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Question:
What is the equation for a bessel function or if there is a lot of different equations what is the most commonly used? Thanks for your time.



Replies:
Bessel functions are *one* family of functions that satisfy Bessel's differential equation:
x^2 * y'' + x * y' + (x^2 - n^2) * y = 0, y' = dy/dx
Bessel functions are denoted by Jn(x) [read "J sub n of x"]
the most commonly used are for n = 0 and 1:
J0(x) = 1 - x^2 / ( 2^2 * (1!)^2 ) + x^4 / ( 2^4 * (2!)^2 ) + ...
J1(x) = x/2 - x^3 / ( 2^3 * 1! * 2! ) + x^5 / ( 2^5 * 2! * 3! ) + ...
the general formula is:
Jn(x) = SUM{ (x/2)^(n+2*k) * (-1)^k / ( k! * n! ) }
summed over k = 0,1,2,...
if you are into recurrence, the relation: Jn-1(x) + Jn+1(x) = 2*n/x * Jn(x)
can be useful.

John Hawley


Note that a large number of modern computers have the bessel functions available in their math library, so you can call them from a program just like sines and cosines. You can also get one of the mathematical manipulation programs (like mathematical) to do all sorts of fun things with them.

Arthur Smith


Watch out for that recurrence, though. It is only stable in one direction.

christopher grayce



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