Integral of Legendre polynomials and Associated Legendre polynomials
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I am trying to find the solution to the following integrals. I am quite sure these are well known, but somehow i did not find them already solved online:
1) $int_0^pi P_l(cos{theta}) P_{l'}^{pm 1} (cos{theta}) , dtheta$
2) $int_0^pi P_l(cos{theta}) frac{d}{dtheta} left[P_{l'}^{pm 1} (cos{theta}) right] sin{theta} , dtheta$
and
3) $int_0^pi P_l(cos{theta}) frac{d}{dtheta} left[P_{l'} (cos{theta}) right] sin{theta} , dtheta$
Where $P_l (cos{theta})$ is the Legendre polynomial of degree $l$, $P_{l'}^{pm 1} (cos{theta})$ is the associate Legendre polynomial of degree $l'$ and order $pm 1$. I am only interested to the case of $l$ and $l'$ integers and $l ge 0$ and $l^{'} ge 0$.
integration definite-integrals legendre-polynomials
asked Jan 8 at 18:11
SSC NapoliSSC Napoli
291113
$endgroup$
add a comment |
$begingroup$
I am trying to find the solution to the following integrals. I am quite sure these are well known, but somehow i did not find them already solved online:
1) $int_0^pi P_l(cos{theta}) P_{l'}^{pm 1} (cos{theta}) , dtheta$
2) $int_0^pi P_l(cos{theta}) frac{d}{dtheta} left[P_{l'}^{pm 1} (cos{theta}) right] sin{theta} , dtheta$
and
3) $int_0^pi P_l(cos{theta}) frac{d}{dtheta} left[P_{l'} (cos{theta}) right] sin{theta} , dtheta$
Where $P_l (cos{theta})$ is the Legendre polynomial of degree $l$, $P_{l'}^{pm 1} (cos{theta})$ is the associate Legendre polynomial of degree $l'$ and order $pm 1$. I am only interested to the case of $l$ and $l'$ integers and $l ge 0$ and $l^{'} ge 0$.
integration definite-integrals legendre-polynomials
asked Jan 8 at 18:11
SSC NapoliSSC Napoli
291113
$endgroup$
add a comment |
$begingroup$
I am trying to find the solution to the following integrals. I am quite sure these are well known, but somehow i did not find them already solved online:
1) $int_0^pi P_l(cos{theta}) P_{l'}^{pm 1} (cos{theta}) , dtheta$
2) $int_0^pi P_l(cos{theta}) frac{d}{dtheta} left[P_{l'}^{pm 1} (cos{theta}) right] sin{theta} , dtheta$
and
3) $int_0^pi P_l(cos{theta}) frac{d}{dtheta} left[P_{l'} (cos{theta}) right] sin{theta} , dtheta$
Where $P_l (cos{theta})$ is the Legendre polynomial of degree $l$, $P_{l'}^{pm 1} (cos{theta})$ is the associate Legendre polynomial of degree $l'$ and order $pm 1$. I am only interested to the case of $l$ and $l'$ integers and $l ge 0$ and $l^{'} ge 0$.
integration definite-integrals legendre-polynomials
asked Jan 8 at 18:11
SSC NapoliSSC Napoli
291113
$endgroup$
I am trying to find the solution to the following integrals. I am quite sure these are well known, but somehow i did not find them already solved online:
1) $int_0^pi P_l(cos{theta}) P_{l'}^{pm 1} (cos{theta}) , dtheta$
2) $int_0^pi P_l(cos{theta}) frac{d}{dtheta} left[P_{l'}^{pm 1} (cos{theta}) right] sin{theta} , dtheta$
and
3) $int_0^pi P_l(cos{theta}) frac{d}{dtheta} left[P_{l'} (cos{theta}) right] sin{theta} , dtheta$
Where $P_l (cos{theta})$ is the Legendre polynomial of degree $l$, $P_{l'}^{pm 1} (cos{theta})$ is the associate Legendre polynomial of degree $l'$ and order $pm 1$. I am only interested to the case of $l$ and $l'$ integers and $l ge 0$ and $l^{'} ge 0$.
integration definite-integrals legendre-polynomials
integration definite-integrals legendre-polynomials
asked Jan 8 at 18:11
SSC NapoliSSC Napoli
291113
asked Jan 8 at 18:11
SSC NapoliSSC Napoli
291113
asked Jan 8 at 18:11
SSC NapoliSSC Napoli
291113
asked Jan 8 at 18:11
SSC NapoliSSC Napoli
291113
asked Jan 8 at 18:11
SSC NapoliSSC Napoli
291113
291113
add a comment |
add a comment |
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