In mathematics, the Jacobi zeta function Z (u ) is the logarithmic derivative of the Jacobi theta function Θ(u). It is also commonly denoted as
zn
(
u
,
k
)
{\displaystyle \operatorname {zn} (u,k)}
[ 1]
Θ
(
u
)
=
Θ
4
(
π
u
2
K
)
{\displaystyle \Theta (u)=\Theta _{4}\left({\frac {\pi u}{2K}}\right)}
Z
(
u
)
=
∂
∂
u
ln
Θ
(
u
)
{\displaystyle Z(u)={\frac {\partial }{\partial u}}\ln \Theta (u)}
=
Θ
′
(
u
)
Θ
(
u
)
{\displaystyle ={\frac {\Theta '(u)}{\Theta (u)}}}
[ 2]
Z
(
ϕ
|
m
)
=
E
(
ϕ
|
m
)
−
E
(
m
)
K
(
m
)
F
(
ϕ
|
m
)
{\displaystyle Z(\phi |m)=E(\phi |m)-{\frac {E(m)}{K(m)}}F(\phi |m)}
[ 3] Where E, K, and F are generic Incomplete Elliptical Integrals of the first and second kind. Jacobi Zeta Functions being kinds of Jacobi theta functions have applications to all their relevant fields and application.
zn
(
u
,
k
)
=
Z
(
u
)
=
∫
0
u
dn
2
v
−
E
K
d
v
{\displaystyle \operatorname {zn} (u,k)=Z(u)=\int _{0}^{u}\operatorname {dn} ^{2}v-{\frac {E}{K}}dv}
[ 1] This relates Jacobi's common notation of,
dn
u
=
1
−
m
sin
θ
2
{\displaystyle \operatorname {dn} {u}={\sqrt {1-m\sin {\theta }^{2}}}}
sn
u
=
sin
θ
{\displaystyle \operatorname {sn} u=\sin {\theta }}
cn
u
=
cos
θ
{\displaystyle \operatorname {cn} u=\cos {\theta }}
[ 1]
Some additional relations include ,
zn
(
u
,
k
)
=
π
2
K
Θ
1
′
π
u
2
K
Θ
1
π
u
2
K
−
cn
u
dn
u
sn
u
{\displaystyle \operatorname {zn} (u,k)={\frac {\pi }{2K}}{\frac {\Theta _{1}'{\frac {\pi u}{2K}}}{\Theta _{1}{\frac {\pi u}{2K}}}}-{\frac {\operatorname {cn} {u}\,\operatorname {dn} {u}}{\operatorname {sn} {u}}}}
[ 1]
zn
(
u
,
k
)
=
π
2
K
Θ
2
′
π
u
2
K
Θ
2
π
u
2
K
−
sn
u
dn
u
cn
u
{\displaystyle \operatorname {zn} (u,k)={\frac {\pi }{2K}}{\frac {\Theta _{2}'{\frac {\pi u}{2K}}}{\Theta _{2}{\frac {\pi u}{2K}}}}-{\frac {\operatorname {sn} {u}\,\operatorname {dn} {u}}{\operatorname {cn} {u}}}}
[ 1]
zn
(
u
,
k
)
=
π
2
K
Θ
3
′
π
u
2
K
Θ
3
π
u
2
K
−
k
2
sn
u
cn
u
dn
u
{\displaystyle \operatorname {zn} (u,k)={\frac {\pi }{2K}}{\frac {\Theta _{3}'{\frac {\pi u}{2K}}}{\Theta _{3}{\frac {\pi u}{2K}}}}-k^{2}{\frac {\operatorname {sn} {u}\,\operatorname {cn} {u}}{\operatorname {dn} {u}}}}
[ 1]
zn
(
u
,
k
)
=
π
2
K
Θ
4
′
π
u
2
K
Θ
4
π
u
2
K
{\displaystyle \operatorname {zn} (u,k)={\frac {\pi }{2K}}{\frac {\Theta _{4}'{\frac {\pi u}{2K}}}{\Theta _{4}{\frac {\pi u}{2K}}}}}
[ 1]
References
^ a b c d e f g Gradshteyn, Ryzhik, I.S., I.M. "Table of Integrals, Series, and Products" (PDF) . booksite.com . {{cite web }}: CS1 maint: multiple names: authors list (link )^ Abramowitz, Milton; Stegun, Irene A. (2012-04-30). Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables ISBN 978-0-486-15824-2 ^ Weisstein, Eric W. "Jacobi Zeta Function" . mathworld.wolfram.com . Retrieved 2019-12-02 .
