Bessel Function of First Kind
Description
Shown is the Bessel's Function of the first kind of order nu. The Bessel's Function is the solution to the second order differential equation shown. It arises in many areas, primarily for cylindrical coordinates. Scaling the equation in x with s can be used to make a Bessel Function Series. If the scaling places the roots of the function at the same location, the functions are orthogonal and can be used to represent piece-wise continuous functions such as an initial condition for a vibrating drum-head.
You can control the order of the Bessel Function and you can scale the Bessel Function with the two sliders nu and s respectively.
Calculation Description
This calculation is split into two parts. For smaller x values the series
(Wikipedia,ND) where N=25 is used. This is good for x not to large.
Another function (Goldstein and Thaler, 1958) which calculates an amplitude and phase, was used for larger x values. The split point equals which was the approximate location where the two calculations were closest to each other.
References:
Wikipedia, (ND), Bessel function, retrieved from https://en.wikipedia.org/wiki/Bessel_function (11/7/21)
Goldstein,M., Thaler,R.M., Jan. 1958, Bessel Functions for Large Arguments, JSTOR, Vol.12, No. 61 (Jan.,1958), pp. 18-26, American Mathematical Society.