Integration of spherical harmonics
$begingroup$
for my research i found myself in place to compute the following integral involving the product of spherical harmonics, $Y_l^m(theta,phi)$, and gradient of spherical harmonics.
$int_R^{infty} int_0^{2pi} int_{0}^{pi} left[ (frac{R}{r})^{l+1} Y_l^m (theta, phi) boldsymbol{nabla} left((frac{R}{r})^{l'+1} Y_{l'}^{-m'}(theta, phi) right) cdot boldsymbol{nabla} left((frac{R}{r})^{l''+1} Y_{l''}^{-m''}(theta, phi) right)right]r^2 sin(theta) , dr , dphi , d theta $
I know that to integrate the product of three associated Legendre polynomials of different order one can use the Gaunt's formulas to obtaine a closed expression. However, I am not sure that the integrand above can be reduced to just a multiplication of associate Legendre polynomials. Is it possible to get a general formula for the expression above for arbitrary indexes? Could you guide me into the derivation?
integration definite-integrals spherical-harmonics
asked Nov 30 '18 at 19:16
SSC NapoliSSC Napoli
286113
$endgroup$
add a comment |
$begingroup$
for my research i found myself in place to compute the following integral involving the product of spherical harmonics, $Y_l^m(theta,phi)$, and gradient of spherical harmonics.
$int_R^{infty} int_0^{2pi} int_{0}^{pi} left[ (frac{R}{r})^{l+1} Y_l^m (theta, phi) boldsymbol{nabla} left((frac{R}{r})^{l'+1} Y_{l'}^{-m'}(theta, phi) right) cdot boldsymbol{nabla} left((frac{R}{r})^{l''+1} Y_{l''}^{-m''}(theta, phi) right)right]r^2 sin(theta) , dr , dphi , d theta $
I know that to integrate the product of three associated Legendre polynomials of different order one can use the Gaunt's formulas to obtaine a closed expression. However, I am not sure that the integrand above can be reduced to just a multiplication of associate Legendre polynomials. Is it possible to get a general formula for the expression above for arbitrary indexes? Could you guide me into the derivation?
integration definite-integrals spherical-harmonics
asked Nov 30 '18 at 19:16
SSC NapoliSSC Napoli
286113
$endgroup$
add a comment |
$begingroup$
for my research i found myself in place to compute the following integral involving the product of spherical harmonics, $Y_l^m(theta,phi)$, and gradient of spherical harmonics.
$int_R^{infty} int_0^{2pi} int_{0}^{pi} left[ (frac{R}{r})^{l+1} Y_l^m (theta, phi) boldsymbol{nabla} left((frac{R}{r})^{l'+1} Y_{l'}^{-m'}(theta, phi) right) cdot boldsymbol{nabla} left((frac{R}{r})^{l''+1} Y_{l''}^{-m''}(theta, phi) right)right]r^2 sin(theta) , dr , dphi , d theta $
I know that to integrate the product of three associated Legendre polynomials of different order one can use the Gaunt's formulas to obtaine a closed expression. However, I am not sure that the integrand above can be reduced to just a multiplication of associate Legendre polynomials. Is it possible to get a general formula for the expression above for arbitrary indexes? Could you guide me into the derivation?
integration definite-integrals spherical-harmonics
asked Nov 30 '18 at 19:16
SSC NapoliSSC Napoli
286113
$endgroup$
for my research i found myself in place to compute the following integral involving the product of spherical harmonics, $Y_l^m(theta,phi)$, and gradient of spherical harmonics.
$int_R^{infty} int_0^{2pi} int_{0}^{pi} left[ (frac{R}{r})^{l+1} Y_l^m (theta, phi) boldsymbol{nabla} left((frac{R}{r})^{l'+1} Y_{l'}^{-m'}(theta, phi) right) cdot boldsymbol{nabla} left((frac{R}{r})^{l''+1} Y_{l''}^{-m''}(theta, phi) right)right]r^2 sin(theta) , dr , dphi , d theta $
I know that to integrate the product of three associated Legendre polynomials of different order one can use the Gaunt's formulas to obtaine a closed expression. However, I am not sure that the integrand above can be reduced to just a multiplication of associate Legendre polynomials. Is it possible to get a general formula for the expression above for arbitrary indexes? Could you guide me into the derivation?
integration definite-integrals spherical-harmonics
integration definite-integrals spherical-harmonics
asked Nov 30 '18 at 19:16
SSC NapoliSSC Napoli
286113
asked Nov 30 '18 at 19:16
SSC NapoliSSC Napoli
286113
asked Nov 30 '18 at 19:16
SSC NapoliSSC Napoli
286113
asked Nov 30 '18 at 19:16
SSC NapoliSSC Napoli
286113
asked Nov 30 '18 at 19:16
SSC NapoliSSC Napoli
286113
286113
add a comment |
add a comment |
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