Bessel functions are a family of solutions to Bessel's differential equation, a second-order linear ordinary differential equation that arises in problems with cylindrical or spherical symmetry. Named after the German astronomer Friedrich Bessel, who systematized their study in the early nineteenth century, they appear throughout mathematical physics, engineering, and applied mathematics - particularly in problems involving wave propagation, heat conduction, and potential theory in cylindrical coordinates.

Bessel's differential equation has the form:

x² y'' + x y' + (x² - α²) y = 0

where α is a real or complex number called the order of the equation. Solutions to this equation are called Bessel functions of order α.

The two linearly independent solutions are conventionally designated:

• Bessel functions of the first kind, J_α(x) - finite at the origin for non-negative integer orders
• Bessel functions of the second kind, Y_α(x) (also called Neumann functions or Weber functions) - singular at the origin

For problems on a domain that excludes the origin, both kinds are needed. For problems on a domain that includes the origin, Y_α is typically excluded on physical grounds. A further pair of combinations, the Hankel functions H_α^(1)(x) and H_α^(2)(x), are constructed from J_α and Y_α and are particularly useful for representing outgoing or incoming waves.

When the argument of the differential equation is imaginary, the solutions are called modified Bessel functions of the first kind (I_α) and second kind (K_α). These arise in problems with exponential rather than oscillatory behavior.

Bessel functions share several important properties that make them useful in applied problems:

• Oscillatory behavior: J_α(x) and Y_α(x) oscillate with decreasing amplitude for large x, resembling damped sinusoids.
• Orthogonality: The functions J_α(k_n x) are orthogonal on [0,1] with respect to a weight function x, where k_n are the positive zeros of J_α. This property underlies their use in eigenfunction expansions (Fourier-Bessel series).
• Recurrence relations: Bessel functions satisfy recurrence relations connecting functions of adjacent orders, facilitating numerical computation.
• Zeros: Each J_α has an infinite number of positive real zeros, whose distribution is important in applications such as drum membrane vibration and optical fiber modes.

Bessel functions appear in a wide range of physical and engineering contexts:

• Acoustics and vibration: The vibrational modes of a circular drum membrane are described by Bessel functions.
• Electromagnetism: Wave propagation in cylindrical waveguides and coaxial cables involves Bessel functions as radial mode functions.
• Heat conduction: Temperature distributions in cylindrical objects under various boundary conditions are expressed as Bessel series.
• Quantum mechanics: The radial wavefunctions for a particle in a cylindrical potential well involve Bessel functions.
• Signal processing: Frequency modulation (FM) spectra are characterized by Bessel function coefficients (Carson's bandwidth rule is an approximation of this).
• Optics: Diffraction patterns from circular apertures (Airy disks) are expressed in terms of J_1.

Although Daniel Bernoulli and Leonhard Euler encountered related functions earlier in the eighteenth century, Friedrich Bessel's 1824 analysis of planetary perturbations gave the first systematic treatment of the functions bearing his name. The theory was subsequently developed by mathematicians including Carl Gottfried Neumann, Hermann Hankel, and Lord Rayleigh. A fuller account is given on the Bessel Functions - History page.

The mathematical theory of Bessel functions is well-established and not subject to significant dispute. The mathematical consensus on their properties, orthogonality, and classification is essentially universal within the mathematical and scientific communities.

• Bessel Functions - History
• Friedrich Bessel
• Ordinary Differential Equations
• Fourier Analysis
• Cylindrical Coordinates
• Spherical Harmonics
• Mathematical Physics

1. Bessel, F. W. (1824). “Untersuchung des Theils der planetarischen Störungen, welcher aus der Bewegung der Sonne entsteht.” Abhandlungen der Berliner Akademie.
2. Watson, G. N. (1944). A Treatise on the Theory of Bessel Functions, 2nd ed. Cambridge University Press.
3. Abramowitz, M., and Stegun, I. A. (1964). Handbook of Mathematical Functions. National Bureau of Standards. Chapter 9.
4. Arfken, G. B., Weber, H. J., and Harris, F. E. (2013). Mathematical Methods for Physicists, 7th ed. Academic Press. Chapter 14.
5. Bowman, F. (1958). Introduction to Bessel Functions. Dover Publications.