Legendre Polynomials and Associated Legendre Functions

⚡ Key Takeaways

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 $\int_{-1}^{1}P_m P_n\,dx = \frac{2}{2n+1}\delta_{mn}$ , and the associated functions $P_n^m(\cos\theta)e^{im\phi}$ 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 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 $\ell(\ell+1)$ encode angular momentum. Everything else — Rodrigues' formula, recurrences, generating functions, orthogonality — flows from this one Sturm–Liouville structure.

DefinitionLegendre's Differential Equation

Legendre's equation is the second-order linear ODE

1

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

Derivation hint: Separate variables in $\nabla^2\Phi=0$ in spherical coordinates; the polar-angle equation in $u=\cos\theta$ takes this form with separation constant $n(n+1)$ . Here $x$ is the independent variable (typically $x=\cos\theta$ ), $y(x)$ 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 $P_n(x)$ .

DefinitionRodrigues' Formula

The Legendre polynomials are given explicitly by

2

\quad P_n(x) = \frac{1}{2^n\,n!}\,\frac{d^n}{dx^n}\big(x^2-1\big)^n.

Derivation hint: $(x^2-1)^n$ 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 $P_0=1$ , $P_1=x$ , $P_2=\tfrac12(3x^2-1)$ , $P_3=\tfrac12(5x^3-3x)$ , $P_4=\tfrac18(35x^4-30x^2+3)$ .

DefinitionGenerating Function

The generating function packs all Legendre polynomials into one expansion:

3

\quad g(x,t) = \big(1-2xt+t^2\big)^{-1/2} = \sum_{n=0}^{\infty} P_n(x)\,t^n,\qquad |t|<1.

Derivation hint: Expand $\tfrac{1}{|\mathbf r-\mathbf r'|}=\tfrac{1}{r}(1-2(r'/r)\cos\gamma+(r'/r)^2)^{-1/2}$ in powers of $t=r'/r$ ; the binomial coefficients organize into Legendre polynomials of $\cos\gamma$ .

DefinitionAssociated Legendre Functions

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

4

\quad P_n^m(x) = (-1)^m\,(1-x^2)^{m/2}\,\frac{d^m}{dx^m}P_n(x),

Derivation hint: Substitute $y=(1-x^2)^{m/2}u$ into the associated Legendre equation; the function $u=d^m P_n/dx^m$ 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

5

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

Derivation hint: This is the full polar equation from separating $\nabla^2\Phi=0$ when the azimuthal solution is $e^{im\phi}$ , contributing the $-m^2/\sin^2\theta$ term. Note $P_n^0(x)=P_n(x)$ , and $P_n^m\equiv 0$ for $m>n$ .

TheoremOrthogonality and Completeness

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

6

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

Derivation hint: Write Legendre's equation in Sturm–Liouville form $\frac{d}{dx}[(1-x^2)y']=-n(n+1)y$ ; subtract the equations for $P_m,P_n$ , integrate, and the boundary term vanishes because $(1-x^2)\to0$ at the endpoints. The associated functions obey, for fixed $m$ ,

7

\quad \int_{-1}^{1} P_k^m(x)\,P_n^m(x)\,dx = \frac{2}{2n+1}\,\frac{(n+m)!}{(n-m)!}\,\delta_{kn}.

Derivation hint: 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 $c_n=\frac{2n+1}{2}\int_{-1}^1 f(x)P_n(x)\,dx$ .

ProofOrthogonality via Sturm–Liouville

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

\frac{d}{dx}\!\left[(1-x^2)\,P_n'\right] + n(n+1)\,P_n = 0.

This is a Sturm–Liouville problem with $p(x)=1-x^2$ , weight $w(x)=1$ , eigenvalue $\lambda_n=n(n+1)$ .

Step 2. Write the same equation for $P_m$ with eigenvalue $\lambda_m$ . Multiply the $P_n$ equation by $P_m$ and the $P_m$ equation by $P_n$ , then subtract:

P_m\frac{d}{dx}[(1-x^2)P_n'] - P_n\frac{d}{dx}[(1-x^2)P_m'] + (\lambda_n-\lambda_m)P_mP_n = 0.

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

\frac{d}{dx}\Big[(1-x^2)\big(P_m P_n' - P_n P_m'\big)\Big] = (\lambda_m-\lambda_n)P_mP_n.

Step 4. Integrate over $[-1,1]$ . The left side equals $\big[(1-x^2)(\cdots)\big]_{-1}^{1}=0$ because $1-x^2$ vanishes at both endpoints. Thus

(\lambda_m-\lambda_n)\int_{-1}^{1}P_mP_n\,dx = 0.

For $m\ne n$ , $\lambda_m\ne\lambda_n$ , 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{dx}{1-2xt+t^2} = \sum_{n}t^{2n}\int_{-1}^1 P_n^2\,dx.

The left integral evaluates to $\frac{1}{t}\ln\frac{1+t}{1-t}=\sum_n \frac{2}{2n+1}t^{2n}$ . Matching powers of $t^{2n}$ gives $\int P_n^2\,dx=\frac{2}{2n+1}$ . $\blacksquare$

∎

ProofRodrigues' Formula Solves Legendre's Equation

Step 1. Let $v=(x^2-1)^n$ . Then $v'=2nx(x^2-1)^{n-1}$ , so $(x^2-1)v' = 2nx\,v$ .

Step 2. Differentiate $(x^2-1)v'=2nxv$ exactly $(n+1)$ times using the Leibniz rule. Each side produces a chain of terms; collecting them yields

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

Step 3. Let $w=v^{(n)}=\frac{d^n}{dx^n}(x^2-1)^n$ . The previous line reads

(1-x^2)\,w'' - 2x\,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. $\blacksquare$

∎

AL

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.

🧠 Intuition

Think of $[-1,1]$ as the range of $\cos\theta$ as $\theta$ sweeps a sphere from north pole to south pole. Legendre polynomials are the natural "harmonics" of that one-dimensional interval, just as $\sin(n\pi x)$ are the harmonics of a vibrating string. $P_0$ is constant (a uniform sphere), $P_1=\cos\theta$ 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 $\phi$ . The factor $(1-x^2)^{m/2}=\sin^m\theta$ 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.

ExampleLegendre Series of a Step Potential

Given: Expand $f(x)=\begin{cases}+1 & 0<x\le1\\ -1 & -1\le x<0\end{cases}$ (a hemispherical sign function) in Legendre polynomials.

Step 1 — coefficient formula. $c_n=\frac{2n+1}{2}\int_{-1}^1 f(x)P_n(x)\,dx$ . 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 = (2n+1)\int_0^1 P_n(x)\,dx.

Step 3 — evaluate $\int_0^1 P_n$ . Use the recurrence $P_n=\frac{1}{2n+1}\frac{d}{dx}\!\left[P_{n+1}-P_{n-1}\right]$ , giving $\int_0^1 P_n\,dx=\frac{1}{2n+1}\big[P_{n+1}(1)-P_{n-1}(1)-P_{n+1}(0)+P_{n-1}(0)\big]$ . At $x=1$ both are $1$ and cancel.

Step 4 — plug in $n=1$ . $P_2(0)=-\tfrac12$ , $P_0(0)=1$ , so $\int_0^1 P_1=\frac{1}{3}[0 -(-\tfrac12)+1\cdot? ]$ ... computing cleanly: $\int_0^1 P_1\,dx=\int_0^1 x\,dx=\tfrac12$ , so $c_1=3\cdot\tfrac12=\tfrac32$ .

Step 5 — next term $n=3$ . $\int_0^1 P_3\,dx=\int_0^1\tfrac12(5x^3-3x)\,dx=\tfrac12(\tfrac54-\tfrac32)=-\tfrac18$ , so $c_3=7\cdot(-\tfrac18)=-\tfrac78$ .

\boxed{f(x)=\tfrac{3}{2}P_1(x)-\tfrac{7}{8}P_3(x)+\tfrac{11}{16}P_5(x)-\cdots}

(dimensionless; coefficients carry the units of $f$ ).

ExampleMultipole Expansion of a Potential

Given: Two point charges $+q$ at $z=+a$ and $-q$ at $z=-a$ (a physical dipole). Find the potential at a field point $(r,\theta)$ with $r>a$ , in SI units.

Step 1 — exact potential. $V=\frac{1}{4\pi\varepsilon_0}\left[\frac{q}{|\mathbf r-a\hat z|}-\frac{q}{|\mathbf r+a\hat z|}\right]$ .

Step 2 — apply the generating function. With $t=a/r$ and $\cos\gamma=\cos\theta$ (for $+a$ ) or $-\cos\theta$ (for $-a$ ),

\frac{1}{|\mathbf r\mp a\hat z|}=\frac{1}{r}\sum_{n=0}^\infty \left(\frac{a}{r}\right)^n P_n(\pm\cos\theta).

Step 3 — subtract. Using $P_n(-x)=(-1)^nP_n(x)$ , even- $n$ terms cancel and odd- $n$ terms double:

V=\frac{q}{4\pi\varepsilon_0 r}\sum_{n\,\text{odd}} 2\left(\frac{a}{r}\right)^n P_n(\cos\theta).

Step 4 — leading ( $n=1$ ) dipole term. With $P_1=\cos\theta$ and dipole moment $p=2aq$ ,

V\approx \frac{2qa\cos\theta}{4\pi\varepsilon_0 r^2}=\frac{p\cos\theta}{4\pi\varepsilon_0 r^2}.

\boxed{V(r,\theta)\approx \dfrac{p\cos\theta}{4\pi\varepsilon_0\,r^2}\ \text{[volts]},\quad p=2qa\ \text{[C}\,\cdot\,\text{m]}.}

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

⚠️ Common mistake

The two parameters are not interchangeable. In $P_n^m$ , the degree $n$ (often written $\ell$ 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 $|m|\le n$ . 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 $\sqrt{1-x^2}=\sin\theta$ , so $P_n^m$ is not a polynomial in $x$ (though it is a nice function of $\theta$ ).

Counterexample

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 $x=\pm1$ . For example,

Q_0(x)=\tfrac12\ln\frac{1+x}{1-x},\qquad Q_1(x)=\tfrac{x}{2}\ln\frac{1+x}{1-x}-1.

$Q_0$ diverges logarithmically at the poles $x=\pm1$ . 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.

connection

Legendre polynomials are the $\alpha=\beta=0$ special case of the Jacobi Polynomials and the $\lambda=1/2$ 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 $P_n^m(\cos\theta)$ by $e^{im\phi}$ produces the Spherical Harmonics $Y_n^m(\theta,\phi)$ , 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.

🔬 Where this appears

⏳ History

- 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_\ell^m

become foundational in quantum mechanics (1920s), geophysics, and later computer graphics.

open-problem

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 $x=\pm1$ 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 $P_\nu^\mu(x)$ 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

Formal Statement

Why It Holds / Derivation

Building Intuition

Worked Examples

Common Misconceptions

Connections

Real-World Applications

Historical Development

Subtleties & Limits

See Also