Bessel Functions

Equivalent forms

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

The circular-geometry cousins of sine and cosine — one family of functions behind drum harmonics, fiber-optic modes, and the Airy diffraction disk.

Symbol	Name	SI unit	Dimension	Range
$J_\nu(x)$	Bessel functionoutput Solution of Bessel's equation	1	1	—
$x$	Argument Radial/scaled coordinate	1	1	0 – 20
$\nu$	Order Order of the Bessel function	1	1	0 – 5

Where it holds

J_\nu

regular at

x=0

; the second solution

Y_\nu

diverges there and is dropped for problems including the axis. Half-integer orders reduce to spherical Bessel functions (elementary sinusoids over x).

Dimensional analysis

x^{2}y'' + xy' + (x^{2}-\nu ^{2})y

0

x and \nu are

dimensionless; every term shares the dimension of y, so the equation is homogeneous.

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Discovery

Friedrich Bessel (earlier Daniel Bernoulli, Euler) · 1824

Daniel Bernoulli and Euler met these functions studying vibrating chains and membranes in the 1700s. Friedrich Bessel systematized them in 1824 while analyzing planetary perturbations (Kepler's equation). They became indispensable once physicists tackled cylindrical and spherical boundary-value problems.

Research status: stable

Real-world applications

Vibrational modes of drums and circular plates
Waveguide and optical-fiber mode profiles
Fraunhofer diffraction from a circular aperture (Airy disk)

Common misconceptions

Bessel zeros are NOT evenly spaced (unlike sine's), though spacing $\to \pi for$ large x
$J_\nu is$ finite at 0 only $for \nu \ge 0$ ; the second solution $Y_\nu$ blows up there
The amplitude decays as $1/\sqrt{x}$ , so they are not periodic

Experimental verification

Drumhead and circular-membrane overtone frequencies match the ratios of Bessel zeros; the Airy diffraction pattern's first dark ring at

1.22\lambda /D

comes directly from the first zero of

J_{1}

.

Derivation

Separate variables in the 2D Helmholtz equation

\nabla ^{2}u+k^{2}u=0

in polar coordinates:

u=R(r)e^{i\nu \varphi }

.

The radial equation becomes

r^{2}R''+rR'+(k^{2}r^{2}-\nu ^{2})R=0

; substituting

x=kr

gives Bessel's equation.

A Frobenius power-series solution about

x=0

yields

J_\nu (x)=\Sigma (-1)^m/(m!\Gamma (m+\nu +1)) (x/2)^{2m+\nu }

.

For large x the asymptotic form is

J_\nu (x)\approx \sqrt{2/\pi x} \cos (x-\nu \pi /2-\pi /4)

, the damped-oscillation behavior.

What if…

What if the order

\nu is

a half-integer?

The Bessel functions become spherical Bessel functions — elementary combinations like sin(x)/x, appearing in 3D scattering and partial waves.

What if x is very large?

$J_\nu (x) \approx \sqrt{2/\pi x} \cos (x - \nu \pi /2 - \pi /4)$ : it looks like a decaying cosine, so far-field circular waves resemble plane waves fading as $1/\sqrt{r}$ .

1

Airy disk radius

Given · first zero of

J_{1} at x\approx 3.8317

Find · angular radius of the first dark ring

Steps

Circular aperture intensity $\propto [2J_{1}(ka \sin \theta )/(ka \sin \theta )]^{2}$
First zero when $ka \sin \theta = 3.8317$
$\sin \theta = 3.8317/(\pi D/\lambda ) = 1.22 \lambda /D$

Result ·

\theta \approx 1.22 \lambda /D

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