Legendre Polynomials

Legendre polynomials $P_n(x)$ are solutions to Legendre's differential equation and arise naturally in problems with spherical symmetry, such as solving the Laplace equation in spherical coordinates.

Three-Term Recurrence

The polynomials can be generated efficiently using the recurrence relation:

$(n+1) P_{n+1}(x) = (2n+1)\, x\, P_n(x) - n\, P_{n-1}(x)$

with initial conditions $P_0(x) = 1$ and $P_1(x) = x$.

The first few polynomials:

• $P_0(x) = 1$
• $P_1(x) = x$
• $P_2(x) = \frac{1}{2}(3x^2 - 1)$
• $P_3(x) = \frac{1}{2}(5x^3 - 3x)$
• $P_4(x) = \frac{1}{8}(35x^4 - 30x^2 + 3)$

Orthogonality

Legendre polynomials are orthogonal on $[-1, 1]$:

$\int_{-1}^{1} P_m(x)\, P_n(x)\, dx = \frac{2}{2n+1}\, \delta_{mn}$

The normalization constant $\|P_n\|^2 = 2/(2n+1)$ is returned by legendre_norm(n).

Rodrigues' Formula

An explicit formula using derivatives:

$P_n(x) = \frac{1}{2^n n!} \frac{d^n}{dx^n} (x^2 - 1)^n$

Legendre Series

Any sufficiently smooth function on $[-1, 1]$ can be expanded in a Legendre series:

$f(x) \approx \sum_{n=0}^{N} c_n P_n(x)$

The coefficients are $c_n = \frac{2n+1}{2} \int_{-1}^{1} f(x) P_n(x)\, dx$.

Your Task

Implement the three functions below. Use only Python's math module and built-in functions.