The Bessel equation (also called Bessel's differential equation) is a second-order linear ordinary differential equation of the form:
$$x^2 \frac{d^2y}{dx^2} + x \frac{d y}{dx} + (x^2 - \alpha^2)y = 0$$
where $\alpha$ is a real or complex number known as the order of the equation. It arises naturally in problems with cylindrical or spherical symmetry, including heat conduction, wave propagation, fluid dynamics, and electromagnetic theory. The equation is named after the German astronomer and mathematician Friedrich Bessel (1784-1846), who systematically analyzed its solutions in the context of planetary perturbation theory, though the equation had appeared earlier in work by Daniel Bernoulli and Leonhard Euler.1)
The general solution to the Bessel equation is expressed as a linear combination of two linearly independent functions. For non-integer $\alpha$, these are the Bessel functions of the first kind, $J_\alpha(x)$, and the Bessel functions of the second kind, $Y_\alpha(x)$ (also called Neumann functions). $J_\alpha(x)$ is finite at the origin for $\alpha \geq 0$; $Y_\alpha(x)$ diverges at $x = 0$ and is therefore excluded from physical solutions in domains that include the origin.2)
For integer orders, $J_{-n}(x) = (-1)^n J_n(x)$, making $J_n$ and $J_{-n}$ linearly dependent; $Y_n$ then serves as the required second independent solution.
Related families of functions are derived from the Bessel equation under coordinate or variable transformations:
The Bessel equation appears whenever the Laplacian is expressed in cylindrical or spherical coordinates and variables are separated. Representative applications include:
Bessel functions satisfy a number of important recurrence relations, integral representations, and orthogonality conditions. The functions $J_\alpha(\lambda_{n} x)$, where $\lambda_n$ are successive zeros of $J_\alpha$, form a complete orthogonal set on $[0,1]$ with respect to the weight function $x$, enabling Bessel-Fourier series expansions analogous to classical Fourier series. The asymptotic behavior for large $x$ is oscillatory and approximately sinusoidal, decaying as $x^{-1/2}$.4)
While the equation bears Bessel's name, its solutions were encountered earlier: Daniel Bernoulli studied oscillations of a hanging chain (1738), and Euler treated related problems in membrane vibration. Bessel's systematic 1824 treatment established the general theory and introduced the functions $J_n$ for arbitrary integer $n$. Later work by Carl Neumann, Hermann Hankel, and others extended the theory to non-integer orders and complex arguments. See Bessel Equation - History for a full account.
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