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Fundamentalsatz für die Entwicklung nach Kugelfunctionen.
−
∫
0
π
S
m
∂
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sin
θ
∂
Q
n
∂
θ
)
∂
θ
d
θ
=
(
s
i
n
θ
S
m
∂
Q
n
∂
θ
)
θ
=
0
−
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s
i
n
θ
S
m
∂
Q
n
∂
θ
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θ
=
π
+
(
s
i
n
θ
Q
n
∂
S
m
∂
θ
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θ
=
π
−
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s
i
n
θ
Q
n
∂
S
m
∂
θ
)
θ
=
0
−
∫
0
π
Q
n
∂
(
sin
θ
∂
S
m
∂
θ
)
∂
θ
d
θ
.
{\displaystyle {\begin{aligned}-\int _{0}^{\pi }S_{m}{\frac {\partial \left(\sin \theta {\frac {\partial Q_{n}}{\partial \theta }}\right)}{\partial \theta }}d\theta &=\left(sin\theta S_{m}{\frac {\partial Q_{n}}{\partial \theta }}\right)_{\theta =0}-\left(sin\theta S_{m}{\frac {\partial Q_{n}}{\partial \theta }}\right)_{\theta =\pi }\\&+\left(sin\theta Q_{n}{\frac {\partial S_{m}}{\partial \theta }}\right)_{\theta =\pi }-\left(sin\theta Q_{n}{\frac {\partial S_{m}}{\partial \theta }}\right)_{\theta =0}\,\\&-\int _{0}^{\pi }Q_{n}{\frac {\partial \left(\sin \theta {\frac {\partial S_{m}}{\partial \theta }}\right)}{\partial \theta }}d\theta .\,\\\end{aligned}}}
sin
θ
=
0
{\displaystyle \sin \theta =0\,}
θ
=
0
{\displaystyle \theta =0\,}
θ
=
π
{\displaystyle \theta =\pi \,}
(9)
−
∫
0
2
π
d
φ
∫
0
π
S
m
∂
(
sin
θ
∂
Q
n
∂
θ
)
∂
θ
d
θ
=
−
∫
0
2
π
d
φ
∫
0
π
Q
n
∂
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sin
θ
∂
S
m
∂
θ
)
∂
θ
d
θ
.
{\displaystyle -\int _{0}^{2\pi }d\varphi \int _{0}^{\pi }S_{m}{\frac {\partial \left(\sin \theta {\frac {\partial Q_{n}}{\partial \theta }}\right)}{\partial \theta }}d\theta =-\int _{0}^{2\pi }d\varphi \int _{0}^{\pi }Q_{n}{\frac {\partial \left(\sin \theta {\frac {\partial S_{m}}{\partial \theta }}\right)}{\partial \theta }}d\theta .}
(10)
n
(
n
+
1
)
∫
0
2
π
d
φ
∫
0
π
Q
n
S
m
s
i
n
θ
d
θ
{\displaystyle n(n+1)\int _{0}^{2\pi }d\varphi \int _{0}^{\pi }Q_{n}S_{m}sin\theta d\theta }
=
−
∫
0
π
d
θ
sin
θ
∫
0
2
π
Q
n
∂
2
S
m
∂
φ
2
d
φ
−
∫
0
2
π
d
φ
∫
0
π
Q
n
∂
(
sin
θ
∂
S
m
∂
θ
)
∂
θ
d
θ
.
{\displaystyle =-\int _{0}^{\pi }{\frac {d\theta }{\sin \theta }}\int _{0}^{2\pi }Q_{n}{\frac {\partial ^{2}S_{m}}{\partial \varphi ^{2}}}d\varphi -\int _{0}^{2\pi }d\varphi \int _{0}^{\pi }Q_{n}{\frac {\partial \left(\sin \theta {\frac {\partial S_{m}}{\partial \theta }}\right)}{\partial \theta }}d\theta .\,}
m
(
m
+
l
)
{\displaystyle m(m+l)\,}
m
(
m
+
1
)
S
m
{\displaystyle m(m+1)S_{m}\,}
−
1
s
i
n
θ
2
∂
2
S
m
∂
φ
2
{\displaystyle -{\frac {1}{\ sin\theta ^{2}}}{\frac {\partial ^{2}S_{m}}{\partial \varphi ^{2}}}}
−
∂
(
sin
θ
∂
S
m
∂
θ
)
sin
θ
∂
θ
,
{\displaystyle -{\frac {\partial \left(\sin \theta {\frac {\partial S_{m}}{\partial \theta }}\right)}{\sin \theta \partial \theta }},}
S
m
{\displaystyle S_{m}\,}
n
(
n
+
1
)
∫
0
2
π
d
φ
∫
0
π
Q
n
S
m
sin
θ
d
θ
=
m
(
m
+
1
)
∫
0
2
π
d
φ
∫
0
π
Q
n
S
m
sin
θ
d
θ
,
{\displaystyle n(n+1)\int _{0}^{2\pi }d\varphi \int _{0}^{\pi }Q_{n}S_{m}\sin \theta d\theta =m(m+1)\int _{0}^{2\pi }d\varphi \int _{0}^{\pi }Q_{n}S_{m}\sin \theta d\theta ,\,}
