ρ {\displaystyle \rho \!\,}
r ~ 2 = r 2 − 2 r ρ c o s θ + ρ 2 {\displaystyle {\tilde {r}}^{2}=r^{2}-2r{\rho }cos\theta +{\rho }^{2}\!\,}
φ ~ = φ {\displaystyle {\tilde {\varphi }}=\varphi }
c o s θ ~ = r c o s θ − ρ r ~ {\displaystyle cos{\tilde {\theta }}={rcos\theta -\rho \over {\tilde {r}}}\!\,}
( T ρ g ) ( p → ) = ( T ρ g ) ( r u → ) = ( T ρ g ) ( r , θ , φ ) = g ( r ′ , θ ′ , φ ′ ) {\displaystyle (T_{\rho }g)({\vec {p}})=(T_{\rho }g)(r{\vec {u}})=(T_{\rho }g)(r,\theta ,\varphi )=g(r',\theta ',\varphi ')}
( T ρ Λ R g ) ( p → ) = ( T ρ Λ R g ) ( r u → ) = ( T ρ Λ R g ) ( r , θ , φ ) = ( Λ R g ) ( r ′ , θ ′ , φ ′ ) {\displaystyle (T_{\rho }\Lambda _{R}g)({\vec {p}})=(T_{\rho }\Lambda _{R}g)(r{\vec {u}})=(T_{\rho }\Lambda _{R}g)(r,\theta ,\varphi )=(\Lambda _{R}g)(r',\theta ',\varphi ')}
F [ T ( ξ , η , ω , ξ ′ , η ′ ) ] = T ~ ( h , m , n , h ′ , m ′ ) = ∑ l l ′ d m h l d h n l d m ′ h ′ l ′ d h ′ n l ′ I m m ′ n l l ′ {\displaystyle F[T(\xi ,\eta ,\omega ,\xi ',\eta ')]={\tilde {T}}(h,m,n,h',m')=\sum _{ll'}d_{mh}^{l}d_{hn}^{l}d_{m'h'}^{l'}d_{h'n}^{l'}I_{mm'n}^{ll'}}
T {\displaystyle T\!\,}
( ξ , η , ω , ξ ′ , η ′ ) {\displaystyle (\xi ,\eta ,\omega ,\xi ',\eta ')\!\,}
c ( R ) = ∑ l m h m ′ d m h l d h m ′ l e x p [ − i ( n ξ + h η + m ω ) ] ⋅ I m m ′ l = T ( ξ , η , ω ) {\displaystyle c(R)=\sum _{lmhm'}d_{mh}^{l}d_{hm'}^{l}exp[-i(n\xi +h\eta +m\omega )]\cdot I_{mm'}^{l}=T(\xi ,\eta ,\omega )}
I m m ′ n l l ′ = ∫ 0 π ∫ 0 ∞ f ^ l m ( r ) ⋅ [ g ^ l ′ m ′ ( r ~ ) ] ∗ d n 0 l ( θ ) d n 0 l ′ ( θ ~ ) ⋅ r 2 d r s i n θ d θ {\displaystyle I_{mm'n}^{ll'}=\int _{0}^{\pi }\int _{0}^{\infty }{\hat {f}}_{lm}(r)\cdot [{\hat {g}}_{l'm'}({\tilde {r}})]^{*}d_{n0}^{l}(\theta )d_{n0}^{l'}({\tilde {\theta }})\cdot r^{2}drsin\theta d\theta }
D n m l ( R ) = ∑ h d n h l ( π / 2 ) d h m l ( π / 2 ) e − i [ n ( φ − π / 2 ) + h ( π − θ ) + m ( ψ − π / 2 ) ] = ∑ h d n h l d h m l e − i [ n ξ + h η + m ω ] {\displaystyle D_{nm}^{l}(R)=\sum _{h}d_{nh}^{l}(\pi /2)d_{hm}^{l}(\pi /2)e^{-i[n(\varphi -\pi /2)+h(\pi -\theta )+m(\psi -\pi /2)]}=\sum _{h}d_{nh}^{l}d_{hm}^{l}e^{-i[n\xi +h\eta +m\omega ]}}
π / 2 {\displaystyle \pi /2\!\,}
ξ = φ − π / 2 {\displaystyle \xi =\varphi -\pi /2\!\,}
η = π − θ {\displaystyle \eta =\pi -\theta \!\,}
ω = ψ − π / 2 {\displaystyle \omega =\psi -\pi /2\!\,}
d m n l = d m n l ( π / 2 ) {\displaystyle d_{mn}^{l}=d_{mn}^{l}(\pi /2)}
D n m l ( R ) = D n m l ( R 1 ⋅ R 2 ) = ∑ h D n h l ( R 1 ) D h m l ( R 2 ) {\displaystyle D_{nm}^{l}(R)=D_{nm}^{l}(R_{1}\cdot R_{2})=\sum _{h}D_{nh}^{l}(R_{1})D_{hm}^{l}(R_{2})}
R 2 = ( π − θ , π / 2 , ψ − π / 2 ) {\displaystyle R_{2}=(\pi -\theta ,\pi /2,\psi -\pi /2)\!\,}
R 1 = ( φ − π / 2 , π / 2 , 0 ) {\displaystyle R_{1}=(\varphi -\pi /2,\pi /2,0)}
R = ( φ , θ , ψ ) {\displaystyle R=(\varphi ,\theta ,\psi )}
c ( R ) = δ l l ′ δ m n ∫ 0 ∞ f ^ l m ( r ) ⋅ [ g ^ l ′ m ′ ( r ) ] ∗ ⋅ r 2 d r {\displaystyle c(R)=\delta _{ll'}\delta _{mn}\int _{0}^{\infty }{\hat {f}}_{lm}(r)\cdot [{\hat {g}}_{l'm'}(r)]^{*}\cdot r^{2}dr}
c ( R ) = ∑ l l ′ m m ′ n n ′ [ D n m l ( R ) D n ′ m ′ l ′ ( R ′ ) ] ∗ ∫ [ ^ f l m ( r ) ] ∗ ⋅ [ g ^ l ′ m ′ ( r ′ ) ] ∗ ⋅ Y l m ( u → ) ⋅ [ Y l ′ m ′ ( u → ′ ) ] ∗ {\displaystyle c(R)=\sum _{ll'mm'nn'}[D_{nm}^{l}(R)D_{n'm'}^{l'}(R')]^{*}\int {\hat {[}}f_{lm}(r)]^{*}\cdot [{\hat {g}}_{l'm'}(r')]^{*}\cdot Y_{lm}({\vec {u}})\cdot [Y_{l'm'}({\vec {u}}')]^{*}}
p → {\displaystyle {\vec {p}}}
p → = r u → {\displaystyle {\vec {p}}=r{\vec {u}}}
| p → | = r {\displaystyle |{\vec {p}}|=r}
| u → | = 1 {\displaystyle |{\vec {u}}|=1}
R {\displaystyle R\!\,}
Λ R {\displaystyle \Lambda _{R}\!\,}
( Λ R g ) ( p → ) = g [ R − 1 ( p → ) ] {\displaystyle (\Lambda _{R}g)({\vec {p}})=g[R^{-1}({\vec {p}})]}
p → {\displaystyle {\vec {p}}\!\,}
f ( r u → ) {\displaystyle f(r{\vec {u}})}
g ( r u → ) {\displaystyle g(r{\vec {u}})}
B {\displaystyle B\!\,}
g ( p → ) = g ( r u → ) ≈ ∑ l = 0 B − 1 ∑ m = − l l g ^ l m ( r ) Y l m ( θ , φ ) {\displaystyle g({\vec {p}})=g(r{\vec {u}})\approx \sum _{l=0}^{B-1}\sum _{m=-l}^{l}{\hat {g}}_{lm}(r)Y_{lm}(\theta ,\varphi )}
f ( p → ) = f ( r u → ) ≈ ∑ l = 0 B − 1 ∑ m = − l l f ^ l m ( r ) Y l m ( θ , φ ) {\displaystyle f({\vec {p}})=f(r{\vec {u}})\approx \sum _{l=0}^{B-1}\sum _{m=-l}^{l}{\hat {f}}_{lm}(r)Y_{lm}(\theta ,\varphi )}
( Λ R g ) ( p → ) = g [ R − 1 ( r u → ) ] = g [ r R − 1 ( u → ) ] = ∑ l = 0 B − 1 ∑ m = − l l g ^ l m ( r ) Y l m [ R − 1 ( u → ) ] {\displaystyle (\Lambda _{R}g)({\vec {p}})=g[R^{-1}(r{\vec {u}})]=g[rR^{-1}({\vec {u}})]=\sum _{l=0}^{B-1}\sum _{m=-l}^{l}{\hat {g}}_{lm}(r)Y_{lm}[R^{-1}({\vec {u}})]}
Y l m [ R − 1 ( u → ) ] = ∑ n D n m l ( R ) Y l n ( θ , φ ) {\displaystyle Y_{lm}[R^{-1}({\vec {u}})]=\sum _{n}D_{nm}^{l}(R)Y_{ln}(\theta ,\varphi )}
D m n l {\displaystyle D_{mn}^{l}\!\,}
D m n l ( φ , θ , ψ ) = e − i ( m φ + n ψ ) d m n l ( θ ) {\displaystyle D_{mn}^{l}(\varphi ,\theta ,\psi )=e^{-i(m\varphi +n\psi )}d_{mn}^{l}(\theta )}
d m n l ( θ ) = ∑ t ( − i ) t ⋅ [ ( l + m ) ! ( l − m ) ! ( l + n ) ! ( l − n ) ! ] 1 2 ( l + m − t ) ! ( l − n − t ) ! t ! ( t + n − m ) ! ⋅ ( c o s θ 2 ) 2 l + m − n − 2 t ( s i n θ 2 ) 2 t + n − m {\displaystyle d_{mn}^{l}(\theta )=\sum _{t}(-i)^{t}\cdot {{[(l+m)!(l-m)!(l+n)!(l-n)!]^{1 \over 2}} \over {(l+m-t)!(l-n-t)!t!(t+n-m)!}}\cdot (cos{\theta \over 2})^{2l+m-n-2t}(sin{\theta \over 2})^{2t+n-m}}
( Λ R g ) ( r u → ) = ∑ l , m , n g ^ l m ( r ) D n m l ( R ) Y l n ( u → ) {\displaystyle (\Lambda _{R}g)(r{\vec {u}})=\sum _{l,m,n}{\hat {g}}_{lm}(r)D_{nm}^{l}(R)Y_{ln}({\vec {u}})}
c ( R , R ′ , ρ ) = ∫ [ Λ R ′ f ( r u → ) ] ∗ ⋅ [ T ρ Λ R g ( r u → ) ] ∗ {\displaystyle c(R,R',\rho )=\int [\Lambda _{R'}f(r{\vec {u}})]^{*}\cdot [T_{\rho }\Lambda _{R}g(r{\vec {u}})]^{*}}
![{\displaystyle F[T(\xi ,\eta ,\omega ,\xi ',\eta ')]={\tilde {T}}(h,m,n,h',m')=\sum _{ll'}d_{mh}^{l}d_{hn}^{l}d_{m'h'}^{l'}d_{h'n}^{l'}I_{mm'n}^{ll'}}](/media/api/rest_v1/media/math/render/svg/98564cad971c6ca9b22d4c50ac879fd82dca1a63)
![{\displaystyle c(R)=\sum _{lmhm'}d_{mh}^{l}d_{hm'}^{l}exp[-i(n\xi +h\eta +m\omega )]\cdot I_{mm'}^{l}=T(\xi ,\eta ,\omega )}](/media/api/rest_v1/media/math/render/svg/c1dd4fa0332c1d8d93d8dce713c57fb3b94d9cff)
![{\displaystyle I_{mm'n}^{ll'}=\int _{0}^{\pi }\int _{0}^{\infty }{\hat {f}}_{lm}(r)\cdot [{\hat {g}}_{l'm'}({\tilde {r}})]^{*}d_{n0}^{l}(\theta )d_{n0}^{l'}({\tilde {\theta }})\cdot r^{2}drsin\theta d\theta }](/media/api/rest_v1/media/math/render/svg/b9b418a87bce0efbee361fc18a49e9a1ec079549)
![{\displaystyle D_{nm}^{l}(R)=\sum _{h}d_{nh}^{l}(\pi /2)d_{hm}^{l}(\pi /2)e^{-i[n(\varphi -\pi /2)+h(\pi -\theta )+m(\psi -\pi /2)]}=\sum _{h}d_{nh}^{l}d_{hm}^{l}e^{-i[n\xi +h\eta +m\omega ]}}](/media/api/rest_v1/media/math/render/svg/956ba92d4f7f6bf31f2b1be4f07de3ea16d4bda3)
![{\displaystyle c(R)=\delta _{ll'}\delta _{mn}\int _{0}^{\infty }{\hat {f}}_{lm}(r)\cdot [{\hat {g}}_{l'm'}(r)]^{*}\cdot r^{2}dr}](/media/api/rest_v1/media/math/render/svg/d3cfb4dd471ca62d14430b7a6ca9dc7203704a8d)
![{\displaystyle c(R)=\sum _{ll'mm'nn'}[D_{nm}^{l}(R)D_{n'm'}^{l'}(R')]^{*}\int {\hat {[}}f_{lm}(r)]^{*}\cdot [{\hat {g}}_{l'm'}(r')]^{*}\cdot Y_{lm}({\vec {u}})\cdot [Y_{l'm'}({\vec {u}}')]^{*}}](/media/api/rest_v1/media/math/render/svg/ef9443f2c903ff800aa173ddbb20d96afd25cd61)
![{\displaystyle (\Lambda _{R}g)({\vec {p}})=g[R^{-1}({\vec {p}})]}](/media/api/rest_v1/media/math/render/svg/4d0aeefb352600886aaf85d33f1be86fb51bbf48)
![{\displaystyle (\Lambda _{R}g)({\vec {p}})=g[R^{-1}(r{\vec {u}})]=g[rR^{-1}({\vec {u}})]=\sum _{l=0}^{B-1}\sum _{m=-l}^{l}{\hat {g}}_{lm}(r)Y_{lm}[R^{-1}({\vec {u}})]}](/media/api/rest_v1/media/math/render/svg/01438051ba626d3bd6d15d8de319bc3a6cd80150)
![{\displaystyle Y_{lm}[R^{-1}({\vec {u}})]=\sum _{n}D_{nm}^{l}(R)Y_{ln}(\theta ,\varphi )}](/media/api/rest_v1/media/math/render/svg/809ba848322dbf7404d1e3a4911766e57f9c4619)
![{\displaystyle d_{mn}^{l}(\theta )=\sum _{t}(-i)^{t}\cdot {{[(l+m)!(l-m)!(l+n)!(l-n)!]^{1 \over 2}} \over {(l+m-t)!(l-n-t)!t!(t+n-m)!}}\cdot (cos{\theta \over 2})^{2l+m-n-2t}(sin{\theta \over 2})^{2t+n-m}}](/media/api/rest_v1/media/math/render/svg/897ed899e52be81d9585ff3f82e5e6153f8f5930)
![{\displaystyle c(R,R',\rho )=\int [\Lambda _{R'}f(r{\vec {u}})]^{*}\cdot [T_{\rho }\Lambda _{R}g(r{\vec {u}})]^{*}}](/media/api/rest_v1/media/math/render/svg/2e3594da90f1e2f48a7d9893b7e51a37ac661ad8)
