Legendre Polynomials

Equivalent forms

(1-x^2)P_n'' - 2x P_n' + n(n+1)P_n = 0

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

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

An orthogonal basis handed to us by the geometry of the sphere — the same polynomials describe planets, atoms, and antenna patterns.

Symbol	Name	SI unit	Dimension	Range
$n$	Degree Order of the polynomial	1	1	0 – 6
$x$	Argument Usually cosθ, on [-1,1]	1	1	-1 – 1
$P_n$	Polynomial valueoutput Value of P_n at x	1	1	-1.2 – 1.2

Unit systems

Symbol	SI	CGS	Imperial
$x$	1	1	1

Where it holds

Defined for all x; the orthogonality and physics live on [-1,1] with

x = \cos \theta

. Solutions bounded at the poles require integer n.

Dimensional analysis

P_n(x) dimensionless

polynomial in x

x = \cos \theta is

a pure number, so every P_n(x) is dimensionless.

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Discovery

Adrien-Marie Legendre · 1782

Legendre introduced them in 1782 studying the gravitational potential of spheroids (like the flattened Earth). They reappear as the polar part of the Laplace equation's solutions and, with an azimuthal factor, form the spherical harmonics central to the hydrogen atom.

Try this

The electric field of a lopsided charge, the shape of a spinning star, the tone of a struck sphere — all get sorted into the same family of wiggly polynomials that are the natural language of roundness.

Step through the degree and watch each polynomial add one more wiggle across [-1, 1] while staying perfectly 'orthogonal' to all the others.

Research status: stable

Real-world applications

Multipole expansion of gravitational and electric potentials
Angular part of the hydrogen atom (spherical harmonics)
Gaussian quadrature nodes (roots of P_n)
Earth-shape (geoid) and antenna radiation patterns

Common misconceptions

They're orthogonal for any weight — the plain P_n are orthogonal with weight 1 on [-1,1]; other intervals/weights give different families.
Higher n just means bigger — it means more angular oscillations (nodes), not larger amplitude.
They're the full spherical harmonic — P_n is only the $m=0$ polar part.

Experimental verification

Multipole expansions of measured gravitational (Earth'

s J_{2}

oblateness) and electric fields use Legendre coefficients that match satellite and lab data; atomic orbital shapes (s, p, d) follow the associated Legendre / spherical-harmonic angular patterns seen in spectroscopy.

Derivation

Rodrigues' formula

P_n = (1/2^{n} n!) d^{n}/dx^{n} (x^{2}-1)^{n}

generates them; equivalently apply Gram–Schmidt to {1,x,

x^{2}

,…} with the

L^{2}[-1

,1] inner product.

They solve Legendre's ODE, the polar equation from separating Laplace's equation in spherical coordinates.

Limiting cases

n = 0

⟶

P_{0} = 1

The monopole / constant term.

n = 1

⟶

P_{1} = x

The dipole term — linear in

\cos \theta

.

x = 1

⟶

P_n(1) = 1

Every Legendre polynomial is normalized to 1 at the pole.

What if…

What if

m \neq 0

(azimuthal dependence)?

You need the associated Legendre functions P_n^m, and together with $e^{im\varphi }$ they form the spherical harmonics Y_n^m.

What if the interval or weight changes?

Orthogonalizing on other domains gives cousins: Chebyshev, Hermite (Gaussian weight), or Laguerre (0.. $\infty )$ .

1

Build P₂ from Rodrigues' formula

Given ·

n:: 2

Find ·

P_{2}(x)

Steps

$(x^{2}-1)^{2} = x^{4}-2x^{2}+1$
$d^{2}/dx^{2} = 12x^{2}-4$
$\div (2^{2}\cdot 2!) = \tfrac{1}{2}(3x^{2}-1)$

Result ·

\tfrac{1}{2}(3x^{2} - 1)

2

Orthogonality check

Given ·

m:: 1
n:: 2

Find ·

\int _{-1}^{1} P_{1}P_{2} dx

Steps

$\int _{-1}^{1} x\cdot \tfrac{1}{2}(3x^{2}-1) dx$
Odd integrand over symmetric interval
$= 0$

Result · 0

laplace equation mathhermite polynomialsgram schmidtorthogonal polynomials