Mathematical function
Color representation of the trigamma function, ψ 1 (z )domain coloring method. In mathematics , the trigamma function , denoted ψ 1 (z )ψ (1) (z )polygamma functions , and is defined by
ψ 1 ( z ) = d 2 d z 2 ln Γ ( z ) {\displaystyle \psi _{1}(z)={\frac {d^{2}}{dz^{2}}}\ln \Gamma (z)} It follows from this definition that
ψ 1 ( z ) = d d z ψ ( z ) {\displaystyle \psi _{1}(z)={\frac {d}{dz}}\psi (z)} where ψ (z )digamma function . It may also be defined as the sum of the series
ψ 1 ( z ) = ∑ n = 0 ∞ 1 ( z + n ) 2 , {\displaystyle \psi _{1}(z)=\sum _{n=0}^{\infty }{\frac {1}{(z+n)^{2}}},} making it a special case of the Hurwitz zeta function
ψ 1 ( z ) = ζ ( 2 , z ) . {\displaystyle \psi _{1}(z)=\zeta (2,z).} Note that the last two formulas are valid when 1 − z is not a natural number .
A double integral representation, as an alternative to the ones given above, may be derived from the series representation:
ψ 1 ( z ) = ∫ 0 1 ∫ 0 x x z − 1 y ( 1 − x ) d y d x {\displaystyle \psi _{1}(z)=\int _{0}^{1}\!\!\int _{0}^{x}{\frac {x^{z-1}}{y(1-x)}}\,dy\,dx} using the formula for the sum of a geometric series . Integration over y
ψ 1 ( z ) = − ∫ 0 1 x z − 1 ln x 1 − x d x {\displaystyle \psi _{1}(z)=-\int _{0}^{1}{\frac {x^{z-1}\ln {x}}{1-x}}\,dx} An asymptotic expansion as a Laurent series can be obtained via the derivative of the asymptotic expansion of the digamma function :
ψ 1 ( z ) ∼ d d z ( ln z − ∑ n = 1 ∞ B n n z n ) = 1 z + ∑ n = 1 ∞ B n z n + 1 = ∑ 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 ⋯ {\displaystyle {\begin{aligned}\psi _{1}(z)&\sim {\operatorname {d} \over \operatorname {d} \!z}\left(\ln z-\sum _{n=1}^{\infty }{\frac {B_{n}}{nz^{n}}}\right)\\&={\frac {1}{z}}+\sum _{n=1}^{\infty }{\frac {B_{n}}{z^{n+1}}}=\sum _{n=0}^{\infty }{\frac {B_{n}}{z^{n+1}}}\\&={\frac {1}{z}}+{\frac {1}{2z^{2}}}+{\frac {1}{6z^{3}}}-{\frac {1}{30z^{5}}}+{\frac {1}{42z^{7}}}-{\frac {1}{30z^{9}}}+{\frac {5}{66z^{11}}}-{\frac {691}{2730z^{13}}}+{\frac {7}{6z^{15}}}\cdots \end{aligned}}} where B n n Bernoulli number and we choose B 1 = 1 / 2
The trigamma function satisfies the recurrence relation
ψ 1 ( z + 1 ) = ψ 1 ( z ) − 1 z 2 {\displaystyle \psi _{1}(z+1)=\psi _{1}(z)-{\frac {1}{z^{2}}}} and the reflection formula
ψ 1 ( 1 − z ) + ψ 1 ( z ) = π 2 sin 2 π z {\displaystyle \psi _{1}(1-z)+\psi _{1}(z)={\frac {\pi ^{2}}{\sin ^{2}\pi z}}\,} which immediately gives the value for z = 1 / 2 : ψ 1 ( 1 2 ) = π 2 2 {\displaystyle \psi _{1}({\tfrac {1}{2}})={\tfrac {\pi ^{2}}{2}}}
At positive half integer values we have that
ψ 1 ( n + 1 2 ) = π 2 2 − 4 ∑ k = 1 n 1 ( 2 k − 1 ) 2 . {\displaystyle \psi _{1}\left(n+{\frac {1}{2}}\right)={\frac {\pi ^{2}}{2}}-4\sum _{k=1}^{n}{\frac {1}{(2k-1)^{2}}}.} Moreover, the trigamma function has the following special values:
ψ 1 ( 1 4 ) = π 2 + 8 G ψ 1 ( 1 2 ) = π 2 2 ψ 1 ( 1 ) = π 2 6 ψ 1 ( 3 2 ) = π 2 2 − 4 ψ 1 ( 2 ) = π 2 6 − 1 ψ 1 ( n ) = π 2 6 − ∑ k = 1 n − 1 1 k 2 {\displaystyle {\begin{aligned}\psi _{1}\left({\tfrac {1}{4}}\right)&=\pi ^{2}+8G\quad &\psi _{1}\left({\tfrac {1}{2}}\right)&={\frac {\pi ^{2}}{2}}&\psi _{1}(1)&={\frac {\pi ^{2}}{6}}\\[6px]\psi _{1}\left({\tfrac {3}{2}}\right)&={\frac {\pi ^{2}}{2}}-4&\psi _{1}(2)&={\frac {\pi ^{2}}{6}}-1\\\psi _{1}(n)&={\frac {\pi ^{2}}{6}}-\sum _{k=1}^{n-1}{\frac {1}{k^{2}}}\end{aligned}}} where G represents Catalan's constant and n is a positive integer.
There are no roots on the real axis of ψ 1 zn , zn Re z < 0 . Each such pair of roots approaches Re zn = −n + 1 / 2 quickly and their imaginary part increases slowly logarithmic with n . For example, z 1 = −0.4121345... + 0.5978119...i z 2 = −1.4455692... + 0.6992608...i Im(z ) > 0 .
Relation to the Clausen function [ edit ] 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]
ψ 1 ( p q ) = π 2 2 sin 2 ( π p / q ) + 2 q ∑ m = 1 ( q − 1 ) / 2 sin ( 2 π m p q ) Cl 2 ( 2 π m q ) . {\displaystyle \psi _{1}\left({\frac {p}{q}}\right)={\frac {\pi ^{2}}{2\sin ^{2}(\pi p/q)}}+2q\sum _{m=1}^{(q-1)/2}\sin \left({\frac {2\pi mp}{q}}\right){\textrm {Cl}}_{2}\left({\frac {2\pi m}{q}}\right).} The trigamma function appears in this sum formula:[ 2]
∑ n = 1 ∞ n 2 − 1 2 ( n 2 + 1 2 ) 2 ( ψ 1 ( n − i 2 ) + ψ 1 ( n + i 2 ) ) = − 1 + 2 4 π coth π 2 − 3 π 2 4 sinh 2 π 2 + π 4 12 sinh 4 π 2 ( 5 + cosh π 2 ) . {\displaystyle \sum _{n=1}^{\infty }{\frac {n^{2}-{\frac {1}{2}}}{\left(n^{2}+{\frac {1}{2}}\right)^{2}}}\left(\psi _{1}{\bigg (}n-{\frac {i}{\sqrt {2}}}{\bigg )}+\psi _{1}{\bigg (}n+{\frac {i}{\sqrt {2}}}{\bigg )}\right)=-1+{\frac {\sqrt {2}}{4}}\pi \coth {\frac {\pi }{\sqrt {2}}}-{\frac {3\pi ^{2}}{4\sinh ^{2}{\frac {\pi }{\sqrt {2}}}}}+{\frac {\pi ^{4}}{12\sinh ^{4}{\frac {\pi }{\sqrt {2}}}}}\left(5+\cosh \pi {\sqrt {2}}\right).} ^ Lewin, L., ed. (1991). Structural properties of polylogarithms . American Mathematical Society. ISBN 978-0821816349 ^ Mező, István (2013). "Some infinite sums arising from the Weierstrass Product Theorem". Applied Mathematics and Computation . 219 (18): 9838–9846. doi :10.1016/j.amc.2013.03.122 . 
From Wikipedia the free encyclopedia
![{\displaystyle {\begin{aligned}\psi _{1}\left({\tfrac {1}{4}}\right)&=\pi ^{2}+8G\quad &\psi _{1}\left({\tfrac {1}{2}}\right)&={\frac {\pi ^{2}}{2}}&\psi _{1}(1)&={\frac {\pi ^{2}}{6}}\\[6px]\psi _{1}\left({\tfrac {3}{2}}\right)&={\frac {\pi ^{2}}{2}}-4&\psi _{1}(2)&={\frac {\pi ^{2}}{6}}-1\\\psi _{1}(n)&={\frac {\pi ^{2}}{6}}-\sum _{k=1}^{n-1}{\frac {1}{k^{2}}}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4d97086322c90ac70241f16c2f7785113a868a15)
