Legendre Polynomials

Legendre polynomials are the polynomial eigenfunctions that appear when rotationally symmetric problems are reduced to an angular equation with azimuthal quantum number $m=0$. They are also the building blocks behind associated Legendre functions, spherical harmonics, multipole expansions, and partial-wave methods.

This page uses the standard convention

$$P_\ell(1)=1, \qquad \ell=0,1,2,\ldots.$$

The full angular-momentum wavefunctions are Spherical Harmonics. Legendre polynomials are the $m=0$ polynomial core and the function $P_\ell(\cos\gamma)$ that appears in rotationally invariant kernels.

Rodrigues Formula

The Legendre polynomial of degree $\ell$ is

$$P_\ell(x) = \frac{1}{2^\ell \ell!} \frac{d^\ell}{dx^\ell} \left(x^2-1\right)^\ell, \qquad \ell=0,1,2,\ldots.$$

The first few are

$$\begin{aligned} P_0(x)&=1,\\ P_1(x)&=x,\\ P_2(x)&=\frac12(3x^2-1),\\ P_3(x)&=\frac12(5x^3-3x). \end{aligned}$$

The parity is

$$P_\ell(-x)=(-1)^\ell P_\ell(x),$$

so even $\ell$ gives an even polynomial and odd $\ell$ gives an odd polynomial.

Generating Function

Legendre polynomials are generated by

$$\frac{1}{\sqrt{1-2xt+t^2}} = \sum_{\ell=0}^{\infty} P_\ell(x)t^\ell, \qquad \lvert t\rvert<1.$$

This is useful in potential theory. For two position vectors $\mathbf r$ and $\mathbf r'$, with angle $\gamma$ between them and

$$r_{<} = \min(r,r'), \qquad r_{>} = \max(r,r'),$$

one has the multipole expansion

$$\frac{1}{ \lvert \mathbf r-\mathbf r'\rvert } = \sum_{\ell=0}^{\infty} \frac{r_{<}^\ell}{r_{>}^{\ell+1}} P_\ell(\cos\gamma), \qquad r_{<}<r_{>}.$$

This formula is one reason Legendre polynomials appear in electrostatics, Coulomb problems, and central-potential calculations.

Differential Equation

Legendre polynomials satisfy Legendre’s differential equation

$$(1-x^2)\frac{d^2P_\ell}{dx^2} -2x\frac{dP_\ell}{dx} +\ell(\ell+1)P_\ell =0.$$

Equivalently,

$$- \frac{d}{dx} \left[ (1-x^2) \frac{dP_\ell}{dx} \right] = \ell(\ell+1)P_\ell.$$

This is a Sturm–Liouville equation on $[-1,1]$ with weight $1$. The endpoints are singular because the coefficient $1-x^2$ vanishes at $x=\pm1$, but the polynomial solutions are regular.

Orthogonality

Legendre polynomials obey

$$\int_{-1}^{1} P_\ell(x)P_{\ell'}(x)\,dx = \frac{2}{2\ell+1} \delta_{\ell\ell'}.$$

The normalized version is therefore

$$\widehat P_\ell(x) = \sqrt{\frac{2\ell+1}{2}}\,P_\ell(x),$$

with

$$\int_{-1}^{1} \widehat P_\ell(x)\widehat P_{\ell'}(x)\,dx = \delta_{\ell\ell'}.$$

For suitable functions on $[-1,1]$, one may expand

$$f(x) = \sum_{\ell=0}^{\infty} a_\ell P_\ell(x),$$

with coefficients

$$a_\ell = \frac{2\ell+1}{2} \int_{-1}^{1} f(x)P_\ell(x)\,dx.$$

As usual, the convergence statement depends on the regularity of $f$ and on whether one means pointwise, mean-square, or distributional convergence.

Recurrence and Derivative Identities

The main three-term recurrence is

$$(\ell+1)P_{\ell+1}(x) = (2\ell+1)xP_\ell(x) -\ell P_{\ell-1}(x), \qquad \ell\ge1.$$

A useful derivative identity is

$$(1-x^2)\frac{dP_\ell}{dx} = \ell \left[ P_{\ell-1}(x)-xP_\ell(x) \right].$$

These identities are useful for manipulating angular integrals, deriving selection rules, and checking symbolic or numerical expressions.

Central-Potential Angular Equation

For a central potential $V(r)$, spherical coordinates match the symmetry. The angular part of the stationary Schrödinger equation involves the angular Laplacian. If the solution is independent of $\phi$, the polar equation can be written as

$$\frac{1}{\sin\theta} \frac{d}{d\theta} \left( \sin\theta \frac{d\Theta}{d\theta} \right) + \ell(\ell+1)\Theta =0.$$

Set

$$x=\cos\theta.$$

Then this equation becomes Legendre’s equation, and the regular solutions are

$$\Theta(\theta) = P_\ell(\cos\theta).$$

For full angular dependence, the functions are spherical harmonics. In the Condon–Shortley convention,

$$Y_\ell^0(\theta,\phi) = \sqrt{\frac{2\ell+1}{4\pi}}\, P_\ell(\cos\theta).$$

For $m\ne0$, one needs Associated Legendre Functions. Those belong to their own page because the phase convention, normalization, and endpoint behavior require separate care.

Addition Theorem Role

The spherical-harmonic addition theorem contains Legendre polynomials:

$$\sum_{m=-\ell}^{\ell} Y_\ell^m(\Omega)^* Y_\ell^m(\Omega') = \frac{2\ell+1}{4\pi} P_\ell(\cos\gamma),$$

where $\gamma$ is the angle between directions $\Omega$ and $\Omega'$. This identity expresses rotational invariance: after summing over $m$, the answer can depend only on the angle between the two directions.

This is why $P_\ell(\cos\gamma)$ appears in multipole expansions, central-force Green functions, scattering partial waves, and angular correlation functions.

Common Mistakes

• Forgetting the normalization convention $P_\ell(1)=1$.
• Using unweighted orthogonality formulas from another polynomial family.
• Confusing Legendre polynomials $P_\ell$ with associated Legendre functions $P_\ell^m$.
• Treating $P_\ell(\cos\theta)$ as the whole spherical harmonic for $m\ne0$.
• Forgetting the spherical measure $\sin\theta\,d\theta\,d\phi$ when converting angular integrals to integrals over $x=\cos\theta$.
• Assuming visual orbital shapes are direct plots of $P_\ell$ alone; physical angular wavefunctions are spherical harmonics.

Cross-Links

• Spherical Harmonics
• Associated Legendre Functions
• Hypergeometric Functions
• Orthogonal Polynomials
• Orbital Angular Momentum
• Spherical Harmonics as Angular-Momentum States
• Angular Momentum Algebra
• Separation of Variables
• Sturm–Liouville Theory
• Hydrogen Atom

References

• NIST Digital Library of Mathematical Functions, Chapters 14 and 18.
• F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, eds., NIST Handbook of Mathematical Functions, Cambridge University Press, 2010.
• M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, Dover, 1965.
• G. B. Arfken, H. J. Weber, and F. E. Harris, Mathematical Methods for Physicists, 7th ed., Academic Press, 2013.
• A. R. Edmonds, Angular Momentum in Quantum Mechanics, Princeton University Press, 1957.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.

Exercises

1. Use Rodrigues’ formula to compute $P_0$, $P_1$, and $P_2$.

Solution

For $\ell=0$,

$$P_0(x)=1.$$

For $\ell=1$,

$$P_1(x) = \frac12 \frac{d}{dx}(x^2-1) = x.$$

For $\ell=2$,

$$P_2(x) = \frac{1}{8} \frac{d^2}{dx^2} (x^2-1)^2.$$

Since

$$(x^2-1)^2 = x^4-2x^2+1,$$

the second derivative is $12x^2-4$. Therefore

$$P_2(x)=\frac12(3x^2-1).$$

2. Use the recurrence relation to compute $P_3$ from $P_2$ and $P_1$.

Solution

Set $\ell=2$ in

$$(\ell+1)P_{\ell+1} = (2\ell+1)xP_\ell-\ell P_{\ell-1}.$$

Then

$$3P_3 = 5xP_2-2P_1 = \frac52 x(3x^2-1)-2x.$$

Thus

$$P_3(x) = \frac12(5x^3-3x).$$

3. Verify the orthogonality of $P_0$ and $P_2$.

Solution

Using $P_0=1$ and $P_2=(3x^2-1)/2$,

$$\int_{-1}^{1}P_0(x)P_2(x)\,dx = \frac12 \int_{-1}^{1} (3x^2-1)\,dx.$$

The integral is

$$\frac12 \left[ 3\cdot\frac{2}{3}-2 \right] =0.$$

4. Show that $Y_\ell^0$ is normalized on the unit sphere if the Legendre orthogonality formula is assumed.

Solution

For

$$Y_\ell^0(\theta,\phi) = \sqrt{\frac{2\ell+1}{4\pi}} P_\ell(\cos\theta),$$

the norm is

$$\int_0^{2\pi}\int_0^\pi \lvert Y_\ell^0\rvert^2 \sin\theta\,d\theta\,d\phi.$$

Set $x=\cos\theta$, so $\sin\theta\,d\theta=-dx$. The integral becomes

$$2\pi \frac{2\ell+1}{4\pi} \int_{-1}^{1} P_\ell(x)^2\,dx.$$

Using

$$\int_{-1}^{1} P_\ell(x)^2\,dx = \frac{2}{2\ell+1},$$

the result is $1$.

5. Explain why $P_\ell(\cos\theta)$ describes only the $m=0$ angular dependence.

Solution

The function $P_\ell(\cos\theta)$ has no $\phi$ dependence, so it is an eigenfunction of $L_z$ with $m=0$. General angular-momentum eigenfunctions also carry the phase factor $e^{im\phi}$ and associated Legendre functions $P_\ell^m(\cos\theta)$. Those full functions are spherical harmonics.

6. Derive the expansion coefficient formula for a Legendre series.

Solution

Assume

$$f(x) = \sum_{\ell=0}^{\infty} a_\ell P_\ell(x).$$

Multiply by $P_{\ell'}(x)$ and integrate from $-1$ to $1$:

$$\int_{-1}^{1} f(x)P_{\ell'}(x)\,dx = \sum_{\ell=0}^{\infty} a_\ell \int_{-1}^{1} P_\ell(x)P_{\ell'}(x)\,dx.$$

Orthogonality leaves only $\ell=\ell'$:

$$\int_{-1}^{1} f(x)P_{\ell'}(x)\,dx = a_{\ell'} \frac{2}{2\ell'+1}.$$

Therefore

$$a_{\ell'} = \frac{2\ell'+1}{2} \int_{-1}^{1} f(x)P_{\ell'}(x)\,dx.$$