User:Favorite game

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\rho\!\,

\tilde r^2=r^2-2r{\rho}cos\theta+{\rho}^2 \!\,

\tilde \varphi=\varphi

cos\tilde \theta={rcos\theta-\rho \over \tilde r}\!\,

(T_{\rho} g)(\vec p)=(T_{\rho} g)(r \vec u)=(T_{\rho} g)(r, \theta, \varphi)=g(r', \theta', \varphi')

(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(\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\!\,

(\xi, \eta, \omega, \xi', \eta')\!\,

c(R)=\sum_{lmhm'} d_{mh}^ld_{hm'}^lexp[-i(n\xi+h\eta+m\omega)] \cdot I_{mm'}^l = T(\xi,\eta,\omega)

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 dr sin\theta d\theta

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}^ld_{hm}^le^{-i[n\xi+h\eta+m\omega]}

\pi/2\!\,

\xi=\varphi-\pi/2\!\,

\eta=\pi-\theta\!\,

\omega=\psi-\pi/2\!\,

d_{mn}^l=d_{mn}^l(\pi/2)

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=(\pi-\theta,\pi/2,\psi-\pi/2)\!\,

R_1=(\varphi-\pi/2,\pi/2,0)

R=(\varphi,\theta,\psi)

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)=\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')]^*

\vec{p}

\vec{p}=r\vec{u}

|\vec{p}|=r

|\vec{u}|=1

R\!\,

\Lambda_R\!\,

(\Lambda_Rg)(\vec{p})=g[R^{-1}(\vec{p})]

\vec{p}\!\,

f(r\vec{u})

g(r\vec{u})

B\!\,

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(\vec{p})=f(r\vec{u}) \approx \sum_{l=0}^{B-1} \sum_{m=-l}^{l} \hat{f}_{lm}(r)Y_{lm}(\theta, \varphi)

(\Lambda_Rg)(\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_{lm}[R^{-1}(\vec{u})]=\sum_n D_{nm}^l(R)Y_{ln}(\theta, \varphi)

D_{mn}^l\!\,

D_{mn}^l(\varphi,\theta,\psi)=e^{-i(m\varphi+n\psi)}d_{mn}^l(\theta)

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}

(\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', \rho)=\int [\Lambda_{R'} f(r\vec{u})]^* \cdot [T_\rho \Lambda_R g(r\vec{u})]^*