5.11: Legendre Polynomials

In this section we cover just enough about Legendre polynomials to be useful in the following section. Before starting, I want you to expand the following expression, by the binomial theorem, for $| x |< 1$, up to $x^4$ :

$$\dfrac{1}{(1-2x \cos θ + x^2)^{1/2}}. \label{5.11.1} \tag{5.11.1}$$

Please do go ahead and do it. Well, you probably won’t, so I’d better do it myself:

I’ll start with

$$(1-X)^{-1/2} = 1 + \dfrac{1}{2} X + \dfrac{3}{8} X^2 + \dfrac{5}{16} X^3 + \dfrac{35}{128} X^4 ... \label{5.11.2} \tag{5.11.2}$$

and therefore

$$[1-x(2\cos θ - x)]^{-1/2} = 1 + \dfrac{1}{2} x (2\cos θ - x) + \dfrac{3}{8} x^2 (2\cos θ - x)^2 + \dfrac{5}{16} x^3 (2\cos θ - x)^3 + \dfrac{35}{128} x^4 (2\cos θ - x)^4 ... \label{5.11.3} \tag{5.11.3}$$

$$= 1 + x \cos θ - \dfrac{1}{2} x^2 + \dfrac{3}{8}x^2 (4 \cos^2 θ - 4x \cos θ + x^2 ) + \dfrac{5}{16} x^3 (8 \cos^3 θ - 12x \cos^2 θ + 6x^2 \cos θ - x^3) + \dfrac{35}{128} x (16 \cos^4 θ - 32x \cos^3 θ + 24x^2 \cos^2 θ - 8x^3 \cos θ + x^4 ) ... \label{5.11.4} \tag{5.11.4}$$

$$= 1 + x \cos θ + x^2 (-\dfrac{1}{2} + \dfrac{3}{2} \cos^2 θ ) + x^3 (-\dfrac{3}{2} \cos θ + \dfrac{5}{2} \cos^3 θ ) + x^4 (\dfrac{3}{8} - \dfrac{15}{4} \cos^2 θ + \dfrac{35}{8} \cos^4 θ)... \label{5.11.5} \tag{5.11.5}$$

The coefficients of the powers of $x$ are the Legendre polynomials $P_l(\cos θ )$, so that

$$\dfrac{1}{(1-2x \cos θ + x^2)^{1/2}} = 1 + x P_1 ( \cos θ) + x^2 P_2 (\cos θ) + x^3 P_3 (\cos θ) + x^4 P_4 (\cos θ) + ... \label{5.11.6} \tag{5.11.6}$$

The Legendre polynomials with argument $\cos θ$ can be written as series of terms in powers of $\cos θ$ by substitution of $\cos θ$ for $x$ in Equations 1.12.5 in Section 1.12 of Chapter 1. Note that $x$ in Section 1 is not the same as $x$ in the present section. Alternatively they can be written as series of cosines of multiples of $θ$ as follows.

$$\begin{array}{l} P_0 = 1 \\ P_1 = \cos θ \\ P_2 = \dfrac{1}{4} (3\cos 2θ + 1) \\ P_3 = \dfrac{1}{8} (5\cos 3θ + 3\cos θ) \\ P_4 = \dfrac{1}{64} (35 \cos 4 θ + 20 \cos 2 θ + 9 ) \\ P_5 = \dfrac{1}{128} (63 \cos 5 θ + 35 \cos 3 θ + 30 \cos θ) \\ P_6 = \dfrac{1}{512} (231 \cos 6 θ + 126 \cos 4θ + 105 \cos 2θ + 50) \\ P_7 = \dfrac{1}{1024} (429 \cos 7θ + 231 \cos 5θ + 189 \cos 3θ + 175 \cos θ) \\ P_8 = (6435 \cos 8θ + 3432 \cos 6θ + 2772 \cos 4θ + 2520 \cos 2θ + 1225)/2^{14} \\ \label{5.11.7} \tag{5.11.7} \end{array}$$

For example, $P_6(\cos θ)$ can be written either as given by Equation $\ref{5.11.7}$, or as given by Equation 1, namely

$$P_6 = \dfrac{1}{16} (231c^6 - 315 c^4 + 105c^2 - 5), \text{ where } c = \cos θ. \label{5.11.8} \tag{5.11.8}$$

The former may look neater, and the latter may look “awkward” because of all the powers. However, the latter is far faster to compute, particularly when written as nested parentheses:

$$P_6 = (-5 + C(105 + C(-315 + 231C)))/16, \text{ where } C = \cos^2 θ. \label{5.11.9} \tag{5.11.9}$$