In mathematics, the Neville theta functions, named after Eric Harold Neville, are defined as follows:
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
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
)
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
)
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
)
where:
K
(
m
)
=
EllipticK
(
m
)
K
′
(
m
)
=
EllipticK
(
1
−
m
)
q
(
m
)
=
e
−
π
K
(
m
)
/
K
′
(
m
)
is the elliptic nome
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
θ
d
(
2.5
,
0.3
)
=
0.95182196661267561994
θ
n
(
2.5
,
0.3
)
=
1.0526693354651613637
θ
s
(
2.5
,
0.3
)
=
0.82086879524530400536
θ
c
(
z
,
m
)
=
θ
c
(
−
z
,
m
)
θ
d
(
z
,
m
)
=
θ
d
(
−
z
,
m
)
θ
n
(
z
,
m
)
=
θ
n
(
−
z
,
m
)
θ
s
(
z
,
m
)
=
−
θ
s
(
−
z
,
m
)
NetvilleThetaC[z,m], NevilleThetaD[z,m], NevilleThetaN[z,m], and NevilleThetaS[z,m] are built-in functions of Mathematica No such functions in Maple.
