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.
NEWTON is an electronic community for Science, Math, and Computer Science K-12 Educators, sponsored and operated by Argonne National Laboratory's Educational Programs, Andrew Skipor, Ph.D., Head of Educational Programs.
