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Bessel Differential Equation

The Bessel differential equation is the linear second-order ordinary differential equation given by

x^2(d^2y)/(dx^2)+x(dy)/(dx)+(x^2-n^2)y=0.
(1)

Equivalently, dividing through by x^2,

(d^2y)/(dx^2)+1/x(dy)/(dx)+(1-(n^2)/(x^2))y=0.
(2)

The solutions to this equation define the Bessel functions J_n(x) and Y_n(x). 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^2y)/(dx^2)+(2p+1)x(dy)/(dx)+(a^2x^(2r)+beta^2)y=0.
(3)

The solution is

y=x^(-p)[C_1J_(q/r)(alpha/rx^r)+C_2Y_(q/r)(alpha/rx^r)],
(4)

where

q=sqrt(p^2-beta^2),
(5)

J_n(x) and Y_n(x) 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^alphaJ_n(betax^gamma), eta=yx^(-alpha), and xi=betax^gamma (Bowman 1958, p. 117), then

(d^2y)/(dx^2)-(2alpha-1)/x(dy)/(dx)+(beta^2gamma^2x^(2gamma-2)+(alpha^2-n^2gamma^2)/(x^2))y=0.
(6)

The solution is

y={x^alpha[AJ_n(betax^gamma)+BY_n(betax^gamma)] for integer n; x^alpha[AJ_n(betax^gamma)+BJ_(-n)(betax^gamma)] for noninteger n.
(7)

See also

Airy Functions, Anger Function, Bei, Ber, Bessel Function, Bessel Function Neumann Series, 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, Parseval's Integral, Poisson Integral, Ramanujan's Integral, Riccati Differential Equation, Sonine's Integral, Struve Function, Weber Functions, Weber's Discontinuous Integrals

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References

Abramowitz, M. and Stegun, I. A. (Eds.). §9.1.1 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, 1972.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.Zwillinger, D. (Ed.). CRC Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, p. 413, 1995.Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 121, 1997.

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Bessel Differential Equation

Cite this as:

Weisstein, Eric W. "Bessel Differential Equation." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/BesselDifferentialEquation.html

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