function y=besJ0(x)
%-----------------------------------------------------------------------
% Approximate the Bessel's function of the first kind of order zero
% via 10-term Taylor's series expansion.
%   J0(x) = sum(sign*x^i/i!) for i=even integers
%   J0(x) = 1 - (x/2)**2/(1!)**2 + (x/2)**4/(2!)**2 - (x/2)**6/(3!)**2 + ...
% Method used: iterate on the following relationship based on Taylor's series:
%   delta = -delta*(x/2)**2/i**2
%   bessel = bessel + delta
% Programming Note:
%   Because Taylor's series is alternating in sign, there is subtractive
%   cancellation.  Nevertheless, convergence is quick enough for a small x.
%   For large values of x, apply the asymptotic approximation formula.
%     J0(x)=sqrt(2/pi/x)*cos(x-pi/4),  for x>25
% Instructor: Nam Sun Wang
%-----------------------------------------------------------------------

% Calculate the Taylor's series expansion
%     Initialize  (The first term is contained in the initialization step.)
      delta = 1. ;
      y = 1. ;

      for i=1 : 99999
        delta = -delta*(x/2.)*(x/2.)/i/i ;
        y = y + delta ;
%       Converge to 5 digits.
%       Since J0 is O(1), this is almost identical to 5 significant digits.
        if(abs(delta) < 1.E-5) break; end
      end