Legendre Polynomials

Legendre polynomials

P_{n} (x)

are an infinite family of orthogonal polynomials on

[- 1, 1]

, with

P_{n}

of degree

n

. Key result: they form a complete orthogonal basis for

L^{2} [- 1, 1]

and are the polynomial eigenfunctions of the Legendre differential operator. Where they apply: solving Laplace's equation in spherical coordinates, multipole expansions, Gaussian quadrature, and approximation theory. Who/when: introduced by Adrien-Marie Legendre in 1782–1785 in studies of gravitational potential.

The Legendre polynomials are the sequence $P_{0}, P_{1}, P_{2}, \dots$ obtained by orthogonalizing the monomials $1, x, x^{2}, \dots$ with respect to the uniform weight on $[- 1, 1]$ , normalized so that $P_{n} (1) = 1$ . They arise naturally as the bounded solutions of Legendre's differential equation, which itself emerges when separating variables in Laplace's equation in spherical coordinates. They matter because they are the canonical tool for expanding functions on a sphere or interval, for computing electrostatic and gravitational potentials, and for designing optimal numerical integration rules. A surprising fact: although defined purely algebraically, they encode the angular structure of the inverse-distance kernel $1/∣ r - r^{'} ∣$ exactly — they are literally the coefficients in the Taylor expansion of that physically fundamental quantity.

The Core Insight

A single generating "seed" — the inverse-distance function $1/ 1 - 2 x t + t^{2}$ — encodes the entire family. Expanding it as a power series in $t$ produces the $P_{n} (x)$ as coefficients. Everything else (orthogonality, recurrences, the ODE, the values at $\pm 1$ ) follows from manipulating this one expression. The deeper unifying theme is orthogonality with respect to the uniform weight on $[- 1, 1]$ : the Legendre polynomials are the answer to "what is the natural orthogonal polynomial basis for the interval?"

Formal Statement

Legendre Polynomials via Rodrigues' Formula

The $n$ -th Legendre polynomial is defined by Rodrigues' formula:

P_{n} (x) = \frac{1}{2 ^{n} n !} \frac{d ^{n}}{d x ^{n}} [(x^{2} - 1)^{n}], n = 0, 1, 2, \dots

⟳

(x^{2} - 1)^{n}

is a degree-

2 n

polynomial whose

n

-th derivative is degree

n

; the prefactor

1/ (2^{n} n!)

fixes the normalization

P_{n} (1) = 1

Here $x \in [- 1, 1]$ is the variable (often $x = cos θ$ in physical applications), $n$ is the degree, and $P_{n}$ is a polynomial of exactly degree $n$ . The first few are:

P_{0} = 1, P_{1} = x, P_{2} = \frac{1}{2} (3 x^{2} - 1), P_{3} = \frac{1}{2} (5 x^{3} - 3 x), P_{4} = \frac{1}{8} (35 x^{4} - 30 x^{2} + 3) .

⟳apply [1] for each

n

, or use the recurrence [7] starting from

P_{0}, P_{1}

Legendre's Differential Equation

The $P_{n} (x)$ are the polynomial solutions of the second-order linear ODE

\frac{d}{d x} [(1 - x^{2}) \frac{d y}{d x}] + n (n + 1) y = 0,

equivalently $(1 - x^{2}) y^{''} - 2 x y^{'} + n (n + 1) y = 0$ .

⟳substitute

y = P_{n}

from Rodrigues' formula and differentiate, or derive from separation of variables in Laplace's equation where

n (n + 1)

is the separation constant.

This is a Sturm–Liouville problem with weight $w (x) = 1$ , eigenvalue $λ_{n} = n (n + 1)$ , and natural boundary conditions of boundedness at $x = \pm 1$ (regular singular points).

Orthogonality and Norm

On $[- 1, 1]$ the Legendre polynomials satisfy

\int_{- 1}^{1} P_{m} (x) P_{n} (x) d x = \frac{2}{2 n + 1} δ_{mn},

where $δ_{mn}$ is the Kronecker delta.

⟳orthogonality (the

m \neq = n

case) follows from the Sturm–Liouville structure of [3]; the norm

2/ (2 n + 1)

follows from integrating Rodrigues' formula by parts

n

times.

Generating Function

For $∣ t ∣ < 1$ and $∣ x ∣ \leq 1$ ,

\frac{1}{1 - 2 x t + t ^{2}} = n = 0 \sum \infty P_{n} (x) t^{n} .

⟳this is the multipole expansion of

1/∣ r - r^{'} ∣

with

x = cos θ

t = r_{<} / r_{>}

; expand the square root as a binomial/Taylor series in

t

and collect powers.

Completeness / Legendre Series

Every $f \in L^{2} [- 1, 1]$ has a convergent (in $L^{2}$ norm) expansion

f (x) = n = 0 \sum \infty c_{n} P_{n} (x), c_{n} = \frac{2 n + 1}{2} \int_{- 1}^{1} f (x) P_{n} (x) d x .

⟳project

f

onto each

P_{n}

using orthogonality [4]; the coefficient formula inverts to give

c_{n}

. Completeness follows from the Stone–Weierstrass theorem (polynomials are dense).

Three-Term Recurrence

The polynomials satisfy

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

with $P_{0} = 1$ , $P_{1} = x$ .

⟳differentiate the generating function [5] with respect to

t

, multiply by

(1 - 2 x t + t^{2})

, and equate coefficients of

t^{n}

Why It Holds / Derivation

Orthogonality from the Sturm–Liouville Form

Step 1. Write Legendre's equation [3] in self-adjoint form:

[(1 - x^{2}) P_{n}^{'}]^{'} = - n (n + 1) P_{n}

. Step 2. Multiply the equation for

P_{n}

P_{m}

and the equation for

P_{m}

P_{n}

P_{m} [(1 - x^{2}) P_{n}^{'}]^{'} = - n (n + 1) P_{m} P_{n},

P_{n} [(1 - x^{2}) P_{m}^{'}]^{'} = - m (m + 1) P_{m} P_{n} .

Step 3. Subtract and integrate over

[- 1, 1]

. The left side is a total derivative:

\int_{- 1}^{1} \frac{d}{d x} [(1 - x^{2}) (P_{m} P_{n}^{'} - P_{n} P_{m}^{'})] d x = [(1 - x^{2}) (P_{m} P_{n}^{'} - P_{n} P_{m}^{'})]_{- 1}^{1} .

Step 4. The boundary term vanishes because

(1 - x^{2}) = 0

x = \pm 1

. Hence

[m (m + 1) - n (n + 1)] \int_{- 1}^{1} P_{m} P_{n} d x = 0.

Step 5. For

m \neq = n

the bracket

m (m + 1) - n (n + 1) \neq = 0

, forcing the integral to be zero. This proves the

δ_{mn}

structure in [4].

■

Norm via Rodrigues' Formula

Step 1. Using [1],

\int_{- 1}^{1} P_{n}^{2} d x = \frac{1}{( 2 ^{n} n ! ) ^{2}} \int_{- 1}^{1} \frac{d ^{n}}{d x ^{n}} (x^{2} - 1)^{n} \cdot \frac{d ^{n}}{d x ^{n}} (x^{2} - 1)^{n} d x

. Step 2. Integrate by parts

n

times. Each boundary term vanishes because

(x^{2} - 1)^{n}

and its first

n - 1

derivatives are zero at

x = \pm 1

. This yields

\int_{- 1}^{1} P_{n}^{2} d x = \frac{( - 1 ) ^{n}}{( 2 ^{n} n ! ) ^{2}} \int_{- 1}^{1} (x^{2} - 1)^{n} \frac{d ^{2 n}}{d x ^{2 n}} (x^{2} - 1)^{n} d x .

Step 3. The

2 n

-th derivative of the degree-

2 n

polynomial

(x^{2} - 1)^{n}

is the constant

(2 n)!

. Step 4. Thus the integral becomes

\frac{( - 1 ) ^{n} ( 2 n )!}{( 2 ^{n} n ! ) ^{2}} \int_{- 1}^{1} (x^{2} - 1)^{n} d x = \frac{( 2 n )!}{( 2 ^{n} n ! ) ^{2}} \int_{- 1}^{1} (1 - x^{2})^{n} d x

. Step 5. Using

\int_{- 1}^{1} (1 - x^{2})^{n} d x = \frac{2 ^{2 n + 1} ( n ! ) ^{2}}{( 2 n + 1 )!}

(a Beta-function integral), everything cancels to give

\frac{2}{2 n + 1}

, confirming [4].

■

Adrien-Marie Legendre | 1752–1833

Legendre was a French mathematician who introduced these polynomials in 1782 while computing the gravitational attraction of spheroids ("Recherches sur l'attraction des sphéroïdes homogènes"). He recognized that the inverse-distance kernel could be expanded in a series whose coefficients are functions of the angle between two position vectors — these "Legendre coefficients" later bore his name. He also made foundational contributions to number theory (the law of quadratic reciprocity), elliptic integrals (the Legendre forms), and the method of least squares. Famously, the only authenticated portrait of him is a caricature, long mistaken for a different Legendre.

Building Intuition

Think of the monomials $1, x, x^{2}, x^{3}, \dots$ as a set of "directions" in function space that are not perpendicular — they overlap a lot (e.g. $\int_{- 1}^{1} 1 \cdot x^{2} d x \neq = 0$ ). The Legendre polynomials are exactly what you get by running Gram–Schmidt orthogonalization on the monomials using the inner product $⟨ f, g ⟩ = \int_{- 1}^{1} f g d x$ . So $P_{n}$ is "the part of $x^{n}$ that is perpendicular to all lower-degree polynomials." Geometrically, expanding a function in Legendre series is like finding its coordinates along mutually perpendicular axes — and because the axes are orthogonal, each coordinate $c_{n}$ is computed independently, with no interference. The oscillation count grows with $n$ : $P_{n}$ has exactly $n$ real roots, all inside $(- 1, 1)$ , so higher-degree terms capture finer wiggles of $f$ , exactly like Fourier modes capture higher frequencies.

:::svg[The first four Legendre polynomials on [-1,1]. Note all pass through (1,1) and that P_n has exactly n roots in (-1,1).]

Worked Examples

Expanding f(x) = x² in Legendre Polynomials

Given:

f (x) = x^{2}

[- 1, 1]

. Express it as

\sum c_{n} P_{n}

. Step 1. Since

f

is degree 2, only

P_{0}, P_{1}, P_{2}

can appear. Write

x^{2} = c_{0} P_{0} + c_{1} P_{1} + c_{2} P_{2}

. Step 2. Use the closed forms:

P_{0} = 1

P_{1} = x

P_{2} = \frac{1}{2} (3 x^{2} - 1)

. Solve

P_{2}

for

x^{2}

x^{2} = \frac{2 P _{2} + 1}{3} = \frac{2}{3} P_{2} + \frac{1}{3} P_{0} .

Step 3. Verify with the projection formula [6]. For

c_{2}

c_{2} = \frac{5}{2} \int_{- 1}^{1} x^{2} \cdot \frac{1}{2} (3 x^{2} - 1) d x = \frac{5}{4} \int_{- 1}^{1} (3 x^{4} - x^{2}) d x = \frac{5}{4} (\frac{6}{5} - \frac{2}{3}) = \frac{5}{4} \cdot \frac{8}{15} = \frac{2}{3} .

✓ Step 4. And

c_{1} = 0

by odd-symmetry,

c_{0} = \frac{1}{2} \int_{- 1}^{1} x^{2} d x = \frac{1}{3}

. ✓

x^{2} = \frac{1}{3} P_{0} (x) + \frac{2}{3} P_{2} (x)

(dimensionless coefficients)

Electrostatic Potential via Multipole Expansion

Given: A point charge

q = 2 \times 1 0^{- 9} C

sits on the

z

-axis at distance

a = 0.10 m

from the origin. Find the potential at a field point at radius

r = 0.50 m

, angle

θ = 6 0^{\circ}

from the

z

-axis, keeping the dipole term. Step 1. The exact potential is

V = \frac{q}{4 π ε _{0}} \frac{1}{∣ r - a ∣}

. With

∣ r - a ∣ = r^{2} - 2 r a cos θ + a^{2}

, factor out

r

and set

t = a / r = 0.20

x = cos 6 0^{\circ} = 0.5

. Step 2. By the generating function [5]:

V = \frac{q}{4 π ε _{0} r} n = 0 \sum \infty (\frac{a}{r})^{n} P_{n} (cos θ) .

Step 3. Compute terms with

k = \frac{q}{4 π ε _{0} r} = \frac{( 8.99 \times 1 0 ^{9} ) ( 2 \times 1 0 ^{- 9} )}{0.50} = 35.96 V

$n = 0$ : $P_{0} = 1$ , term $= k (1) (1) = 35.96 V$ .
$n = 1$ : $P_{1} (0.5) = 0.5$ , term $= k (0.20) (0.5) = 3.596 V$ .

Step 4. Sum monopole + dipole:

V \approx 35.96 + 3.60 = 39.56 V

V \approx 39.6 V

(The next term,

n = 2

P_{2} (0.5) = - 0.125

, contributes only

k (0.04) (- 0.125) \approx - 0.18

V, so the expansion converges quickly.)

Common Misconceptions

The normalization is NOT to unit norm. A frequent error is assuming

\int_{- 1}^{1} P_{n}^{2} d x = 1

. In fact it equals

\frac{2}{2 n + 1}

— the Legendre polynomials are orthogonal but not orthonormal. If you need an orthonormal set, use

P_{n} = \frac{2 n + 1}{2} P_{n}

. Forgetting the

\frac{2 n + 1}{2}

factor in the coefficient formula [6] is the single most common source of wrong Legendre series. Legendre polynomials live on $[- 1, 1]$ , not arbitrary intervals. To expand a function on

[a, b]

you must first change variables: let

x = \frac{2 u - ( a + b )}{b - a}

to map

[a, b] \to [- 1, 1]

, then apply the standard formulas. Skipping this rescaling destroys orthogonality.

One might guess that since $P_{n}$ are "nice" polynomials, the Legendre series of any continuous function converges pointwise everywhere. False. Like Fourier series, Legendre series can fail to converge pointwise at points of discontinuity or even at isolated continuous points for badly behaved functions; near a jump discontinuity they exhibit a Gibbs phenomenon with overshoot. Convergence is guaranteed in the $L^{2}$ norm (by completeness [6]), and pointwise only under additional smoothness (e.g. piecewise $C^{1}$ ). The mere fact that the basis functions are smooth polynomials does not transfer smoothness to the partial sums of a rough target.

Connections

Legendre polynomials sit at the center of a web of related ideas. They are the $m = 0$ case of the Associated Legendre Functions, which combine with $e^{im ϕ}$ to build the Spherical Harmonics $Y_{ℓ}^{m}$ — the angular eigenfunctions on the sphere. They are a special case ( $α = β = 0$ ) of the Jacobi Polynomials and the limiting case ( $α \to \frac{1}{2}$ ) of the Gegenbauer Polynomials. As orthogonal polynomials they belong to the general theory of Sturm-Liouville Theory, and their roots are the optimal nodes in Gaussian Quadrature. The expansion [6] is the analogue, for the interval $[- 1, 1]$ , of Fourier Series on a circle, both being instances of eigenfunction expansions for self-adjoint operators in Hilbert Spaces.

Real-World Applications

Electrostatics & gravitation (multipole expansion): Potentials of charge/mass distributions are expanded in $P_{n} (cos θ)$ via [5]; the $n = 0, 1, 2$ terms are the monopole, dipole, and quadrupole.
Geodesy & geophysics: Earth's gravitational and magnetic fields are modeled with spherical harmonics (built from $P_{n}$ ); satellite missions like GRACE report results as Legendre/spherical-harmonic coefficients.
Gaussian quadrature (Gauss–Legendre): The $n$ roots of $P_{n}$ give nodes that integrate polynomials up to degree $2 n - 1$ exactly — the backbone of high-accuracy numerical integration in engineering software.
Quantum mechanics: Angular parts of hydrogen-atom wavefunctions and rigid-rotor states are spherical harmonics; the orbital angular momentum eigenvalue $ℓ (ℓ + 1) ℏ^{2}$ is the $n (n + 1)$ of equation [3].
Computer graphics & rendering: "Spherical harmonic lighting" precomputes diffuse illumination by projecting environment light onto low-order $Y_{ℓ}^{m}$ , enabling real-time global illumination.
Antenna theory & acoustics: Radiation patterns and scattering from spheres are decomposed into Legendre/spherical-harmonic modes.
Statistics & approximation: Legendre series provide orthogonal polynomial regression and serve as a building block for spectral methods in solving PDEs.

Historical Development

1782: Adrien-Marie Legendre introduces the polynomials in his memoir on the attraction of spheroids, as coefficients in the expansion of the inverse-distance potential.
1785: Legendre publishes further results, establishing key properties and the connection to gravitational potentials.
1782–1799: Pierre-Simon Laplace develops spherical harmonics in the Mécanique Céleste, embedding Legendre's functions in the broader theory of harmonic analysis on the sphere.
1816: Olinde Rodrigues publishes the differentiation formula [1] (now called Rodrigues' formula) in his doctoral work.
1837: Charles Sturm and Joseph Liouville develop the general eigenvalue theory of second-order ODEs, retroactively explaining the orthogonality and completeness of $P_{n}$ .
1814 / 1826: Gauss (1814) and later Jacobi (1826) develop Gaussian quadrature, identifying the roots of $P_{n}$ as optimal integration nodes.
1875–1880s: Heine and others systematize the theory of orthogonal polynomials, placing Legendre polynomials within a general framework.
1885: Edward Heine's treatise consolidates results on spherical and Legendre functions.
20th century: With the rise of quantum mechanics (1920s) and digital computation (1950s onward), Legendre polynomials become indispensable in physics and numerical analysis; modern stable algorithms (e.g. via the recurrence [7]) make Gauss–Legendre quadrature routine.

Subtleties & Limits

While the theory of Legendre polynomials on $[- 1, 1]$ is classically complete, several research-adjacent subtleties persist. (1) Efficient computation of high-order Gauss–Legendre nodes: classical Newton iteration on $P_{n}$ becomes ill-conditioned for very large $n$ ; asymptotic-expansion-based methods (Bogaert, 2014) achieving $O (1)$ work per node are still being refined. (2) Optimal convergence rates of Legendre spectral methods for PDEs with corner singularities, and how to mitigate the Gibbs phenomenon via filtering or reprojection (e.g. Gegenbauer reconstruction). (3) Sharp Lebesgue-constant and pointwise-convergence estimates for Legendre series of functions of limited regularity remain an active topic in approximation theory. These are practical/analytical refinements rather than open foundational questions — the existence, orthogonality, and completeness are fully settled.