Associated Legendre function > Catalogs

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An informational catalog, or several catalogs, about Associated Legendre function.

The associated Legendre functions through l = 6 are:

$\begin{align} P_0^0(x) &= 1 \\ \\ P_1^0(x) &= x \\ P_1^1(x) & = (1-x^2)^{1/2} \\ \\ P_2^0(x) &= \tfrac{1}{2}(3x^2-1)\\ P_2^1(x) &= 3(1-x^2)^{1/2} x\\ P_2^2(x) &= 3(1-x^2) \\ \\ P_3^0(x) &= \tfrac{1}{2}(5x^3 -3x)\\ P_3^1(x) &= \tfrac{1}{2}(1-x^2)^{1/2} (15x^2-3) \\ P_3^2(x) &= 15(1-x^2)x \\ P_3^3(x) &= 15 (1-x^2)^{3/2} \\ \\ P_4^0(x) &= \tfrac{1}{8}(35x^4- 30x^2 + 3)\\ P_4^1(x) &= \tfrac{1}{2}(1-x^2)^{1/2} (35x^3 - 15x) \\ P_4^2(x) &= \tfrac{1}{2}(1-x^2)(105x^2 -15) \\ P_4^3(x) &= 105 (1-x^2)^{3/2} x \\ P_4^4(x) &= 105 (1-x^2)^{2} \\ \\ P_5^0(x) &= \tfrac{1}{8}(63x^5- 70x^3 + 15x)\\ P_5^1(x) &= \tfrac{1}{8}(1-x^2)^{1/2} (315x^4 - 210x^2 + 15) \\ P_5^2(x) &= \tfrac{1}{2}(1-x^2)(315x^3 -105x) \\ P_5^3(x) &= \tfrac{1}{2} (1-x^2)^{3/2} (945x^2 -105) \\ P_5^4(x) &= 945 (1-x^2)^{2} x \\ P_5^5(x) &= 945 (1-x^2)^{5/2} \\ \\ P_6^0(x) &= \tfrac{1}{16}(231x^6- 315x^4 + 105x^2 -5)\\ P_6^1(x) &= \tfrac{1}{8}(1-x^2)^{1/2} (693x^5 - 630x^3 + 105x) \\ P_6^2(x) &= \tfrac{1}{8}(1-x^2)(3465 x^4 - 1890 x^2 +105) \\ P_6^3(x) &= \tfrac{1}{2} (1-x^2)^{3/2} (3465x^3-945x) \\ P_6^4(x) &= \tfrac{1}{2} (1-x^2)^{2} (10395x^2-945) \\ P_6^5(x) &= 10395 (1-x^2)^{5/2} x \\ P_6^6(x) &= 10395 (1-x^2)^{3} \\ \end{align}$