ルジャンドル多項式

最初のいくつかの値：

$$P_0(z) = 1,$$ (C.2.1)

$$P_1(z) = z,$$ (C.2.2)

$$P_2(z) = \frac{3z^2 - 1}{2}$$ (C.2.3)

$$P_3(z) = \frac{5z^3 - 3z}{2}$$ (C.2.4)

$$P_4(z) = \frac{35z^4 - 30z^2 + 3}{8}$$ (C.2.5)

巾乗をルジャンドル多項式で表すと次のようになる：

$$1 = P_0(z),$$ (C.2.6)

$$z = P_1(z),$$ (C.2.7)

$$z^2 = \frac23 P_2(z) + \frac13 P_0(z),$$ (C.2.8)

$$z^3 = \frac25 P_3(z) + \frac35 P_1(z),$$ (C.2.9)

$$z^4 = \frac{8}{35} P_4(z) + \frac47 P_2(z) + \frac15 P_0(z)$$ (C.2.10)

直交性と完全性：

$$\int_{-1}^{1} d\mu P_m(\mu) P_n(\mu) = \frac{2\delta_{mn}}{2m + 1}$$ (C.2.11)

$$\sum_{l = 0}^\infty (2l+1) P_l(\mu) P_l(\mu') = 2\delta(\mu - \mu')$$ (C.2.12)