Bessel Differential Equation

$$x^2{d^2y\over dx^2} + x {dy\over dx} + (x^2-m^2)y = 0.$$ (1)

Equivalently, dividing through by $x^2$,

$${d^2y\over dx^2} + {1\over x}{dy\over dx} + \left({1-{m^2\over x^2}}\right)y = 0.$$ (2)

The solutions to this equation define the Bessel Functions. The equation has a regular Singularity at 0 and an irregular Singularity at $\infty$.

A transformed version of the Bessel differential equation given by Bowman (1958) is

$$x^2 {d^2 y\over dx^2}+(2p+1)x{dy\over dx}+(a^2 x^{2r}+\beta^2)y = 0.$$ (3)

The solution is

$$y=x^{-p}\biggl[C_1 J_{q/r}\left({{\alpha\over r} x^r}\right) +C_2 Y_{q/r}\left({{\alpha\over r} x^r}\right)\biggr],$$ (4)

where

$$q\equiv\sqrt{p^2-\beta^2},$$ (5)

$J$ and $Y$ are the Bessel Functions of the First and Second Kinds, and $C_1$ and $C_2$ are constants. Another form is given by letting $y=x^\alpha J_n(\beta x^\gamma)$, $\eta=yx^{-\alpha}$, and $\xi=\beta x^\gamma$ (Bowman 1958, p. 117), then

$${d^2y\over dx^2}-{2\alpha -1\over x}{dy\over dx}+\left({\bet... ...^2 x^{2\gamma-2} +{\alpha^2-n^2\gamma^2\over x^2}}\right)y=0.$$ (6)

The solution is

$$y=\cases{x^\alpha [AJ_n(\beta x^\gamma)+BY_n(\beta x^\gamma)... ...a x^\gamma)+BJ_{-n}(\beta x^\gamma)] & for noninteger $n$.\cr}$$ (7)

See also Airy Functions, Anger Function, Bei, Ber, Bessel Function, Bourget's Hypothesis, Catalan Integrals, Cylindrical Function, Dini Expansion, Hankel Function, Hankel's Integral, Hemispherical Function, Kapteyn Series, Lipschitz's Integral, Lommel Differential Equation, Lommel Function, Lommel's Integrals, Neumann Series (Bessel Function), Parseval's Integral, Poisson Integral, Ramanujan's Integral, Riccati Differential Equation, Sonine's Integral, Struve Function, Weber Functions, Weber's Discontinuous Integrals

References

Bowman, F. Introduction to Bessel Functions. New York: Dover, 1958.

Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, p. 550, 1953.