2}{3}}\pi )}}\\&\psi _{1}\left({\tfrac {3}{4}}\right)&={}&\pi ^{2}-8G\\&\psi _{1}\,(1)&={}&\zeta (2)={\tfrac {1}{6}}\pi ^{2}\\&\psi _{1}\left({\tfrac {5}{4}}\right)&={}&\pi ^{2}+8G-16\\&\psi _{1}\left({\tfrac {3}{2}}\right)&={}&{\tfrac {1}{2}}\pi ^{2}-4\\&\psi _{1}\,(2)&={}&{\tfrac {1}{6}}\pi ^{2}-1\end{aligned}}}$

credentials

1. ↑ Eric W. Weisstein : Polygamma Function . In: MathWorld (English).
2. ↑ Eric W. Weisstein : Hurwitz Zeta Function . In: MathWorld (English).
3. ↑ Eric W. Weisstein : Clausen Function . In: MathWorld (English).

• 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).