Radial Symmetry and Bessel Functions

The Hankel transform is essentially a radial version of the Fourier transform. For a circularly symmetric function $f\left(r\right)$, its zero-order Hankel transform is $F\left(k\right)={\int }_0^{\infty }f\left(r\right)J{0}_{}\left(k,r\right)r\mathrm{dr}$ where $J_0$ denotes the Bessel function of the first kind. This transform preserves radial information and appears in solutions of wave and diffusion equations with cylindrical symmetry.

The ordinary Fourier transform decomposes a function into plane waves $e^{ikx}$. In polar coordinates, radial waves involve Bessel functions because they satisfy the differential equation $r^2{\partial }_{}^{2}f+r{\partial }_{}^{1}f+(k^2)f=0$. The Hankel transform inherits orthogonality properties from these Bessel functions.

Series Representation of J₀

The Bessel function $J_0$ can be expressed as a power series $J_0\left(z\right)={\sum }_m^0\frac{\left(-1\right)m^2{z}^{2m}}{m^2\left(m,\left(\right)\right)}$. In practice we approximate $J_0$ using the first several terms. Because the integrand also contains $r$, higher powers contribute less for moderate arguments, making the approximation effective.

Numerical Integration

To compute $F\left(k\right)$ numerically, we evaluate the integrand $f\left(r\right)J_0\left(k,r\right)r$ on a grid from $0$ to a chosen upper limit. Simpson's rule averages pairs of intervals to approximate the integral with good accuracy. Because the Hankel transform integrates to infinity, we rely on the user to pick an upper limit large enough that the tail contributions are negligible for the function in question.

Choosing a larger limit may improve accuracy at the cost of more function evaluations. Similarly, decreasing the step size improves the Simpson approximation but requires more computation. The default values provide a reasonable trade-off for smooth, rapidly decaying functions like Gaussians.

Using the Calculator

Enter your radial function in terms of r. Good examples include exp(-r^2) or 1/(1+r^2). Specify the parameter $k$ where you want the transform evaluated and select an upper integration limit. When you press the button, the script samples the function at evenly spaced points, computes the Bessel factor with a short series, and applies Simpson's rule to produce an approximate value for $F\left(k\right)$.

The output appears below the form with six decimal places. You can explore how $F\left(k\right)$ changes with $k$ or how different functions decay. Many problems in optics, acoustics, and electromagnetism reduce to Hankel transforms when radial symmetry is present. Understanding these transforms provides insight into diffraction patterns, beam propagation, and solutions of Bessel-type equations.

Implementation Highlights

The integral is approximated as

$F\left(k\right)\approx {\sum }_j^{=}w{j}_{}f\left(r,j_{}\right)J_0\left(k,r,j_{}\right)r_j$

where $w_j$ are Simpson weights and $r_j$ are equally spaced sample points. The series used for $J_0$ includes ten terms, providing good accuracy for arguments less than about $20$. You can adjust the step size or extend the series for higher precision.

Because everything runs in your browser, no data leaves your computer. This makes it convenient for quick checks or classroom demonstrations. For heavy computational tasks, specialized numerical libraries might be preferable.

Further Exploration

Try computing the transform of exp(-r^2) at several $k$ values and compare it to the known analytic result $F\left(k\right)=\frac{1}{2}{e}^{-\frac{k^2}{4}}$. Seeing this numerical agreement reinforces both the theory and the correctness of the algorithm. The Hankel transform provides a window into radial phenomena ranging from sound waves in a cylindrical duct to the diffraction of laser beams.

Beyond physics, Hankel transforms also arise in probability theory and machine learning when dealing with radial basis functions. Because radial kernels depend only on distance, their Fourier transforms reduce to Hankel transforms. Understanding how these transforms behave helps in designing efficient algorithms for pattern recognition and density estimation on high-dimensional data sets.

Worked Example

Consider the Gaussian function $f\left(r\right)=e^{-r^2}$. The zero‑order Hankel transform of this function has the analytic solution $\frac{1}{2}{e}^{-\frac{k^2}{4}}$. Enter exp(-r^2) in the form, keep $k$ at 1, and set the upper limit to 10. The calculator samples the integrand on 200 points and produces approximately 0.303265. The exact value is 0.303265, demonstrating the accuracy of the numerical approach for smooth functions. Try decreasing the upper limit to 5 to see how truncating the domain introduces slight error.

Common Functions and Transforms

f(r)	Analytic F(k)
exp(-r^2)	$\frac{1}{2}{e}^{-\frac{k^2}{4}}$
1/(1+r^2)	$e^{-k}$
rect(r/a)

These examples provide benchmarks for checking numerical results. Comparing the calculator’s output to known solutions helps build intuition about how rapidly different functions decay in the transform domain.

Limitations

The implementation assumes the integrand remains well behaved over the interval and that the upper limit adequately captures the function’s tail. Highly oscillatory functions may require more integration points or specialized quadrature methods. The Bessel series used for $J_0$ includes ten terms; while sufficient for many arguments, extreme values of $\mathrm{k\; r}$ can reduce accuracy. Users needing high precision should cross‑check with scientific computing packages or increase the series length and step count in the script.

Related Calculators

To explore other transform tools, try the Radon Transform Calculator for tomographic projections or the Laplace Equation Solver for potential problems in cylindrical coordinates.