In mathematics , the trigamma function , denoted ψ 1 (z )ψ (1) (z )polygamma functions , and is defined by
Color representation of the trigamma function, ψ 1 (z )domain coloring method.
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 .
Calculation
edit
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 is
ψ
1
(
z
)
=
1
z
+
1
2
z
2
+
∑
k
=
1
∞
B
2
k
z
2
k
+
1
=
∑
k
=
0
∞
B
k
z
k
+
1
{\displaystyle \psi _{1}(z)={\frac {1}{z}}+{\frac {1}{2z^{2}}}+\sum _{k=1}^{\infty }{\frac {B_{2k}}{z^{2k+1}}}=\sum _{k=0}^{\infty }{\frac {B_{k}}{z^{k+1}}}}
if we have chosen B 1 = 1 / 2 Bernoulli numbers of the second kind.
Recurrence and reflection formulae
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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
edit
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).}
Computation and approximation
edit
An easy method to approximate the trigamma function is to take the derivative of the asymptotic expansion of the digamma function .
ψ
1
(
x
)
≈
1
x
+
1
2
x
2
+
1
6
x
3
−
1
30
x
5
+
1
42
x
7
−
1
30
x
9
+
5
66
x
11
−
691
2730
x
13
+
7
6
x
15
{\displaystyle \psi _{1}(x)\approx {\frac {1}{x}}+{\frac {1}{2x^{2}}}+{\frac {1}{6x^{3}}}-{\frac {1}{30x^{5}}}+{\frac {1}{42x^{7}}}-{\frac {1}{30x^{9}}}+{\frac {5}{66x^{11}}}-{\frac {691}{2730x^{13}}}+{\frac {7}{6x^{15}}}}
Appearance
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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).}
See also
edit
^ 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 .
References
edit
![{\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)
