Differential Equations and Linear Algebra, 4th Edition

11.4 Bessel Functions

We have already seen several cases of Bessel’s equation of order $p\geqq 0$,

$$x^{2}y″+xy\text{′}+(x^2-p^2)y=0.$$ (1)

Its solutions are now called Bessel functions of order p. Such functions first appeared in the 1730s in the work of Daniel Bernoulli and Euler on the oscillations of a vertically suspended chain. The equation itself appears in a 1764 article by Euler on the vibrations of a circular drumhead, and Fourier used Bessel functions in his classical treatise on heat (1822). But their general properties were first studied systematically in an 1824 memoir by the German astronomer and mathematician Friedrich W. Bessel (1784–1846), who was ...