Bessel Function

Bessel functions are canonical solutions $y(x)$ to Bessel’s differential equation:

$$ x^2 \frac{d^2 y}{dx^2} + x \frac{dy}{dx} + (x^2 - \alpha^2) y = 0 $$

where $\alpha$ is a constant (the order of the Bessel function).

Bessel Functions of the First Kind

The Bessel function of the first kind of order $n$, denoted $J_n(x)$, is defined by the series:

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

For integer order $n$, this simplifies to:

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

Integral Representation

The zeroth-order Bessel function has the integral representation:

$$ J_0(x) = \frac{1}{\pi} \int_0^{\pi} \cos(x \sin \theta) \, d\theta $$

Applications in Fourier Optics

In the context of circularly symmetric functions, the two-dimensional Fourier transform reduces to a Hankel transform (or Fourier-Bessel transform):

$$ G(\rho) = 2\pi \int_0^\infty g(r) J_0(2\pi \rho r) \, r \, dr $$

where $J_0$ is the zeroth-order Bessel function of the first kind.