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Statements
4
π
∫
0
π
/
2
𝒞
2
ℓ
+
1
(
j
)
(
2
h
R
)
cos
(
(
2
ℓ
+
1
)
ϕ
)
ce
2
m
+
1
(
t
,
h
2
)
d
t
=
(
-
1
)
ℓ
+
m
A
2
ℓ
+
1
2
m
+
1
(
h
2
)
Mc
2
m
+
1
(
j
)
(
z
,
h
)
,
4
𝜋
superscript
subscript
0
𝜋
2
subscript
superscript
𝒞
𝑗
2
ℓ
1
2
ℎ
𝑅
2
ℓ
1
italic-ϕ
Mathieu-ce
2
𝑚
1
𝑡
superscript
ℎ
2
𝑡
superscript
1
ℓ
𝑚
subscript
superscript
𝐴
2
𝑚
1
2
ℓ
1
superscript
ℎ
2
modified-Mathieu-Mc
𝑗
2
𝑚
1
𝑧
ℎ
{\displaystyle{\displaystyle\dfrac{4}{\pi}\int_{0}^{\pi/2}\mathcal{C}^{(j)}_{2%
\ell+1}(2hR)\cos\left((2\ell+1)\phi\right)\mathrm{ce}_{2m+1}\left(t,h^{2}%
\right)\mathrm{d}t=(-1)^{\ell+m}A^{2m+1}_{2\ell+1}(h^{2}){\mathrm{Mc}^{(j)}_{2%
m+1}}\left(z,h\right),}}
ce
n
(
z
,
q
)
Mathieu-ce
𝑛
𝑧
𝑞
{\displaystyle{\displaystyle\mathrm{ce}_{\NVar{n}}\left(\NVar{z},\NVar{q}%
\right)}}
π
{\displaystyle{\displaystyle\pi}}
cos
z
𝑧
{\displaystyle{\displaystyle\cos\NVar{z}}}
d
x
𝑥
{\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
∫
{\displaystyle{\displaystyle\int}}
Mc
n
(
j
)
(
z
,
h
)
modified-Mathieu-Mc
𝑗
𝑛
𝑧
ℎ
{\displaystyle{\displaystyle{\mathrm{Mc}^{(\NVar{j})}_{\NVar{n}}}\left(\NVar{z%
},\NVar{h}\right)}}
m
𝑚
{\displaystyle{\displaystyle m}}
h
ℎ
{\displaystyle{\displaystyle h}}
j
𝑗
{\displaystyle{\displaystyle j}}
z
𝑧
{\displaystyle{\displaystyle z}}
𝒞
μ
(
j
)
superscript
subscript
𝒞
𝜇
𝑗
{\displaystyle{\displaystyle\mathcal{C}_{\mu}^{(j)}}}
ϕ
(
z
,
t
)
italic-ϕ
𝑧
𝑡
{\displaystyle{\displaystyle\phi(z,t)}}
R
(
z
,
t
)
𝑅
𝑧
𝑡
{\displaystyle{\displaystyle R(z,t)}}
A
m
(
q
)
subscript
𝐴
𝑚
𝑞
{\displaystyle{\displaystyle A_{m}(q)}}
