28 year-old Painting Investments Worker Truman from Regina, usually spends time with pastimes for instance interior design, property developers in new launch ec Singapore and writing. Last month just traveled to City of the Renaissance.
Trigamma function
ψ
1
(
z
)
{\displaystyle \psi _{1}(z)}
complex plane . The color of a point
z
{\displaystyle z}
ψ
1
(
z
)
{\displaystyle \psi _{1}(z)}
argument . In mathematics , the trigamma function , denoted
ψ
1
(
z
)
{\displaystyle \psi _{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
)
{\displaystyle \psi (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).{\frac {}{}}}
Note that the last two formulæ are valid when
1
−
z
{\displaystyle 1-z}
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
y
x
z
−
1
y
1
−
x
d
x
d
y
{\displaystyle \psi _{1}(z)=\int _{0}^{1}\int _{0}^{y}{\frac {x^{z-1}y}{1-x}}\,dx\,dy}
using the formula for the sum of a geometric series . Integration by parts 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 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
{\displaystyle B_{1}=1/2}
Bernoulli numbers of the second kind.
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.
Special values The trigamma function has the following special values:
ψ
1
(
1
4
)
=
π
2
+
8
K
{\displaystyle \psi _{1}\left({\frac {1}{4}}\right)=\pi ^{2}+8K}
ψ
1
(
1
2
)
=
π
2
2
{\displaystyle \psi _{1}\left({\frac {1}{2}}\right)={\frac {\pi ^{2}}{2}}}
ψ
1
(
1
)
=
π
2
6
{\displaystyle \psi _{1}(1)={\frac {\pi ^{2}}{6}}}
ψ
1
(
3
2
)
=
π
2
2
−
4
{\displaystyle \psi _{1}\left({\frac {3}{2}}\right)={\frac {\pi ^{2}}{2}}-4}
ψ
1
(
2
)
=
π
2
6
−
1
{\displaystyle \psi _{1}(2)={\frac {\pi ^{2}}{6}}-1}
where K represents Catalan's constant .
There are no roots on the real axis of
ψ
1
{\displaystyle \psi _{1}}
z
n
,
z
n
¯
{\displaystyle z_{n},{\overline {z_{n}}}}
ℜ
(
z
)
<
0
{\displaystyle \Re (z)<0}
ℜ
(
z
n
)
=
−
n
+
1
/
2
{\displaystyle \Re (z_{n})=-n+1/2}
z
1
=
−
0.4121345
…
+
i
0.5978119
…
{\displaystyle z_{1}=-0.4121345\ldots +i0.5978119\ldots }
z
2
=
−
1.4455692
…
+
i
0.6992608
…
{\displaystyle z_{2}=-1.4455692\ldots +i0.6992608\ldots }
ℑ
(
z
)
>
0
{\displaystyle \Im (z)>0}
Appearance The trigamma function appears in the next surprising sum formula:[1]
∑
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}\left(n-{\frac {i}{\sqrt {2}}}\right)+\psi _{1}\left(n+{\frac {i}{\sqrt {2}}}\right)\right]=-1+{\frac {\sqrt {2}}{4}}\pi \coth \left({\frac {\pi }{\sqrt {2}}}\right)-{\frac {3\pi ^{2}}{4\sinh ^{2}\left({\frac {\pi }{\sqrt {2}}}\right)}}+{\frac {\pi ^{4}}{12\sinh ^{4}\left({\frac {\pi }{\sqrt {2}}}\right)}}\left(5+\cosh \left(\pi {\sqrt {2}}\right)\right).}
See also Notes 43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.
References æ
hu:Trigamma-függvény
↑ Cite error: Invalid <ref> tag; no text was provided for refs named mezo 
![{\displaystyle \sum _{n=1}^{\infty }{\frac {n^{2}-{\frac {1}{2}}}{\left(n^{2}+{\frac {1}{2}}\right)^{2}}}\left[\psi _{1}\left(n-{\frac {i}{\sqrt {2}}}\right)+\psi _{1}\left(n+{\frac {i}{\sqrt {2}}}\right)\right]=-1+{\frac {\sqrt {2}}{4}}\pi \coth \left({\frac {\pi }{\sqrt {2}}}\right)-{\frac {3\pi ^{2}}{4\sinh ^{2}\left({\frac {\pi }{\sqrt {2}}}\right)}}+{\frac {\pi ^{4}}{12\sinh ^{4}\left({\frac {\pi }{\sqrt {2}}}\right)}}\left(5+\cosh \left(\pi {\sqrt {2}}\right)\right).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b2e5fc1734f15e086cc222ae134cc567092a7d4b)