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In mathematics, the Neville theta functions , named after Eric Harold Neville [citation needed , are defined as follows:[ 1] [ 2]
θ
c
(
z
,
m
)
=
2
q
(
m
)
1
/
4
∑
k
=
0
∞
q
(
m
)
k
(
k
+
1
)
cos
(
1
2
(
2
k
+
1
)
π
z
K
(
m
)
)
K
(
m
)
m
1
/
4
{\displaystyle \theta _{c}(z,m)={\frac {{\sqrt {2}}\,q(m)^{1/4}\sum _{k=0}^{\infty }q(m)^{k(k+1)}\cos \left({\dfrac {{\frac {1}{2}}(2k+1)\pi z}{K(m)}}\right)}{{\sqrt {K(m)}}\,m^{1/4}}}}
θ
c
(
z
,
m
)
=
2
π
q
(
m
)
1
/
4
∑
k
=
0
∞
q
(
m
)
k
(
k
+
1
)
cos
(
1
2
(
2
k
+
1
)
π
z
K
(
m
)
)
K
(
m
)
m
1
/
4
{\displaystyle \theta _{c}(z,m)={\frac {{\sqrt {2\pi }}\,q(m)^{1/4}\sum _{k=0}^{\infty }q(m)^{k(k+1)}\cos \left({\dfrac {{\frac {1}{2}}(2k+1)\pi z}{K(m)}}\right)}{{\sqrt {K(m)}}\,m^{1/4}}}}
θ
d
(
z
,
m
)
=
π
/
2
(
1
+
2
∑
k
=
1
∞
q
(
m
)
k
2
cos
(
k
π
z
K
(
m
)
)
)
K
(
m
)
{\displaystyle \theta _{d}(z,m)={\frac {{\sqrt {\pi /2}}\left(1+2\sum _{k=1}^{\infty }q(m)^{k^{2}}\cos \left({\dfrac {k\pi z}{K(m)}}\right)\right)}{\sqrt {K(m)}}}}
θ
n
(
z
,
m
)
=
π
(
1
+
2
(
∑
k
=
1
∞
(
−
1
)
k
q
(
m
)
k
2
cos
(
k
π
z
)
K
(
m
)
)
)
2
(
1
−
m
)
1
/
4
K
(
m
)
{\displaystyle \theta _{n}(z,m)={\frac {{\sqrt {\pi }}\left(1+2\left(\sum _{k=1}^{\infty }(-1)^{k}q(m)^{k^{2}}{\dfrac {\cos(k\pi z)}{K(m)}}\right)\right)}{{\sqrt {2}}(1-m)^{1/4}{\sqrt {K(m)}}}}}
[There is an error above: Two of the subscripts are "c " and one of those should be an "s ". I'd fix it if I knew which one.]
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)}}
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)}
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. 
