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 ψ -->
1
(
z
)
=
d
2
d
z
2
ln
-->
Γ Γ -->
(
z
)
{\displaystyle \psi _{1}(z)={\frac {d^{2}}{dz^{2}}}\ln \Gamma (z)}
It follows from this definition that
ψ ψ -->
1
(
z
)
=
d
d
z
ψ 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 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 ψ -->
1
(
z
)
=
ζ 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 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 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 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 ψ -->
1
(
z
+
1
)
=
ψ ψ -->
1
(
z
)
− − -->
1
z
2
{\displaystyle \psi _{1}(z+1)=\psi _{1}(z)-{\frac {1}{z^{2}}}}
and the reflection formula
ψ ψ -->
1
(
1
− − -->
z
)
+
ψ ψ -->
1
(
z
)
=
π π -->
2
sin
2
-->
π 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 ψ -->
1
(
1
2
)
=
π π -->
2
2
{\displaystyle \psi _{1}({\tfrac {1}{2}})={\tfrac {\pi ^{2}}{2}}}
Special values
At positive integer values we have that
ψ ψ -->
1
(
n
)
=
π π -->
2
6
− − -->
∑ ∑ -->
k
=
1
n
− − -->
1
1
k
2
,
ψ ψ -->
1
(
1
)
=
π π -->
2
6
,
ψ ψ -->
1
(
2
)
=
π π -->
2
6
− − -->
1
,
ψ ψ -->
1
(
3
)
=
π π -->
2
6
− − -->
5
4
.
{\displaystyle \psi _{1}(n)={\frac {\pi ^{2}}{6}}-\sum _{k=1}^{n-1}{\frac {1}{k^{2}}},\qquad \psi _{1}(1)={\frac {\pi ^{2}}{6}},\qquad \psi _{1}(2)={\frac {\pi ^{2}}{6}}-1,\qquad \psi _{1}(3)={\frac {\pi ^{2}}{6}}-{\frac {5}{4}}.}
ψ ψ -->
1
(
n
+
1
2
)
=
π π -->
2
2
− − -->
4
∑ ∑ -->
k
=
1
n
1
(
2
k
− − -->
1
)
2
,
ψ ψ -->
1
(
1
2
)
=
π π -->
2
2
,
ψ ψ -->
1
(
3
2
)
=
π π -->
2
2
− − -->
4.
{\displaystyle \psi _{1}\left(n+{\frac {1}{2}}\right)={\frac {\pi ^{2}}{2}}-4\sum _{k=1}^{n}{\frac {1}{(2k-1)^{2}}},\qquad \psi _{1}\left({\tfrac {1}{2}}\right)={\frac {\pi ^{2}}{2}},\qquad \psi _{1}\left({\tfrac {3}{2}}\right)={\frac {\pi ^{2}}{2}}-4.}
The trigamma function has other special values such as:
ψ ψ -->
1
(
1
4
)
=
π π -->
2
+
8
G
{\displaystyle \psi _{1}\left({\tfrac {1}{4}}\right)=\pi ^{2}+8G}
where G represents Catalan's constant .
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
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 ψ -->
1
(
p
q
)
=
π π -->
2
2
sin
2
-->
(
π π -->
p
/
q
)
+
2
q
∑ ∑ -->
m
=
1
(
q
− − -->
1
)
/
2
sin
-->
(
2
π π -->
m
p
q
)
Cl
2
(
2
π 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:[ 2]
∑ ∑ -->
n
=
1
∞ ∞ -->
n
2
− − -->
1
2
(
n
2
+
1
2
)
2
(
ψ ψ -->
1
(
n
− − -->
i
2
)
+
ψ ψ -->
1
(
n
+
i
2
)
)
=
− − -->
1
+
2
4
π π -->
coth
-->
π π -->
2
− − -->
3
π π -->
2
4
sinh
2
-->
π π -->
2
+
π π -->
4
12
sinh
4
-->
π π -->
2
(
5
+
cosh
-->
π 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
Notes
^ 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
