Bessel Functions

Bessel functions are the radial special functions of cylindrical symmetry. Spherical Bessel functions are the free radial special functions of three-dimensional central problems. Together they appear in cylindrical wells, waveguides, diffraction, Green functions, and partial-wave scattering.

The useful mental model is simple: Fourier modes handle the angle, while Bessel functions handle the radius. The radial measure and the boundary condition decide which member of the family is physically allowed.

Bessel Equation

Bessel’s equation of order $\nu$ is

$$x^2y'' +xy' +(x^2-\nu^2)y =0.$$

Two standard independent solutions are the Bessel function of the first kind $J_\nu(x)$ and the Bessel function of the second kind $Y_\nu(x)$, also called a Neumann function:

$$y(x) = A J_\nu(x)+B Y_\nu(x).$$

The first-kind solution has the series

$$J_\nu(x) = \sum_{s=0}^{\infty} \frac{(-1)^s}{s!\,\Gamma(s+\nu+1)} \left(\frac{x}{2}\right)^{2s+\nu}.$$

The gamma factor extends the coefficient formula beyond integer order; the relevant identities are collected in Gamma and Beta Functions.

For nonnegative integer order $m$, this becomes

$$J_m(x) = \sum_{s=0}^{\infty} \frac{(-1)^s}{s!\,(s+m)!} \left(\frac{x}{2}\right)^{2s+m}.$$

Thus

$$J_m(x)\sim \frac{1}{m!} \left(\frac{x}{2}\right)^m \qquad x\to0.$$

For $m>0$, $J_m(0)=0$ but the solution is still regular. The second-kind solution $Y_m$ is singular at the origin. For example,

$$Y_0(x) \sim \frac{2}{\pi} \left[ \log\left(\frac{x}{2}\right) +\gamma_{\mathrm E} \right], \qquad x\to0,$$

where $\gamma_{\mathrm E}$ is Euler’s constant.

Recurrence Identities

For $Z_\nu=J_\nu,Y_\nu,H_\nu^{(1)}$, or $H_\nu^{(2)}$, the standard recurrences include

$$Z_{\nu-1}(x)+Z_{\nu+1}(x) = \frac{2\nu}{x}Z_\nu(x),$$

and

$$Z_{\nu-1}(x)-Z_{\nu+1}(x) = 2\frac{dZ_\nu}{dx}(x).$$

These identities are useful for angular-momentum matrix elements, boundary conditions involving derivatives, and numerical checks.

Cylindrical Separation

In cylindrical coordinates, the two-dimensional Helmholtz equation has radial structure

$$\left[ \frac{1}{r} \frac{\partial}{\partial r} \left( r\frac{\partial}{\partial r} \right) + \frac{1}{r^2} \frac{\partial^2}{\partial\phi^2} + k_\perp^2 \right]\psi =0.$$

Try a separated angular mode

$$\psi(r,\phi)=R(r)e^{im\phi}, \qquad m\in\mathbb Z.$$

The radial equation is

$$r^2R'' +rR' +\left(k_\perp^2r^2-m^2\right)R =0.$$

With $x=k_\perp r$, this becomes Bessel’s equation of integer order $m$. Regularity at the axis usually selects

$$R(r)=J_m(k_\perp r).$$

A hard-wall cylinder of radius $a$ with Dirichlet boundary condition $R(a)=0$ therefore requires

$$J_m(k_\perp a)=0.$$

If $\alpha_{m,s}$ is the $s$th positive zero of $J_m$, then

$$k_\perp = \frac{\alpha_{m,s}}{a}, \qquad s=1,2,\ldots.$$

For a Neumann boundary condition $R'(a)=0$, the allowed values instead come from zeros of $J_m'$. The boundary condition, not the differential equation alone, decides which zeros are relevant.

Radial Orthogonality

Bessel functions in a finite cylinder form a weighted orthogonal family. For Dirichlet zeros $\alpha_{m,s}$ of $J_m$,

$$\int_0^a r\, J_m\left(\frac{\alpha_{m,s}r}{a}\right) J_m\left(\frac{\alpha_{m,t}r}{a}\right) \,dr = \frac{a^2}{2} \left[ J_{m+1}(\alpha_{m,s}) \right]^2 \delta_{st}.$$

The factor $r\,dr$ is not optional. It is the radial part of the cylindrical area element. This is the Sturm–Liouville weight for the radial problem.

Hankel Functions and Cylindrical Waves

The Hankel functions are

$$H_\nu^{(1)}(x) = J_\nu(x)+iY_\nu(x), \qquad H_\nu^{(2)}(x) = J_\nu(x)-iY_\nu(x).$$

For large positive $x$,

$$J_\nu(x) \sim \sqrt{\frac{2}{\pi x}} \cos\left( x-\frac{\nu\pi}{2}-\frac{\pi}{4} \right),$$

and

$$Y_\nu(x) \sim \sqrt{\frac{2}{\pi x}} \sin\left( x-\frac{\nu\pi}{2}-\frac{\pi}{4} \right).$$

Thus

$$H_\nu^{(1)}(x) \sim \sqrt{\frac{2}{\pi x}} \exp\left[ i\left( x-\frac{\nu\pi}{2}-\frac{\pi}{4} \right) \right].$$

With the common time convention $e^{-i\omega t}$, $H_\nu^{(1)}$ represents an outgoing cylindrical wave and $H_\nu^{(2)}$ an incoming cylindrical wave. With the opposite time convention, the labels are reversed.

Spherical Bessel Functions

Three-dimensional central problems use spherical Bessel functions. They solve

$$x^2z'' +2xz' +\left[ x^2-\ell(\ell+1) \right]z =0, \qquad \ell=0,1,2,\ldots.$$

They are related to ordinary Bessel functions by

$$j_\ell(x) = \sqrt{\frac{\pi}{2x}}\, J_{\ell+1/2}(x), \qquad y_\ell(x) = \sqrt{\frac{\pi}{2x}}\, Y_{\ell+1/2}(x).$$

Some books write $n_\ell$ instead of $y_\ell$ for the spherical Neumann function. This page uses $y_\ell$ but notes $n_\ell$ when discussing common scattering notation.

The first examples are

$$j_0(x)=\frac{\sin x}{x}, \qquad y_0(x)=-\frac{\cos x}{x},$$

and

$$j_1(x) = \frac{\sin x}{x^2} - \frac{\cos x}{x}.$$

As $x\to0$, the regular spherical Bessel function behaves as

$$j_\ell(x) \sim \frac{x^\ell}{(2\ell+1)!!}.$$

The irregular solution $y_\ell$ is singular at the origin.

Free Radial Motion and Partial Waves

For a free particle in a central partial wave, the radial equation for $R_\ell(r)$ is

$$R_\ell'' +\frac{2}{r}R_\ell' +\left[ k^2-\frac{\ell(\ell+1)}{r^2} \right]R_\ell =0.$$

The regular free radial function is

$$R_\ell(r) \propto j_\ell(kr).$$

If $u_\ell(r)=rR_\ell(r)$ is the reduced radial function, then the convenient free basis is given by Riccati-Bessel functions

$$\hat j_\ell(x)=xj_\ell(x), \qquad \hat y_\ell(x)=xy_\ell(x).$$

For large $x$,

$$\hat j_\ell(x) \sim \sin\left( x-\frac{\ell\pi}{2} \right), \qquad \hat y_\ell(x) \sim - \cos\left( x-\frac{\ell\pi}{2} \right).$$

This is the free standing-wave reference used in partial-wave scattering. A short-range central potential changes the relative combination to

$$u_\ell(r) \propto \hat j_\ell(kr)\cos\delta_\ell - \hat y_\ell(kr)\sin\delta_\ell,$$

or equivalently shifts the asymptotic sine by $\delta_\ell$. The detailed scattering interpretation belongs to Partial-Wave Expansion and Phase Shifts.

Common Mistakes

• Confusing ordinary Bessel functions $J_m$ with spherical Bessel functions $j_\ell$.
• Using the irregular solution $Y_m$ or $y_\ell$ at the origin when regularity is required.
• Forgetting the radial measure: $r\,dr$ in cylindrical problems and $r^2\,dr$ in spherical problems.
• Quantizing a cylinder with zeros of $J_m$ when the boundary condition actually involves $J_m'$.
• Treating Bessel-function zeros as exactly evenly spaced; their spacing is only asymptotic.
• Mixing outgoing-wave conventions for Hankel functions without stating the time dependence.
• Matching reduced radial functions $u_\ell$ to $j_\ell$ instead of to the Riccati-Bessel functions $\hat j_\ell$ and $\hat y_\ell$.

Cross-Links

• Separation of Variables
• Sturm–Liouville Theory
• Boundary Conditions
• Legendre Polynomials
• Spherical Harmonics
• Hypergeometric Functions
• Gamma and Beta Functions
• Partial-Wave Expansion
• Phase Shifts
• Phase Shift Extraction Notebook

References

• NIST Digital Library of Mathematical Functions, Chapter 10, Bessel 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.
• G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed., Cambridge University Press, 1944.
• 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.
• J. R. Taylor, Scattering Theory: The Quantum Theory of Nonrelativistic Collisions, Dover, 2006.

Exercises

1. Use the series for $J_m$ to show that $J_m$ is regular at the origin for nonnegative integer $m$.

Solution

The first nonzero term is

$$J_m(x) = \frac{1}{m!} \left(\frac{x}{2}\right)^m + O(x^{m+2}).$$

For $m=0$, this tends to $1$. For $m>0$, it tends to $0$ as a positive power of $x$. In either case it is finite at the origin.

2. A circular two-dimensional infinite well has radius $a$ and wavefunctions $\psi(r,\phi)=R(r)e^{im\phi}$. If $\psi(a,\phi)=0$, what quantization condition determines the radial wave numbers?

Solution

Regularity at the origin selects

$$R(r)=J_m(kr).$$

The wall requires

$$J_m(ka)=0.$$

Thus

$$k=\frac{\alpha_{m,s}}{a},$$

where $\alpha_{m,s}$ is the $s$th positive zero of $J_m$.

3. Verify directly that $j_0(x)=\sin x/x$ solves the spherical Bessel equation for $\ell=0$.

Solution

For $\ell=0$, the equation is

$$x^2z'' +2xz' +x^2z =0.$$

Let

$$z=\frac{\sin x}{x}.$$

Then

$$x^2z' = x\cos x-\sin x,$$

so

$$\frac{d}{dx} \left( x^2z' \right) = -x\sin x.$$

But

$$x^2z=x\sin x.$$

Therefore

$$\frac{d}{dx} \left( x^2z' \right) +x^2z =0,$$

which is the same equation.

4. Show that the phase-shift matching form

$$u_\ell(r) \propto \hat j_\ell(kr)\cos\delta_\ell - \hat y_\ell(kr)\sin\delta_\ell$$

has asymptotic phase $kr-\ell\pi/2+\delta_\ell$.

Solution

Use

$$\hat j_\ell(x) \sim \sin\left( x-\frac{\ell\pi}{2} \right), \qquad \hat y_\ell(x) \sim - \cos\left( x-\frac{\ell\pi}{2} \right).$$

Let

$$\chi=x-\frac{\ell\pi}{2}.$$

Then

$$\hat j_\ell(x)\cos\delta_\ell - \hat y_\ell(x)\sin\delta_\ell \sim \sin\chi\cos\delta_\ell + \cos\chi\sin\delta_\ell.$$

The sine addition formula gives

$$\sin\chi\cos\delta_\ell + \cos\chi\sin\delta_\ell = \sin(\chi+\delta_\ell).$$

Thus

$$u_\ell(r) \sim \sin\left( kr-\frac{\ell\pi}{2}+\delta_\ell \right).$$