Legendre Polynomials and Associated Legendre Functions

One line: Legendre polynomials

P_{n} (x)

are the orthogonal polynomial solutions to Legendre's differential equation on

[- 1, 1]

, and the associated Legendre functions

P_{n}^{m} (x)

generalize them to handle the angular part of problems with azimuthal dependence. Key result:

{P_{n} (x)}

form a complete orthogonal basis for

L^{2} [- 1, 1]

with

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

, and the associated functions

P_{n}^{m} (cos θ) e^{im ϕ}

build the spherical harmonics. Where it applies: Solutions of Laplace's and Helmholtz's equations in spherical coordinates — electrostatics, gravitation, quantum mechanics (hydrogen atom), geophysics, and antenna theory. Who/when: Introduced by Adrien-Marie Legendre (1782) studying gravitational potentials; the associated functions developed through the 19th century by Ferrers, Heine, and others.

The Legendre polynomials $P_{n} (x)$ are a sequence of polynomials, one for each non-negative integer $n$ , of degree $n$ , defined on $[- 1, 1]$ and mutually orthogonal under the uniform weight. They arise naturally whenever one separates variables in Laplace's Equation in spherical coordinates with no azimuthal ( $ϕ$ ) dependence. The associated Legendre functions $P_{n}^{m} (x)$ extend this to nonzero azimuthal quantum numbers $m$ and are the polar-angle ingredients of the Spherical Harmonics. A surprising fact: although they look like specialized special functions, $P_{n} (x)$ appear as the coefficients in the generating function $(1 - 2 x t + t^{2})^{- 1/2} = \sum_{n} P_{n} (x) t^{n}$ — exactly the expansion of a Coulomb/Newton potential in powers of distance ratios, which is why multipole expansions exist.

The Core Insight

A self-adjoint second-order differential operator on a finite interval, with the right boundary behavior, produces a complete orthogonal family of eigenfunctions. For the angular Laplacian on the sphere, that operator is the Legendre operator, its eigenfunctions are the Legendre polynomials (when $m = 0$ ) and associated Legendre functions (when $m \neq = 0$ ), and their eigenvalues $ℓ (ℓ + 1)$ encode angular momentum. Everything else — Rodrigues' formula, recurrences, generating functions, orthogonality — flows from this one Sturm–Liouville structure.

Formal Statement

Legendre's Differential Equation

Legendre's equation is the second-order linear ODE

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

⟳Separate variables in

\nabla^{2} Φ = 0

in spherical coordinates; the polar-angle equation in

u = cos θ

takes this form with separation constant

n (n + 1)

Here $x$ is the independent variable (typically $x = cos θ$ ), $y (x)$ the unknown function, and $n$ the degree (a real parameter; bounded polynomial solutions exist only for $n = 0, 1, 2, \dots$ ). When $n$ is a non-negative integer, the polynomial solution normalized so that $P_{n} (1) = 1$ is the Legendre polynomial $P_{n} (x)$ .

Rodrigues' Formula

The Legendre polynomials are given explicitly by

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

⟳

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

is a polynomial annihilated by a clever operator; differentiating

n

times produces a degree-

n

solution of Legendre's equation; the prefactor enforces

P_{n} (1) = 1

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

Generating Function

The generating function packs all Legendre polynomials into one expansion:

g (x, t) = (1 - 2 x t + t^{2})^{- 1/2} = n = 0 \sum \infty P_{n} (x) t^{n}, ∣ t ∣ < 1.

⟳Expand

\frac{1}{∣ r - r ^{'} ∣} = \frac{1}{r} (1 - 2 (r^{'} / r) cos γ + (r^{'} / r)^{2})^{- 1/2}

in powers of

t = r^{'} / r

; the binomial coefficients organize into Legendre polynomials of

cos γ

Associated Legendre Functions

For integers $0 \leq m \leq n$ , the associated Legendre functions are

P_{n}^{m} (x) = (- 1)^{m} (1 - x^{2})^{m /2} \frac{d ^{m}}{d x ^{m}} P_{n} (x),

⟳Substitute

y = (1 - x^{2})^{m /2} u

into the associated Legendre equation; the function

u = d^{m} P_{n} / d x^{m}

satisfies the resulting ODE, recovering

P_{n}^{m}

up to the Condon–Shortley phase

(- 1)^{m}

with the $(- 1)^{m}$ factor being the Condon–Shortley phase (conventions vary). They satisfy the associated Legendre equation

(1 - x^{2}) y^{''} - 2 x y^{'} + [n (n + 1) - \frac{m ^{2}}{1 - x ^{2}}] y = 0.

⟳This is the full polar equation from separating

\nabla^{2} Φ = 0

when the azimuthal solution is

e^{im ϕ}

, contributing the

- m^{2} / sin^{2} θ

term.

Note $P_{n}^{0} (x) = P_{n} (x)$ , and $P_{n}^{m} \equiv 0$ for $m > n$ .

Orthogonality and Completeness

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

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

⟳Write Legendre's equation in Sturm–Liouville form

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

; subtract the equations for

P_{m}, P_{n}

, integrate, and the boundary term vanishes because

(1 - x^{2}) \to 0

at the endpoints.

The associated functions obey, for fixed $m$ ,

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

⟳Same Sturm–Liouville argument on Eq. [5] (with weight 1 since the

m^{2}

term has no

x

-derivative); the factorial factor comes from repeated integration by parts in Rodrigues' form.

Both families are complete in $L^{2} [- 1, 1]$ (for fixed $m$ in the associated case), so any square-integrable function admits a convergent Legendre series $f (x) = \sum_{n} c_{n} P_{n} (x)$ with $c_{n} = \frac{2 n + 1}{2} \int_{- 1}^{1} f (x) P_{n} (x) d x$ .

Why It Holds / Derivation

Orthogonality via Sturm–Liouville

Step 1. Cast Legendre's equation into self-adjoint form. Multiplying Eq. [1] is unnecessary; it is already exact:

\frac{d}{d x} [(1 - x^{2}) P_{n}^{'}] + n (n + 1) P_{n} = 0.

This is a Sturm–Liouville problem with

p (x) = 1 - x^{2}

, weight

w (x) = 1

, eigenvalue

λ_{n} = n (n + 1)

. Step 2. Write the same equation for

P_{m}

with eigenvalue

λ_{m}

. Multiply the

P_{n}

equation by

P_{m}

and the

P_{m}

equation by

P_{n}

, then subtract:

P_{m} \frac{d}{d x} [(1 - x^{2}) P_{n}^{'}] - P_{n} \frac{d}{d x} [(1 - x^{2}) P_{m}^{'}] + (λ_{n} - λ_{m}) P_{m} P_{n} = 0.

Step 3. The first two terms combine into a total derivative (Lagrange's identity):

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

Step 4. Integrate over

[- 1, 1]

. The left side equals

[(1 - x^{2}) (\dots)]_{- 1}^{1} = 0

because

1 - x^{2}

vanishes at both endpoints. Thus

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

For

m \neq = n

λ_{m} \neq = λ_{n}

, forcing the integral to vanish. Step 5. For the normalization

m = n

, use the generating function. Squaring Eq. [3] and integrating:

\int_{- 1}^{1} \frac{d x}{1 - 2 x t + t ^{2}} = n \sum t^{2 n} \int_{- 1}^{1} P_{n}^{2} d x .

The left integral evaluates to

\frac{1}{t} ln \frac{1 + t}{1 - t} = \sum_{n} \frac{2}{2 n + 1} t^{2 n}

. Matching powers of

t^{2 n}

gives

\int P_{n}^{2} d x = \frac{2}{2 n + 1}

■

Rodrigues' Formula Solves Legendre's Equation

Step 1. Let

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

. Then

v^{'} = 2 n x (x^{2} - 1)^{n - 1}

, so

(x^{2} - 1) v^{'} = 2 n x v

. Step 2. Differentiate

(x^{2} - 1) v^{'} = 2 n xv

exactly

(n + 1)

times using the Leibniz rule. Each side produces a chain of terms; collecting them yields

(x^{2} - 1) v^{(n + 2)} + 2 x v^{(n + 1)} - n (n + 1) v^{(n)} = 0.

Step 3. Let

w = v^{(n)} = \frac{d ^{n}}{d x ^{n}} (x^{2} - 1)^{n}

. The previous line reads

(1 - x^{2}) w^{''} - 2 x w^{'} + n (n + 1) w = 0,

which is exactly Legendre's equation [1]. Hence

w

— and any constant multiple — solves it. Step 4. The prefactor

1/ (2^{n} n!)

normalizes so that

P_{n} (1) = 1

: evaluating the highest-order contribution near

x = 1

gives the factor

2^{n} n!

, which the prefactor cancels.

■

Adrien-Marie Legendre | 1752–1833

A French mathematician whose name attaches to an extraordinary range of results — least squares, elliptic integrals, the prime-counting conjecture, and these polynomials. In 1782 he introduced what he called fonctions de Legendre (later named coefficients) while studying the Newtonian gravitational attraction of spheroids — exactly the multipole problem encoded in Eq. [3]. Legendre worked partly in the shadow of Laplace, who extended the angular theory to the full sphere (Laplace's coefficients, now Spherical Harmonics). A famous portrait long thought to be Legendre was discovered in 2005 to depict a different man; the only authentic likeness is a 1820 caricature.

Building Intuition

Think of $[- 1, 1]$ as the range of $cos θ$ as $θ$ sweeps a sphere from north pole to south pole. Legendre polynomials are the natural "harmonics" of that one-dimensional interval, just as $sin (nπ x)$ are the harmonics of a vibrating string. $P_{0}$ is constant (a uniform sphere), $P_{1} = cos θ$ is a dipole pattern (top vs. bottom), $P_{2}$ is a quadrupole (squashing/stretching), and higher $n$ pack more zeros — exactly $n$ of them inside $(- 1, 1)$ — meaning finer angular oscillation.

The index $m$ in the associated functions counts oscillations around the equator (azimuthal wiggles), which is why $P_{n}^{m}$ enters only when the problem depends on $ϕ$ . The factor $(1 - x^{2})^{m /2} = sin^{m} θ$ in Eq. [4] forces these functions to vanish at the poles when $m > 0$ — physically, a pattern with azimuthal structure cannot have a well-defined value exactly on the rotation axis.

The generating function [3] is the deepest intuition: it is the statement that the inverse distance between two points expands cleanly in the ratio of their radii. Multipole expansions in physics are not a trick layered on top — they are the Legendre generating function in disguise.

:::svg[The first four Legendre polynomials on $[- 1, 1]$ , showing increasing numbers of interior zeros]

Worked Examples

Legendre Series of a Step Potential

Given: Expand

f (x) = {+ 1 - 1 0 < x \leq 1 - 1 \leq x < 0

(a hemispherical sign function) in Legendre polynomials. Step 1 — coefficient formula.

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

. Since

f

is odd, only odd

n

survive. Step 2 — exploit symmetry. For odd

n

f (x) P_{n} (x)

is even, so

c_{n} = (2 n + 1) \int_{0}^{1} P_{n} (x) d x .

Step 3 — evaluate $\int_{0}^{1} P_{n}$ . Use the recurrence

P_{n} = \frac{1}{2 n + 1} \frac{d}{d x} [P_{n + 1} - P_{n - 1}]

, giving

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

. At

x = 1

both are

1

and cancel. Step 4 — plug in $n = 1$ .

P_{2} (0) = - \frac{1}{2}

P_{0} (0) = 1

, so

\int_{0}^{1} P_{1} = \frac{1}{3} [0 - (- \frac{1}{2}) + 1 \cdot ?]

... computing cleanly:

\int_{0}^{1} P_{1} d x = \int_{0}^{1} x d x = \frac{1}{2}

, so

c_{1} = 3 \cdot \frac{1}{2} = \frac{3}{2}

. Step 5 — next term $n = 3$ .

\int_{0}^{1} P_{3} d x = \int_{0}^{1} \frac{1}{2} (5 x^{3} - 3 x) d x = \frac{1}{2} (\frac{5}{4} - \frac{3}{2}) = - \frac{1}{8}

, so

c_{3} = 7 \cdot (- \frac{1}{8}) = - \frac{7}{8}

f (x) = \frac{3}{2} P_{1} (x) - \frac{7}{8} P_{3} (x) + \frac{11}{16} P_{5} (x) - \dots

(dimensionless; coefficients carry the units of

f

Multipole Expansion of a Potential

Given: Two point charges

+ q

z = + a

and

- q

z = - a

(a physical dipole). Find the potential at a field point

(r, θ)

with

r > a

, in SI units. Step 1 — exact potential.

V = \frac{1}{4 π ε _{0}} [\frac{q}{∣ r - a z ^ ∣} - \frac{q}{∣ r + a z ^ ∣}]

. Step 2 — apply the generating function. With

t = a / r

and

cos γ = cos θ

(for

+ a

) or

- cos θ

(for

- a

\frac{1}{∣ r \mp a z ^ ∣} = \frac{1}{r} n = 0 \sum \infty (\frac{a}{r})^{n} P_{n} (\pm cos θ) .

Step 3 — subtract. Using

P_{n} (- x) = (- 1)^{n} P_{n} (x)

, even-

n

terms cancel and odd-

n

terms double:

V = \frac{q}{4 π ε _{0} r} n odd \sum 2 (\frac{a}{r})^{n} P_{n} (cos θ) .

Step 4 — leading ( $n = 1$ ) dipole term. With

P_{1} = cos θ

and dipole moment

p = 2 a q

V \approx \frac{2 q a cos θ}{4 π ε _{0} r ^{2}} = \frac{p cos θ}{4 π ε _{0} r ^{2}} .

V (r, θ) \approx \frac{p cos θ}{4 π ε _{0} r ^{2}} [volts], p = 2 q a [C\cdotm] .

The Legendre expansion automatically names the dipole, octupole, ... terms.

Common Misconceptions

The two parameters are not interchangeable. In

P_{n}^{m}

, the degree

n

(often written

ℓ

in physics) and the order

m

play distinct roles:

n

sets the eigenvalue

n (n + 1)

and total angular structure;

m

sets azimuthal oscillation and must satisfy

∣ m ∣ \leq n

. Writing

P_{n}^{m}

with

m > n

gives identically zero, not an error to "fix." The Condon–Shortley phase $(- 1)^{m}$ is a convention, not a theorem. Different textbooks (and software like older versions of Mathematica vs. SciPy) include or omit it, and may also fold it into the spherical harmonics instead. Always check which convention a formula assumes before combining results. "Associated Legendre functions are polynomials" — false for odd $m$ . The factor

(1 - x^{2})^{m /2}

is a polynomial only when

m

is even; for odd

m

it contains

1 - x^{2} = sin θ

, so

P_{n}^{m}

is not a polynomial in

x

(though it is a nice function of

θ

One might guess that every solution of Legendre's equation [1] is a polynomial. It is not. For integer $n$ , the equation has a second, linearly independent solution $Q_{n} (x)$ — the Legendre function of the second kind — which is singular at $x = \pm 1$ . For example,

Q_{0} (x) = \frac{1}{2} ln \frac{1 + x}{1 - x}, Q_{1} (x) = \frac{x}{2} ln \frac{1 + x}{1 - x} - 1.

Q_{0}

diverges logarithmically at the poles

x = \pm 1

. The physical requirement of boundedness on the whole sphere is what discards

Q_{n}

and quantizes

n

to integers; without that boundary condition the polynomial family alone would be an incomplete answer.

Connections

Legendre polynomials are the $α = β = 0$ special case of the Jacobi Polynomials and the $λ = 1/2$ case of the Gegenbauer Polynomials, placing them in the broad family of Classical Orthogonal Polynomials. Their defining ODE is one instance of the Sturm-Liouville Theory framework, which guarantees the orthogonality and completeness used throughout. Multiplying $P_{n}^{m} (cos θ)$ by $e^{im ϕ}$ produces the Spherical Harmonics $Y_{n}^{m} (θ, ϕ)$ , the angular eigenfunctions of the Laplace's Equation operator on the sphere and of the quantum Angular Momentum operator $\hat{L}^{2}$ . The generating function [3] is the engine of the Multipole Expansion in electromagnetism and gravitation, and the recurrence relations connect to the theory of Three-Term Recurrence Relations satisfied by all orthogonal polynomials.

Real-World Applications

Electrostatic & gravitational multipoles. Expanding $1/∣ r - r^{'} ∣$ via Eq. [3] yields the monopole/dipole/quadrupole series used to characterize charge distributions, molecular interactions, and planetary gravity fields.
Hydrogen atom (quantum mechanics). The angular wavefunctions are spherical harmonics $Y_{ℓ}^{m}$ built from $P_{ℓ}^{m} (cos θ)$ ; $ℓ$ and $m$ are the orbital and magnetic quantum numbers labeling s, p, d, f orbitals.
Geodesy & geomagnetism. Earth's gravity field (the geoid) and magnetic field are tabulated as spherical harmonic coefficients — directly the associated Legendre expansion — in models like EGM2008 and IGRF.
Gauss–Legendre quadrature. The $n$ roots of $P_{n} (x)$ are the optimal sampling nodes for numerical integration, exactly integrating polynomials up to degree $2 n - 1$ .
Antenna radiation patterns. Far-field angular distributions of spherical and conical antennas are expanded in Legendre/associated Legendre functions.
Computer graphics — precomputed radiance transfer. Real-time global illumination encodes lighting environments in low-order spherical-harmonic (Legendre-based) coefficients.
Heat conduction & diffusion in spheres. Steady-state temperature in a ball with prescribed surface temperature is a Legendre series in $cos θ$ .

Historical Development

1782 — Adrien-Marie Legendre introduces the polynomials studying the gravitational attraction of spheroids; he calls them coefficients.
1785 — Pierre-Simon Laplace develops the full theory of "Laplace coefficients" (spherical harmonics) for the same potential problems, generalizing to azimuthal dependence.
1816 — Olinde Rodrigues publishes the differentiation formula now bearing his name (Eq. [2]), though it circulated and was rediscovered by others.
1837 — Heine and contemporaries systematize the associated Legendre functions and the function of the second kind $Q_{n}$ .
1877 — N.M. Ferrers gives the associated functions in the modern form used on $[- 1, 1]$ (Ferrers functions).
1875–1894 — E.W. Hobson's work culminates in his treatise The Theory of Spherical and Ellipsoidal Harmonics (1931), the definitive 19th-century synthesis.
1929 — Condon and Shortley fix the phase convention $(- 1)^{m}$ in their book The Theory of Atomic Spectra, standardizing usage in physics.
1814–1826 — Carl Friedrich Gauss develops Gauss–Legendre quadrature, exploiting the roots of $P_{n}$ for numerical integration.
20th c. — Spherical harmonics built from $P_{ℓ}^{m}$ become foundational in quantum mechanics (1920s), geophysics, and later computer graphics.

Subtleties & Limits

Convergence of Legendre series at endpoints and jumps. Like Fourier series, Legendre expansions of discontinuous functions exhibit a Gibbs phenomenon near jumps — an overshoot that does not vanish as more terms are added — and convergence at

x = \pm 1

is delicate, depending on the function's endpoint behavior. The precise pointwise convergence theory (Haar-type conditions, rates of decay of

c_{n}

) is subtle and parallels but is not identical to the trigonometric case. Non-integer and complex degree. For non-integer

n

, Eq. [1] has no polynomial solution; the bounded-on-the-sphere requirement is what quantizes

n

. The general Legendre functions

P_{ν}^{μ} (x)

of complex degree and order are full-fledged hypergeometric special functions, with branch cuts and analytic-continuation issues that go well beyond the polynomial story — important in scattering theory and the Regge-pole formalism of high-energy physics. Numerical stability at high order. The factorial factors

(n + m)! / (n - m)!

in normalization (Eq. [7]) overflow rapidly; high-degree geomagnetic and gravity models (degree > 2000) require carefully designed recursive, normalized algorithms to evaluate

P_{n}^{m}

without loss of precision — an active engineering concern.

Legendre Polynomials and Associated Legendre Functions

The Core Insight

Formal Statement

Legendre's Differential Equation

Rodrigues' Formula

Generating Function

Associated Legendre Functions

Orthogonality and Completeness

Why It Holds / Derivation

Orthogonality via Sturm–Liouville

Rodrigues' Formula Solves Legendre's Equation

Adrien-Marie Legendre | 1752–1833

Building Intuition

Worked Examples

Legendre Series of a Step Potential

Multipole Expansion of a Potential

Common Misconceptions

Connections

Real-World Applications

Historical Development

Subtleties & Limits

See Also

Before this

Where this leads