derivative of bessel function of the first kind !!

Hello! I would like to check if my implementation of the derivative of bessel function of the first kind is working properly or not , how can I check?! this is the code that I have implemented, please correct me if it is wrong!

    c         =     sqrt(pi./(2.x));
    D_bessel  = c.*besselj(n-0.5,x)-c.*besselj(n+0.5,x).*(n+1)./(x);

Answer 1

[Morgan]

I've found a slightly easier implementation of the derivative for you based on this link. It essentially says that

In MATLAB, this would look like

function dJndx = dbesselj(n,x)
% DBESSELJ      A function that will generically calculate the
%               the derivative of a Bessel function of the first
%               kind of order n for all values of x.
%
% Example usage: dJndx = dbesselj(n,x);
% 
% INPUT ARGUMENTS
% ================
% n             Order of the Bessel function of the first kind
% x             Input variable to Bessel function
% 
% OUTPUT ARGUMENTS
% ================
% dJndx         Derivative of nth order Bessel function of the first 
%               kind at all values of x

    dJndx = n*besselj(n,x)./x - besselj(n+1,x);
    
end

Hopefully this answers your question, let me know if this helps!

Answer 2

[Feruza]

Is there way to compute third order derivatives of Bessel functions? I could find second order derivative from bessel equation but how to proceed with the third order derivative

[Morgan]

Should be able to do

d3Jndx = dbesselj(n,dbesselj(n,dbesselj(n,x)));

For third order derivatives with the function I wrote above.

[Feruza]

Thanks, I also used MATLAB Symbolics to get analytical formulars for derivatives:

syms nu z

b = besselj(nu,z);

db = diff(b)

bj = besselj(nu,z);

ddbj = diff(bj,z,2)

dddbj = diff(bj,z,3)