From Wikipedia, the free encyclopedia
In mathematics, the Neville theta functions , named after Eric Harold Neville ,[ 1] [ 2] [ 3] [ 4]
θ
c
(
z
,
m
)
=
2
π
q
(
m
)
4
∑
k
=
0
∞
(
q
(
m
)
)
k
(
k
+
1
)
cos
(
1
2
⋅
(
2
k
+
1
)
π
z
K
(
m
)
)
1
K
(
m
)
1
m
4
{\displaystyle \theta _{c}(z,m)={\sqrt {2}}{\sqrt {\pi }}{\sqrt[{4}]{q(m)}}\sum _{k=0}^{\infty }(q(m))^{k(k+1)}\cos \left({\frac {1}{2}}\cdot {\frac {(2k+1)\pi z}{K(m)}}\right){\frac {1}{\sqrt {K(m)}}}{\frac {1}{\sqrt[{4}]{m}}}}
θ
d
(
z
,
m
)
=
1
/
2
2
π
(
1
+
2
∑
k
=
1
∞
(
q
(
m
)
)
k
2
cos
(
π
z
k
K
(
m
)
)
)
1
K
(
m
)
{\displaystyle \theta _{d}(z,m)=1/2\,{\sqrt {2}}{\sqrt {\pi }}\left(1+2\,\sum _{k=1}^{\infty }(q(m))^{k^{2}}\cos \left({\frac {\pi zk}{K(m)}}\right)\right){\frac {1}{\sqrt {K(m)}}}}
θ
n
(
z
,
m
)
=
1
/
2
π
2
(
1
+
2
∑
k
=
1
∞
(
−
1
)
k
(
q
(
m
)
)
k
2
cos
(
π
z
k
K
(
m
)
)
)
1
1
−
m
4
1
K
(
m
)
{\displaystyle \theta _{n}(z,m)=1/2\,{\sqrt {\pi }}{\sqrt {2}}\left(1+2\sum _{k=1}^{\infty }(-1)^{k}(q(m))^{k^{2}}\cos \left({\frac {\pi zk}{K(m)}}\right)\right){\frac {1}{\sqrt[{4}]{1-m}}}{\frac {1}{\sqrt {K(m)}}}}
θ
s
(
z
,
m
)
=
π
2
q
(
m
)
4
∑
k
=
0
∞
(
−
1
)
k
(
q
(
m
)
)
k
(
k
+
1
)
sin
(
1
/
2
(
2
k
+
1
)
π
z
K
(
m
)
)
1
1
−
m
4
1
m
4
1
K
(
m
)
{\displaystyle \theta _{s}(z,m)={\sqrt {\pi }}{\sqrt {2}}{\sqrt[{4}]{q(m)}}\sum _{k=0}^{\infty }(-1)^{k}(q(m))^{k(k+1)}\sin \left(1/2\,{\frac {(2k+1)\pi z}{K(m)}}\right){\frac {1}{\sqrt[{4}]{1-m}}}{\frac {1}{\sqrt[{4}]{m}}}{\frac {1}{\sqrt {K(m)}}}}
where:
K
(
m
)
=
EllipticK
(
m
)
{\displaystyle K(m)=\operatorname {EllipticK} ({\sqrt {m}})}
K
′
(
m
)
=
EllipticK
(
1
−
m
)
{\displaystyle K'(m)=\operatorname {EllipticK} ({\sqrt {1-m}})}
q
(
m
)
=
e
−
π
K
(
m
)
/
K
′
(
m
)
{\displaystyle q(m)=e^{-\pi K(m)/K'(m)}}
Relationship to other functions
The Neville theta functions may be expressed in terms of the Jacobi theta functions[ 5]
θ
s
(
z
|
τ
)
=
θ
23
(
0
|
τ
)
θ
1
(
u
|
τ
)
/
θ
1
′
(
0
|
τ
)
{\displaystyle \theta _{s}(z|\tau )=\theta _{23}(0|\tau )\theta _{1}(u|\tau )/\theta '_{1}(0|\tau )}
θ
c
(
z
|
τ
)
=
θ
2
(
u
|
τ
)
/
θ
2
(
0
|
τ
)
{\displaystyle \theta _{c}(z|\tau )=\theta _{2}(u|\tau )/\theta _{2}(0|\tau )}
θ
n
(
z
|
τ
)
=
θ
4
(
u
|
τ
)
/
θ
4
(
0
|
τ
)
{\displaystyle \theta _{n}(z|\tau )=\theta _{4}(u|\tau )/\theta _{4}(0|\tau )}
θ
d
(
z
|
τ
)
=
θ
3
(
u
|
τ
)
/
θ
3
(
0
|
τ
)
{\displaystyle \theta _{d}(z|\tau )=\theta _{3}(u|\tau )/\theta _{3}(0|\tau )}
The Neville theta functions are related to the Jacobi elliptic functions . If pq(u,m) is a Jacobi elliptic function (p and q are one of s,c,n,d), then
p
q
(
u
,
m
)
=
θ
p
(
u
,
m
)
θ
q
(
u
,
m
)
{\displaystyle \operatorname {pq(u,m)} ={\frac {\theta _{p}(u,m)}{\theta _{q}(u,m)}}}
Examples
Substitute z = 2.5, m = 0.3 into the above definitions of Neville theta functions (using Maple ) once obtain the following (consistent with results from wolfram math) .
θ
c
(
2.5
,
0.3
)
=
−
0.65900466676738154967
{\displaystyle \theta _{c}(2.5,0.3)=-0.65900466676738154967}
[ 6]
θ
d
(
2.5
,
0.3
)
=
0.95182196661267561994
{\displaystyle \theta _{d}(2.5,0.3)=0.95182196661267561994}
θ
n
(
2.5
,
0.3
)
=
1.0526693354651613637
{\displaystyle \theta _{n}(2.5,0.3)=1.0526693354651613637}
θ
s
(
2.5
,
0.3
)
=
0.82086879524530400536
{\displaystyle \theta _{s}(2.5,0.3)=0.82086879524530400536}
Symmetry
θ
c
(
z
,
m
)
=
θ
c
(
−
z
,
m
)
{\displaystyle \theta _{c}(z,m)=\theta _{c}(-z,m)}
θ
d
(
z
,
m
)
=
θ
d
(
−
z
,
m
)
{\displaystyle \theta _{d}(z,m)=\theta _{d}(-z,m)}
θ
n
(
z
,
m
)
=
θ
n
(
−
z
,
m
)
{\displaystyle \theta _{n}(z,m)=\theta _{n}(-z,m)}
θ
s
(
z
,
m
)
=
−
θ
s
(
−
z
,
m
)
{\displaystyle \theta _{s}(z,m)=-\theta _{s}(-z,m)}
Complex 3D plots
Implementation
NetvilleThetaC[z,m], NevilleThetaD[z,m], NevilleThetaN[z,m], and NevilleThetaS[z,m] are built-in functions of Mathematica[ 7]
Notes
References
![{\displaystyle \theta _{c}(z,m)={\sqrt {2}}{\sqrt {\pi }}{\sqrt[{4}]{q(m)}}\sum _{k=0}^{\infty }(q(m))^{k(k+1)}\cos \left({\frac {1}{2}}\cdot {\frac {(2k+1)\pi z}{K(m)}}\right){\frac {1}{\sqrt {K(m)}}}{\frac {1}{\sqrt[{4}]{m}}}}](/media/api/rest_v1/media/math/render/svg/f53fce6a7c1aa15af6dd3c7e3f9db669f6df15a6)
![{\displaystyle \theta _{n}(z,m)=1/2\,{\sqrt {\pi }}{\sqrt {2}}\left(1+2\sum _{k=1}^{\infty }(-1)^{k}(q(m))^{k^{2}}\cos \left({\frac {\pi zk}{K(m)}}\right)\right){\frac {1}{\sqrt[{4}]{1-m}}}{\frac {1}{\sqrt {K(m)}}}}](/media/api/rest_v1/media/math/render/svg/8a28fa907e4093140539a9857a8d4576af2d1a4c)
![{\displaystyle \theta _{s}(z,m)={\sqrt {\pi }}{\sqrt {2}}{\sqrt[{4}]{q(m)}}\sum _{k=0}^{\infty }(-1)^{k}(q(m))^{k(k+1)}\sin \left(1/2\,{\frac {(2k+1)\pi z}{K(m)}}\right){\frac {1}{\sqrt[{4}]{1-m}}}{\frac {1}{\sqrt[{4}]{m}}}{\frac {1}{\sqrt {K(m)}}}}](/media/api/rest_v1/media/math/render/svg/28d4c040618d852d180380c6c96649e4af4a4f7d)
