Legendre polynomial

The polynomial solution of degree l normalized by P_l(1)=1 is the Legendre polynomial P_l.

The Legendre polynomials are orthogonal on

[- 1, 1]

with unit weight:

\int_{- 1}^{1} P_{ℓ} (x) P_{m} (x) d x = \frac{2}{2 ℓ + 1} δ_{ℓ m} .

Legendre polynomial recurrence relation

The three-term recurrence

(2 ℓ + 1) x P_{ℓ} (x) = (ℓ + 1) P_{ℓ + 1} (x) + ℓ P_{ℓ - 1} (x)

x

into coupling between adjacent Legendre multipoles.

For a contour enclosing

t

P_{n} (t) = \frac{1}{2 ^{n + 1} πi} \oint \frac{( z ^{2} - 1 ) ^{n}}{( z - t ) ^{n + 1}} d z .

For fixed

0 < θ < π

P_{n} (cos θ) \sim \frac{2}{πn s i n θ} cos ((n + \frac{1}{2}) θ - \frac{π}{4}) .

For fixed

ξ > 0

P_{n} (cosh ξ) \sim \frac{e ^{(n + 1/2) ξ}}{2 πn s i n h ξ} .

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