- Source: Trigamma function
In mathematics, the trigamma function, denoted ψ1(z) or ψ(1)(z), is the second of the 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) is the 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.
Calculation
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 yields:
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 Bn is the nth Bernoulli number and we choose B1 = 1/2.
= Recurrence and reflection formulae
=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}}}
.
= Special values
=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 ψ1, but there exist infinitely many pairs of roots zn, zn for 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, z1 = −0.4121345... + 0.5978119...i and z2 = −1.4455692... + 0.6992608...i are the first two roots with Im(z) > 0.
= 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,
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).}
Appearance
The trigamma function appears in this sum formula:
∑
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).}
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 section §6.4
Eric W. Weisstein. Trigamma Function -- from MathWorld--A Wolfram Web Resource
Kata Kunci Pencarian:
- Fungsi poligamma
- Trigamma function
- Polygamma function
- Psi function
- Digamma function
- Beta distribution
- Catalan's constant
- Wishart distribution
- Dirichlet distribution
- Generalized Pareto distribution
- Generalized normal distribution
