From Wikipedia, the free encyclopedia
In mathematics, the Neville theta functions , named after Eric Harold Neville [citation needed , are defined as follows:[ 1] [ 2]
θ
c
(
z
,
m
)
=
2
π
q
(
m
)
4
∑
k
=
0
I
N
F
(
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\left(m\right)}}\sum _{k=0}^{\it {INF}}\left(q\left(m\right)\right)^{k\left(k+1\right)}\cos \left(1/2\,{\frac {\left(2\,k+1\right)\pi \,z}{K\left(m\right)}}\right){\frac {1}{\sqrt {K\left(m\right)}}}{\frac {1}{\sqrt[{4}]{m}}}}
θ
d
(
z
,
m
)
=
1
/
2
2
π
(
1
+
2
∑
k
=
1
I
N
F
(
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}^{\it {INF}}\left(q\left(m\right)\right)^{{k}^{2}}\cos \left({\frac {\pi \,zk}{K\left(m\right)}}\right)\right){\frac {1}{\sqrt {K\left(m\right)}}}}
θ
n
(
z
,
m
)
=
1
/
2
π
2
(
1
+
2
∑
k
=
1
I
N
F
(
−
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}^{\it {INF}}\left(-1\right)^{k}\left(q\left(m\right)\right)^{{k}^{2}}\cos \left({\frac {\pi \,zk}{K\left(m\right)}}\right)\right){\frac {1}{\sqrt[{4}]{1-m}}}{\frac {1}{\sqrt {K\left(m\right)}}}}
θ
s
(
z
,
m
)
=
π
2
q
(
m
)
4
∑
k
=
0
I
N
F
(
−
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\left(m\right)}}\sum _{k=0}^{\it {INF}}\left(-1\right)^{k}\left(q\left(m\right)\right)^{k\left(k+1\right)}\sin \left(1/2\,{\frac {\left(2\,k+1\right)\pi \,z}{K\left(m\right)}}\right){\frac {1}{\sqrt[{4}]{1-m}}}{\frac {1}{\sqrt[{4}]{m}}}{\frac {1}{\sqrt {K\left(m\right)}}}}
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)}}
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}
[ 3]
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)}
In terms of other special functions
θ
c
(
z
,
m
)
=
2
π
e
−
π
EllipticK
(
1
−
m
)
EllipticK
(
m
)
4
∑
k
=
0
∞
(
e
−
π
EllipticK
(
1
−
m
)
E
l
l
i
p
t
i
c
K
(
m
)
)
k
(
k
+
1
)
(
1
/
2
(
2
k
+
1
)
π
z
EllipticK
(
m
)
+
1
/
2
π
)
M
(
1
,
2
,
2
i
(
1
/
2
(
2
k
+
1
)
π
z
EllipticK
(
m
)
+
1
/
2
π
)
)
(
e
i
(
1
/
2
(
2
k
+
1
)
π
z
EllipticK
(
m
)
+
1
/
2
π
)
)
−
1
1
EllipticK
(
m
)
1
m
4
{\displaystyle \theta _{c}(z,m)={\sqrt {2}}{\sqrt {\pi }}{\sqrt[{4}]{{\rm {e}}^{-{\frac {\pi \,{\operatorname {EllipticK} }\left({\sqrt {1-m}}\right)}{\operatorname {EllipticK} \left({\sqrt {m}}\right)}}}}}\sum _{k=0}^{\infty }\left({{\rm {e}}^{-{\frac {\pi \,{\operatorname {EllipticK} }\left({\sqrt {1-m}}\right)}{{\it {EllipticK}}\left({\sqrt {m}}\right)}}}}\right)^{k\left(k+1\right)}\left(1/2\,{\frac {\left(2\,k+1\right)\pi \,z}{\operatorname {EllipticK} \left({\sqrt {m}}\right)}}+1/2\,\pi \right){{\rm {M}}\left(1,\,2,\,2\,i\left(1/2\,{\frac {\left(2\,k+1\right)\pi \,z}{\operatorname {EllipticK} \left({\sqrt {m}}\right)}}+1/2\,\pi \right)\right)}\left({{\rm {e}}^{i\left(1/2\,{\frac {\left(2\,k+1\right)\pi \,z}{\operatorname {EllipticK} \left({\sqrt {m}}\right)}}+1/2\,\pi \right)}}\right)^{-1}{\frac {1}{\sqrt {\operatorname {EllipticK} \left({\sqrt {m}}\right)}}}{\frac {1}{\sqrt[{4}]{m}}}}
θ
d
(
z
,
n
)
=
1
/
2
2
π
(
1
+
2
∑
k
=
1
∞
(
e
−
π
EllipticK
(
1
−
m
)
EllipticK
(
m
)
)
k
2
(
k
π
z
EllipticK
(
m
)
+
1
/
2
π
)
M
(
1
,
2
,
2
i
(
k
π
z
EllipticK
(
m
)
+
1
/
2
π
)
)
(
e
i
(
k
π
z
EllipticK
(
m
)
+
1
/
2
π
)
)
−
1
)
1
EllipticK
(
m
)
{\displaystyle \theta _{d}(z,n)=1/2\,{\sqrt {2}}{\sqrt {\pi }}\left(1+2\,\sum _{k=1}^{\infty }\left({{\rm {e}}^{-{\frac {\pi \operatorname {EllipticK} \left({\sqrt {1-m}}\right)}{\operatorname {EllipticK} \left({\sqrt {m}}\right)}}}}\right)^{{k}^{2}}\left({\frac {k\pi \,z}{\operatorname {EllipticK} \left({\sqrt {m}}\right)}}+1/2\,\pi \right){{\rm {M}}\left(1,\,2,\,2\,i\left({\frac {k\pi \,z}{\operatorname {EllipticK} \left({\sqrt {m}}\right)}}+1/2\,\pi \right)\right)}\left({{\rm {e}}^{i\left({\frac {k\pi \,z}{\operatorname {EllipticK} \left({\sqrt {m}}\right)}}+1/2\,\pi \right)}}\right)^{-1}\right){\frac {1}{\sqrt {\operatorname {EllipticK} \left({\sqrt {m}}\right)}}}}
θ
n
(
z
,
m
)
=
1
/
2
2
π
(
1
+
2
∑
k
=
1
∞
(
−
1
)
k
(
e
−
π
EllipticK
(
1
−
m
)
EllipticK
(
m
)
)
k
2
(
k
π
z
EllipticK
(
m
)
+
1
/
2
π
)
M
(
1
,
2
,
2
i
(
k
π
z
EllipticK
(
m
)
+
1
/
2
π
)
)
(
e
i
(
k
π
z
EllipticK
(
m
)
+
1
/
2
π
)
)
−
1
)
1
1
−
m
4
1
EllipticK
(
m
)
{\displaystyle \theta _{n}(z,m)=1/2\,{\sqrt {2}}{\sqrt {\pi }}\left(1+2\,\sum _{k=1}^{\infty }\left(-1\right)^{k}\left({{\rm {e}}^{-{\frac {\pi \operatorname {EllipticK} \left({\sqrt {1-m}}\right)}{\operatorname {EllipticK} \left({\sqrt {m}}\right)}}}}\right)^{{k}^{2}}\left({\frac {k\pi \,z}{\operatorname {EllipticK} \left({\sqrt {m}}\right)}}+1/2\,\pi \right){{\rm {M}}\left(1,\,2,\,2\,i\left({\frac {k\pi \,z}{\operatorname {EllipticK} \left({\sqrt {m}}\right)}}+1/2\,\pi \right)\right)}\left({{\rm {e}}^{i\left({\frac {k\pi \,z}{\operatorname {EllipticK} \left({\sqrt {m}}\right)}}+1/2\,\pi \right)}}\right)^{-1}\right){\frac {1}{\sqrt[{4}]{1-m}}}{\frac {1}{\sqrt {\operatorname {EllipticK} \left({\sqrt {m}}\right)}}}}
θ
s
(
z
,
m
)
=
2
π
e
−
π
EllipticK
(
1
−
m
)
EllipticK
(
m
)
4
∑
k
=
0
∞
1
/
2
(
−
1
)
k
(
e
−
π
EllipticK
(
1
−
m
)
EllipticK
(
m
)
)
k
(
k
+
1
)
(
2
k
+
1
)
π
z
M
(
1
,
2
,
i
π
z
(
2
k
+
1
)
EllipticK
(
m
)
)
(
EllipticK
(
m
)
)
−
1
(
e
1
/
2
i
π
z
(
2
k
+
1
)
EllipticK
(
m
)
)
−
1
1
1
−
m
4
1
m
4
1
EllipticK
(
m
)
{\displaystyle \theta _{s}(z,m)={\sqrt {2}}{\sqrt {\pi }}{\sqrt[{4}]{{\rm {e}}^{-{\frac {\pi \operatorname {EllipticK} \left({\sqrt {1-m}}\right)}{\operatorname {EllipticK} \left({\sqrt {m}}\right)}}}}}\sum _{k=0}^{\infty }1/2\,\left(-1\right)^{k}\left({{\rm {e}}^{-{\frac {\pi \operatorname {EllipticK} \left({\sqrt {1-m}}\right)}{\operatorname {EllipticK} \left({\sqrt {m}}\right)}}}}\right)^{k\left(k+1\right)}\left(2\,k+1\right)\pi \,z{{\rm {M}}\left(1,\,2,\,{\frac {i\pi \,z\left(2\,k+1\right)}{\operatorname {EllipticK} \left({\sqrt {m}}\right)}}\right)}\left(\operatorname {EllipticK} \left({\sqrt {m}}\right)\right)^{-1}\left({{\rm {e}}^{\frac {1/2\,i\pi \,z\left(2\,k+1\right)}{\operatorname {EllipticK} \left({\sqrt {m}}\right)}}}\right)^{-1}{\frac {1}{\sqrt[{4}]{1-m}}}{\frac {1}{\sqrt[{4}]{m}}}{\frac {1}{\sqrt {\operatorname {EllipticK} \left({\sqrt {m}}\right)}}}}
Series expansions
θ
c
(
z
,
1
/
2
)
=
0.9998
−
0.3641
z
2
+
0.2466
e
−
1
z
4
−
0.1210
e
−
2
z
6
+
0.8707
e
−
4
z
8
+
O
(
z
1
0
)
{\displaystyle \theta _{c}(z,1/2)=0.9998-0.3641z^{2}+0.2466e-1z^{4}-0.1210e-2z^{6}+0.8707e-4z^{8}+O(z^{1}0)}
θ
d
(
z
,
1
/
2
)
=
0.9995
−
0.1143
z
2
+
0.2736
e
−
1
z
4
−
0.2629
e
−
2
z
6
+
0.1368
e
−
3
z
8
+
O
(
z
1
0
)
{\displaystyle \theta _{d}(z,1/2)=0.9995-0.1143z^{2}+0.2736e-1z^{4}-0.2629e-2z^{6}+0.1368e-3z^{8}+O(z^{1}0)}
θ
n
(
z
,
1
/
2
)
=
1.000
+
0.1358
z
2
−
0.3244
e
−
1
z
4
+
0.3093
e
−
2
z
6
−
0.1561
e
−
3
z
8
+
O
(
z
1
0
)
{\displaystyle \theta _{n}(z,1/2)=1.000+0.1358z^{2}-0.3244e-1z^{4}+0.3093e-2z^{6}-0.1561e-3z^{8}+O(z^{1}0)}
θ
s
(
z
,
1
/
2
)
=
1.000
z
−
0.1142
z
3
+
0.2358
e
−
2
z
5
+
0.2276
e
−
3
z
7
−
0.2630
e
−
4
z
9
+
O
(
z
1
1
)
{\displaystyle \theta _{s}(z,1/2)=1.000z-0.1142z^{3}+0.2358e-2z^{5}+0.2276e-3z^{7}-0.2630e-4z^{9}+O(z^{1}1)}
2-dimensional plot
Neville ThetaC function Maple plot Neville ThetaD function Maple plot Neville ThetaD function Maple plot Neville ThetaS function Maple plot
Complex 3D plots
Implementation
NetvilleThetaC[z,m], NevilleThetaD[z,m], NevilleThetaN[z,m], and NevilleThetaS[z,m] are built-in functions of Mathematica [ 4] Maple .
External links
References
Milton Abramowitz and Irene Stegun,Handbook of Mathematical Functions, p578, National Bureau of Standards, 1972. 
![{\displaystyle \theta _{c}(z,m)={\sqrt {2}}{\sqrt {\pi }}{\sqrt[{4}]{q\left(m\right)}}\sum _{k=0}^{\it {INF}}\left(q\left(m\right)\right)^{k\left(k+1\right)}\cos \left(1/2\,{\frac {\left(2\,k+1\right)\pi \,z}{K\left(m\right)}}\right){\frac {1}{\sqrt {K\left(m\right)}}}{\frac {1}{\sqrt[{4}]{m}}}}](/media/api/rest_v1/media/math/render/svg/459d60091d97c039bb963e9292ed42dca63968d0)
![{\displaystyle \theta _{n}(z,m)=1/2\,{\sqrt {\pi }}{\sqrt {2}}\left(1+2\,\sum _{k=1}^{\it {INF}}\left(-1\right)^{k}\left(q\left(m\right)\right)^{{k}^{2}}\cos \left({\frac {\pi \,zk}{K\left(m\right)}}\right)\right){\frac {1}{\sqrt[{4}]{1-m}}}{\frac {1}{\sqrt {K\left(m\right)}}}}](/media/api/rest_v1/media/math/render/svg/a97fa8f3a297748f9af2b25cb42c67772ab83255)
![{\displaystyle \theta _{s}(z,m)={\sqrt {\pi }}{\sqrt {2}}{\sqrt[{4}]{q\left(m\right)}}\sum _{k=0}^{\it {INF}}\left(-1\right)^{k}\left(q\left(m\right)\right)^{k\left(k+1\right)}\sin \left(1/2\,{\frac {\left(2\,k+1\right)\pi \,z}{K\left(m\right)}}\right){\frac {1}{\sqrt[{4}]{1-m}}}{\frac {1}{\sqrt[{4}]{m}}}{\frac {1}{\sqrt {K\left(m\right)}}}}](/media/api/rest_v1/media/math/render/svg/97fd185a89249ec3575d4c8e38ad19a26760b40c)
![{\displaystyle \theta _{c}(z,m)={\sqrt {2}}{\sqrt {\pi }}{\sqrt[{4}]{{\rm {e}}^{-{\frac {\pi \,{\operatorname {EllipticK} }\left({\sqrt {1-m}}\right)}{\operatorname {EllipticK} \left({\sqrt {m}}\right)}}}}}\sum _{k=0}^{\infty }\left({{\rm {e}}^{-{\frac {\pi \,{\operatorname {EllipticK} }\left({\sqrt {1-m}}\right)}{{\it {EllipticK}}\left({\sqrt {m}}\right)}}}}\right)^{k\left(k+1\right)}\left(1/2\,{\frac {\left(2\,k+1\right)\pi \,z}{\operatorname {EllipticK} \left({\sqrt {m}}\right)}}+1/2\,\pi \right){{\rm {M}}\left(1,\,2,\,2\,i\left(1/2\,{\frac {\left(2\,k+1\right)\pi \,z}{\operatorname {EllipticK} \left({\sqrt {m}}\right)}}+1/2\,\pi \right)\right)}\left({{\rm {e}}^{i\left(1/2\,{\frac {\left(2\,k+1\right)\pi \,z}{\operatorname {EllipticK} \left({\sqrt {m}}\right)}}+1/2\,\pi \right)}}\right)^{-1}{\frac {1}{\sqrt {\operatorname {EllipticK} \left({\sqrt {m}}\right)}}}{\frac {1}{\sqrt[{4}]{m}}}}](/media/api/rest_v1/media/math/render/svg/fa1b70ab3830e22cefca8553542631e1a4cf8d13)
![{\displaystyle \theta _{n}(z,m)=1/2\,{\sqrt {2}}{\sqrt {\pi }}\left(1+2\,\sum _{k=1}^{\infty }\left(-1\right)^{k}\left({{\rm {e}}^{-{\frac {\pi \operatorname {EllipticK} \left({\sqrt {1-m}}\right)}{\operatorname {EllipticK} \left({\sqrt {m}}\right)}}}}\right)^{{k}^{2}}\left({\frac {k\pi \,z}{\operatorname {EllipticK} \left({\sqrt {m}}\right)}}+1/2\,\pi \right){{\rm {M}}\left(1,\,2,\,2\,i\left({\frac {k\pi \,z}{\operatorname {EllipticK} \left({\sqrt {m}}\right)}}+1/2\,\pi \right)\right)}\left({{\rm {e}}^{i\left({\frac {k\pi \,z}{\operatorname {EllipticK} \left({\sqrt {m}}\right)}}+1/2\,\pi \right)}}\right)^{-1}\right){\frac {1}{\sqrt[{4}]{1-m}}}{\frac {1}{\sqrt {\operatorname {EllipticK} \left({\sqrt {m}}\right)}}}}](/media/api/rest_v1/media/math/render/svg/4e2524cd9421f668c0d33bb825a32cfe752ab10f)
![{\displaystyle \theta _{s}(z,m)={\sqrt {2}}{\sqrt {\pi }}{\sqrt[{4}]{{\rm {e}}^{-{\frac {\pi \operatorname {EllipticK} \left({\sqrt {1-m}}\right)}{\operatorname {EllipticK} \left({\sqrt {m}}\right)}}}}}\sum _{k=0}^{\infty }1/2\,\left(-1\right)^{k}\left({{\rm {e}}^{-{\frac {\pi \operatorname {EllipticK} \left({\sqrt {1-m}}\right)}{\operatorname {EllipticK} \left({\sqrt {m}}\right)}}}}\right)^{k\left(k+1\right)}\left(2\,k+1\right)\pi \,z{{\rm {M}}\left(1,\,2,\,{\frac {i\pi \,z\left(2\,k+1\right)}{\operatorname {EllipticK} \left({\sqrt {m}}\right)}}\right)}\left(\operatorname {EllipticK} \left({\sqrt {m}}\right)\right)^{-1}\left({{\rm {e}}^{\frac {1/2\,i\pi \,z\left(2\,k+1\right)}{\operatorname {EllipticK} \left({\sqrt {m}}\right)}}}\right)^{-1}{\frac {1}{\sqrt[{4}]{1-m}}}{\frac {1}{\sqrt[{4}]{m}}}{\frac {1}{\sqrt {\operatorname {EllipticK} \left({\sqrt {m}}\right)}}}}](/media/api/rest_v1/media/math/render/svg/f2f27b44099728de7617b6c69897a7054a5d61b8)
