Bessel Function

Bessel functions are a family of special functions that arise as solutions to Bessel's differential equation, a second-order linear equation that appears repeatedly whenever a physical problem involving circular or cylindrical symmetry, such as the vibration of a circular drumhead or the propagation of heat in a cylindrical rod, is solved using the method of separation of variables. They are named after the German astronomer and mathematician Friedrich Bessel, who studied them systematically in the 1820s in connection with problems in celestial mechanics, though related series had already appeared earlier in the work of Daniel Bernoulli and others. Bessel functions come in several standard families distinguished by their behavior, most notably the functions of the first kind, which remain finite at the origin, and the functions of the second kind, which instead become infinite there, with a further pair of modified Bessel functions used for problems whose governing equation has the opposite sign from the ordinary case. Beyond their origin in mathematical physics, Bessel functions appear throughout applied mathematics and engineering, including in signal processing, antenna theory, and the analysis of frequency modulation, wherever a system's natural symmetry is circular or cylindrical rather than the rectangular symmetry more commonly assumed.