Associated Legendre Functions

Associated Legendre functions are the polar-angle functions that appear in spherical harmonics when the azimuthal quantum number $m$ is nonzero. They extend Legendre Polynomials by differentiating $P_\ell(x)$ and multiplying by the endpoint factor $(1-x^2)^{m/2}$.

This page uses the convention already used in the angular-momentum pages:

$$P_\ell^m(x) = (1-x^2)^{m/2} \frac{d^mP_\ell}{dx^m}, \qquad 0\le m\le \ell.$$

With this convention, the Condon–Shortley phase appears explicitly in the spherical harmonic:

$$Y_\ell^m(\theta,\phi) = (-1)^m \sqrt{ \frac{2\ell+1}{4\pi} \frac{(\ell-m)!}{(\ell+m)!} } P_\ell^m(\cos\theta)e^{im\phi}, \qquad m\ge0.$$

Many references instead absorb the factor $(-1)^m$ into the definition of $P_\ell^m$. Always check which convention a table uses before comparing signs.

Definition on the Interval

For integer $\ell\ge0$ and $0\le m\le\ell$, define

$$P_\ell^m(x) = (1-x^2)^{m/2} \frac{d^m}{dx^m}P_\ell(x), \qquad -1\le x\le1.$$

These are often called associated Legendre functions on the cut, or Ferrers functions, because the interval $[-1,1]$ is the natural domain for angular problems with $x=\cos\theta$.

For $m=0$, one recovers the Legendre polynomial:

$$P_\ell^0(x)=P_\ell(x).$$

The first few nontrivial examples are

$$P_1^1(x)=\sqrt{1-x^2},$$ $$P_2^1(x)=3x\sqrt{1-x^2}, \qquad P_2^2(x)=3(1-x^2).$$

If a reference includes the Condon–Shortley phase inside $P_\ell^m$, these examples acquire signs $(-1)^m$.

Differential Equation

Associated Legendre functions satisfy

$$(1-x^2)\frac{d^2y}{dx^2} -2x\frac{dy}{dx} + \left[ \ell(\ell+1) - \frac{m^2}{1-x^2} \right]y =0.$$

The term $m^2/(1-x^2)$ is singular at $x=\pm1$. The regular angular solutions are selected by endpoint behavior and by the integer labels $\ell$ and $m$.

When $m=0$, this reduces to Legendre’s differential equation.

Endpoint Behavior

For $m>0$, the factor $(1-x^2)^{m/2}$ controls behavior at the poles:

$$P_\ell^m(x) \sim (1-x^2)^{m/2} \quad \text{near }x=\pm1,$$

up to a finite nonzero factor when the derivative does not vanish there.

In angular variables, $x=\cos\theta$, so

$$1-x^2=\sin^2\theta.$$

The factor $(1-x^2)^{m/2}$ is therefore a power of $\sin\theta$, which keeps the spherical harmonic regular at the north and south poles when combined with the azimuthal phase $e^{im\phi}$.

Orthogonality for Fixed m

For fixed $m$, the functions $P_\ell^m$ with $\ell=m,m+1,\ldots$ are orthogonal on $[-1,1]$:

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

This is the polar-angle part of spherical-harmonic orthonormality. The factorial ratio is exactly what the normalization constant in $Y_\ell^m$ cancels.

For the broader weighted-inner-product viewpoint, see Orthogonal Polynomials.

Spherical Harmonics

The spherical harmonics separate the angular dependence into a polar function and an azimuthal phase:

$$Y_\ell^m(\theta,\phi) = N_{\ell m} P_\ell^m(\cos\theta)e^{im\phi}.$$

With this page’s convention,

$$N_{\ell m} = (-1)^m \sqrt{ \frac{2\ell+1}{4\pi} \frac{(\ell-m)!}{(\ell+m)!} }, \qquad m\ge0.$$

For negative $m$, this site uses the standard relation

$$Y_\ell^{-m} = (-1)^m \left(Y_\ell^m\right)^*, \qquad m>0.$$

The physical angular-momentum content belongs to the spherical harmonics: they diagonalize $L^2$ and $L_z$. Associated Legendre functions are the polar special functions used to build them.

How the Phase Convention Changes

Some references define

$$\widetilde P_\ell^m(x) = (-1)^mP_\ell^m(x).$$

In that convention, the spherical-harmonic formula is usually written without a separate $(-1)^m$:

$$Y_\ell^m(\theta,\phi) = \sqrt{ \frac{2\ell+1}{4\pi} \frac{(\ell-m)!}{(\ell+m)!} } \widetilde P_\ell^m(\cos\theta)e^{im\phi}.$$

Both conventions describe the same normalized spherical harmonics if the sign is handled consistently. Mixing the two conventions changes signs in tables, selection-rule calculations, and Clebsch–Gordan coefficient comparisons.

Relation to Angular Separation

For a central potential, the angular equation after separation of variables includes the term

$$\frac{1}{\sin^2\theta} \frac{\partial^2}{\partial\phi^2}.$$

Taking the azimuthal dependence as

$$\Phi(\phi)=e^{im\phi}$$

turns this term into

$$-\frac{m^2}{\sin^2\theta}.$$

After the substitution $x=\cos\theta$, the polar equation becomes the associated Legendre equation. Regularity and single-valuedness require

$$\ell=0,1,2,\ldots, \qquad m=-\ell,\ldots,\ell.$$

This is the analytic origin of the spherical-harmonic labels.

Common Mistakes

• Mixing conventions for whether $(-1)^m$ belongs to $P_\ell^m$ or to $Y_\ell^m$.
• Treating $P_\ell^m$ as a full angular wavefunction without the azimuthal factor $e^{im\phi}$.
• Forgetting that $\ell\ge m\ge0$ in the derivative definition for positive $m$.
• Applying the Legendre-polynomial orthogonality formula without the factorial ratio for fixed $m$.
• Ignoring endpoint behavior at $x=\pm1$.
• Confusing associated Legendre functions on $[-1,1]$ with other Legendre functions used for non-angular domains.

Cross-Links

• Legendre Polynomials
• Spherical Harmonics
• Orthogonal Polynomials
• Spherical Harmonics as Angular-Momentum States
• Notation and Conventions
• Separation of Variables
• Angular Momentum Algebra

References

• NIST Digital Library of Mathematical Functions, Chapter 14, Legendre and Related Functions.
• 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.
• A. R. Edmonds, Angular Momentum in Quantum Mechanics, Princeton University Press, 1957.
• D. A. Varshalovich, A. N. Moskalev, and V. K. Khersonskii, Quantum Theory of Angular Momentum, World Scientific, 1988.
• G. B. Arfken, H. J. Weber, and F. E. Harris, Mathematical Methods for Physicists, 7th ed., Academic Press, 2013.

Exercises

1. Compute $P_1^1(x)$ and $P_2^1(x)$ using this page’s convention.

Solution

Since $P_1(x)=x$,

$$P_1^1(x) = (1-x^2)^{1/2} \frac{dP_1}{dx} = \sqrt{1-x^2}.$$

Since

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

one has

$$\frac{dP_2}{dx}=3x.$$

Therefore

$$P_2^1(x) = 3x\sqrt{1-x^2}.$$

2. Show that the associated Legendre equation reduces to Legendre’s equation when $m=0$.

Solution

Set $m=0$ in

$$(1-x^2)y'' -2xy' + \left[ \ell(\ell+1) - \frac{m^2}{1-x^2} \right]y =0.$$

The singular term disappears, leaving

$$(1-x^2)y'' -2xy' +\ell(\ell+1)y =0,$$

which is Legendre’s equation.

3. Verify the normalization of $Y_\ell^m$ for $m\ge0$ using the fixed-$m$ orthogonality formula.

Solution

The squared normalization factor is

$$\lvert N_{\ell m}\rvert^2 = \frac{2\ell+1}{4\pi} \frac{(\ell-m)!}{(\ell+m)!}.$$

The angular norm is

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

Set $x=\cos\theta$. The $\phi$ integral gives $2\pi$, and the polar integral gives

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

Multiplying these factors gives $1$.

4. Translate to the convention $\widetilde P_\ell^m=(-1)^mP_\ell^m$. What is $\widetilde P_1^1(x)$?

Solution

This page gives

$$P_1^1(x)=\sqrt{1-x^2}.$$

The alternate convention gives

$$\widetilde P_1^1(x) = (-1)^1P_1^1(x) = -\sqrt{1-x^2}.$$

5. Why is $P_\ell^m(\cos\theta)$ not by itself an eigenfunction of $L_z$ with eigenvalue $\hbar m$?

Solution

The operator $L_z$ acts on the azimuthal dependence:

$$L_z=-i\hbar\frac{\partial}{\partial\phi}.$$

The function $P_\ell^m(\cos\theta)$ has no $\phi$ dependence, so $L_z$ acting on it gives zero. The full spherical harmonic includes $e^{im\phi}$, and

$$-i\hbar\frac{\partial}{\partial\phi}e^{im\phi} = \hbar m e^{im\phi}.$$