Special Functions

Abstract

The Legendre polynomial of degree l (l = 0,1,2,..., ∞) is defined by the formula

$$ P_l (u) = \frac{1} {{2^l l!}}{\mathbf{ }}\frac{{d^l }} {{du^l }}(u^2 - 1)^l {\mathbf{ }}. $$

(7)

(C.1) It is a polynomial with l zeros in the range (−1, +1) and with parity (− 1)^l. The first five Legendre polynomials are P ₀ = 1, (C.2) P ₁ = u, (C.3) P ₂ = 1/2(3u ²-1), (C.4) P ₃ = 1/2(5u ³−3u), (C.5) P ₄ = 1/8 (35u ⁴ − 30u ² + 3). (C.6)