Bessel Differential Equation
المؤلف:
Abramowitz, M. and Stegun, I. A
المصدر:
in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover
الجزء والصفحة:
...
30-5-2018
3060
Bessel Differential Equation
The Bessel differential equation is the linear second-order ordinary differential equation given by
 |
(1)
|
Equivalently, dividing through by 
,
 |
(2)
|
The solutions to this equation define the Bessel functions 
and 
. The equation has a regular singularity at 0 and an irregular singularity at 
.
A transformed version of the Bessel differential equation given by Bowman (1958) is
 |
(3)
|
The solution is
![y=x^(-p)[C_1J_(q/r)(alpha/rx^r)+C_2Y_(q/r)(alpha/rx^r)],](http://mathworld.wolfram.com/images/equations/BesselDifferentialEquation/NumberedEquation4.gif) |
(4)
|
where
 |
(5)
|
and 
are the Bessel functions of the first and second kinds, and 
and 
are constants. Another form is given by letting 
, 
, and 
(Bowman 1958, p. 117), then
 |
(6)
|
The solution is
![y=<span style=]() {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. " src="http://mathworld.wolfram.com/images/equations/BesselDifferentialEquation/NumberedEquation7.gif" style="height:41px; width:311px" /> |
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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