Associated Legendre Polynomials Orthogonality Proof: $\int_{-1}^1 P_k^m(x) \cdot P_l^m(x) \; \mathrm{d} x = \frac{2(l+m)!}{(2l+1)(l-m)!} \delta_{k,l}$

score 114 · Answer 8 · 2026-03-26 20:41:56

114

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Understanding part of the proof that $\int_{x=-1}^1P_L(x)P_M(x)\,\rm{d} x =0\quad\text{(for}\,\, L\ne M)$

Published on 26 Mar 2026 - 20:41

score 366 · Answer 9 · 2016-04-15 11:52:45

366

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Integral of Legendre Polynomial

Published on 15 Apr 2016 - 11:52

score 161 · Answer 10 · 2016-04-17 23:57:16

161

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Prove that $\displaystyle\int_{x=-1}^{1}P_L(x)P_{L-1}\acute (x)\,\mathrm{d}x=\int_{x=-1}^{1}P_L\acute(x)P_{L+1} (x)\,\mathrm{d}x=0$

Published on 17 Apr 2016 - 23:57

score 324 · Answer 11 · 2026-03-26 08:08:55

324

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Need help understanding part of the proof that $\displaystyle\int_{x=-1}^{1}[P_L(x)]^2\,\mathrm{d}x=\frac{2}{2L+1}$

Published on 26 Mar 2026 - 8:08

score 76 · Answer 12 · 2016-04-18 06:42:08

76

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How does one simplify this series expression?

Published on 18 Apr 2016 - 6:42

score 404 · Answer 13 · 2016-04-20 22:51:59

404

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Asymptotic behavior of integrals of Legendre polynomials

Published on 20 Apr 2016 - 22:51

score 1.4k · Answer 14 · 2016-04-24 05:16:03

1.4k

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Deriving the Normalization formula for Associated Legendre functions: Stage $1$ of $4$

Published on 24 Apr 2016 - 5:16

score 199 · Answer 15 · 2016-05-01 17:52:59

199

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How to proof the following properties about legendre function?

Published on 01 May 2016 - 17:52

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Associated Legendre Polynomials!?

Associated Legendre Polynomials Orthogonality Proof: $\int_{-1}^1 P_k^m(x) \cdot P_l^m(x) \; \mathrm{d} x = \frac{2(l+m)!}{(2l+1)(l-m)!} \delta_{k,l}$

Understanding part of the proof that $\int_{x=-1}^1P_L(x)P_M(x)\,\rm{d} x =0\quad\text{(for}\,\, L\ne M)$

Integral of Legendre Polynomial

Prove that $\displaystyle\int_{x=-1}^{1}P_L(x)P_{L-1}\acute (x)\,\mathrm{d}x=\int_{x=-1}^{1}P_L\acute(x)P_{L+1} (x)\,\mathrm{d}x=0$

Need help understanding part of the proof that $\displaystyle\int_{x=-1}^{1}[P_L(x)]^2\,\mathrm{d}x=\frac{2}{2L+1}$

How does one simplify this series expression?

Asymptotic behavior of integrals of Legendre polynomials

Deriving the Normalization formula for Associated Legendre functions: Stage $1$ of $4$

How to proof the following properties about legendre function?

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