Legendre Polynomials

⚡ Key Takeaways

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 Core Insight

A single generating "seed" — the inverse-distance function $1/\sqrt{1-2xt+t^2}$ — 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 $\pm1$ ) 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?"

DefinitionLegendre Polynomials via Rodrigues' Formula

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

1

\quad P_n(x) = \frac{1}{2^n\, n!}\,\frac{d^n}{dx^n}\left[(x^2-1)^n\right], \qquad n = 0,1,2,\dots

Derivation hint: $(x^2-1)^n$ is a degree- $2n$ 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 $x = \cos\theta$ in physical applications), $n$ is the degree, and $P_n$ is a polynomial of exactly degree $n$ . The first few are:

2

\quad P_0 = 1,\; P_1 = x,\; P_2 = \tfrac{1}{2}(3x^2-1),\; P_3 = \tfrac{1}{2}(5x^3-3x),\; P_4 = \tfrac{1}{8}(35x^4-30x^2+3).

Derivation hint: apply [1] for each $n$ , or use the recurrence [7] starting from $P_0,P_1$ .

DefinitionLegendre's Differential Equation

The $P_n(x)$ are the polynomial solutions of the second-order linear ODE

3

\quad \frac{d}{dx}\!\left[(1-x^2)\frac{dy}{dx}\right] + n(n+1)\,y = 0,

equivalently $(1-x^2)y'' - 2x\,y' + n(n+1)y = 0$ . Derivation hint: 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 $\lambda_n = n(n+1)$ , and natural boundary conditions of boundedness at $x = \pm 1$ (regular singular points).

TheoremOrthogonality and Norm

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

4

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

where $\delta_{mn}$ is the Kronecker delta. Derivation hint: orthogonality (the $m\neq n$ case) follows from the Sturm–Liouville structure of [3]; the norm $2/(2n+1)$ follows from integrating Rodrigues' formula by parts $n$ times.

TheoremGenerating Function

For $|t| < 1$ and $|x| \le 1$ ,

5

\quad \frac{1}{\sqrt{1 - 2xt + t^2}} = \sum_{n=0}^{\infty} P_n(x)\, t^n.

Derivation hint: this is the multipole expansion of $1/|\mathbf{r}-\mathbf{r}'|$ with $x=\cos\theta$ , $t = r_</r_>$ ; expand the square root as a binomial/Taylor series in $t$ and collect powers.

TheoremCompleteness / Legendre Series

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

6

\quad f(x) = \sum_{n=0}^{\infty} c_n P_n(x), \qquad c_n = \frac{2n+1}{2}\int_{-1}^{1} f(x)\,P_n(x)\,dx.

Derivation hint: 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).

DefinitionThree-Term Recurrence

The polynomials satisfy

7

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

with $P_0 = 1$ , $P_1 = x$ . Derivation hint: differentiate the generating function [5] with respect to $t$ , multiply by $(1-2xt+t^2)$ , and equate coefficients of $t^n$ .

ProofOrthogonality from the Sturm–Liouville Form

Step 1. Write Legendre's equation [3] in self-adjoint form: $\big[(1-x^2)P_n'\big]' = -n(n+1)P_n$ .

Step 2. Multiply the equation for $P_n$ by $P_m$ and the equation for $P_m$ by $P_n$ :

P_m\big[(1-x^2)P_n'\big]' = -n(n+1)P_mP_n,

P_n\big[(1-x^2)P_m'\big]' = -m(m+1)P_mP_n.

Step 3. Subtract and integrate over $[-1,1]$ . The left side is a total derivative:

\int_{-1}^{1}\frac{d}{dx}\Big[(1-x^2)(P_mP_n' - P_nP_m')\Big]dx = \big[(1-x^2)(P_mP_n'-P_nP_m')\big]_{-1}^{1}.

Step 4. The boundary term vanishes because $(1-x^2)=0$ at $x=\pm1$ . Hence

\big[m(m+1) - n(n+1)\big]\int_{-1}^{1}P_mP_n\,dx = 0.

Step 5. For $m\neq n$ the bracket $m(m+1)-n(n+1)\neq0$ , forcing the integral to be zero. This proves the $\delta_{mn}$ structure in [4]. $\blacksquare$

∎

ProofNorm via Rodrigues' Formula

Step 1. Using [1], $\int_{-1}^{1}P_n^2\,dx = \frac{1}{(2^n n!)^2}\int_{-1}^{1}\frac{d^n}{dx^n}(x^2-1)^n \cdot \frac{d^n}{dx^n}(x^2-1)^n\,dx$ .

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 $x=\pm1$ . This yields

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

Step 3. The $2n$ -th derivative of the degree- $2n$ polynomial $(x^2-1)^n$ is the constant $(2n)!$ .

Step 4. Thus the integral becomes $\frac{(-1)^n(2n)!}{(2^n n!)^2}\int_{-1}^{1}(x^2-1)^n\,dx = \frac{(2n)!}{(2^n n!)^2}\int_{-1}^{1}(1-x^2)^n\,dx$ .

Step 5. Using $\int_{-1}^{1}(1-x^2)^n\,dx = \frac{2^{2n+1}(n!)^2}{(2n+1)!}$ (a Beta-function integral), everything cancels to give $\frac{2}{2n+1}$ , confirming [4]. $\blacksquare$

∎

AL

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.

🧠 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. $\int_{-1}^1 1\cdot x^2\,dx \neq 0$ ). The Legendre polynomials are exactly what you get by running Gram–Schmidt orthogonalization on the monomials using the inner product $\langle f,g\rangle = \int_{-1}^1 fg\,dx$ . 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.

ExampleExpanding f(x) = x² in Legendre Polynomials

Given: $f(x) = x^2$ on $[-1,1]$ . 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: $P_0=1$ , $P_1=x$ , $P_2=\frac{1}{2}(3x^2-1)$ . Solve $P_2$ for $x^2$ :

x^2 = \frac{2P_2 + 1}{3} = \frac{2}{3}P_2 + \frac{1}{3}P_0.

Step 3. Verify with the projection formula [6]. For $c_2$ : $c_2 = \frac{5}{2}\int_{-1}^1 x^2 \cdot \frac{1}{2}(3x^2-1)\,dx = \frac{5}{4}\int_{-1}^1(3x^4 - x^2)\,dx = \frac{5}{4}\left(\frac{6}{5} - \frac{2}{3}\right) = \frac{5}{4}\cdot\frac{8}{15} = \frac{2}{3}.$ ✓

Step 4. And $c_1 = 0$ by odd-symmetry, $c_0 = \frac{1}{2}\int_{-1}^1 x^2\,dx = \frac{1}{3}$ . ✓

\boxed{x^2 = \tfrac{1}{3}P_0(x) + \tfrac{2}{3}P_2(x)}

(dimensionless coefficients)

ExampleElectrostatic Potential via Multipole Expansion

Given: A point charge $q = 2\times10^{-9}\,\text{C}$ sits on the $z$ -axis at distance $a = 0.10\,\text{m}$ from the origin. Find the potential at a field point at radius $r = 0.50\,\text{m}$ , angle $\theta = 60^\circ$ from the $z$ -axis, keeping the dipole term.

Step 1. The exact potential is $V = \frac{q}{4\pi\varepsilon_0}\frac{1}{|\mathbf{r}-\mathbf{a}|}$ . With $|\mathbf{r}-\mathbf{a}| = \sqrt{r^2 - 2ra\cos\theta + a^2}$ , factor out $r$ and set $t = a/r = 0.20$ , $x=\cos60^\circ = 0.5$ .

Step 2. By the generating function [5]:

V = \frac{q}{4\pi\varepsilon_0 r}\sum_{n=0}^\infty \left(\frac{a}{r}\right)^n P_n(\cos\theta).

Step 3. Compute terms with $k = \frac{q}{4\pi\varepsilon_0 r} = \frac{(8.99\times10^9)(2\times10^{-9})}{0.50} = 35.96\,\text{V}$ :

$n=0$ : $P_0=1$ , term $= k(1)(1) = 35.96\,\text{V}$ .
$n=1$ : $P_1(0.5)=0.5$ , term $= k(0.20)(0.5) = 3.596\,\text{V}$ .

Step 4. Sum monopole + dipole: $V \approx 35.96 + 3.60 = 39.56\,\text{V}$ .

\boxed{V \approx 39.6\,\text{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 mistake

The normalization is NOT to unit norm. A frequent error is assuming $\int_{-1}^1 P_n^2\,dx = 1$ . In fact it equals $\frac{2}{2n+1}$ — the Legendre polynomials are orthogonal but not orthonormal. If you need an orthonormal set, use $\widetilde{P}_n = \sqrt{\frac{2n+1}{2}}\,P_n$ . Forgetting the $\frac{2n+1}{2}$ factor in the coefficient formula [6] is the single most common source of wrong Legendre series.

⚠️ Common mistake

Legendre polynomials live on $[-1,1]$ , not arbitrary intervals. To expand a function on $[a,b]$ you must first change variables: let $x = \frac{2u - (a+b)}{b-a}$ to map $[a,b]\to[-1,1]$ , then apply the standard formulas. Skipping this rescaling destroys orthogonality.

Counterexample

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.

connection

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 $e^{im\phi}$ to build the Spherical Harmonics $Y_\ell^m$ — the angular eigenfunctions on the sphere. They are a special case ( $\alpha=\beta=0$ ) of the Jacobi Polynomials and the limiting case ( $\alpha\to\frac{1}{2}$ ) 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.

🔬 Where this appears

⏳ History

- 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.

open-problem

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.

Formal Statement

Why It Holds / Derivation

Building Intuition

Worked Examples

Common Misconceptions

Connections

Real-World Applications

Historical Development

Subtleties & Limits

See Also