MATLAB: How to solve this equation which contains complex number, bessel equation and its derivative

bessel equationbesseljcomplex numberdifferential equationsMATLABnewton raphsonnewton's methodode

1i*besselj(1,x) – x.*(besselj(0,x) – besselj(2,x))/2 = 0;
I can't solve this equation because it contains complex number. This equation is very important for my dissertation. If anyone knows the solution, please help me out… Thanks in advance!

Best Answer

Hi bohan shen,
The first six roots are shown below. First, since J1 is odd and J0,J2 are even, if z is a root then so is -z. The contour plot of abs(f(z)) shows roots in quadrant IV, meaning there also roots in quadrant II. No roots in quadrants I or III.
Roots are determined by Newton's method. Each initial estimate has to be within an enclosed contour. From the contour plot the estimate of 3 looks a bit sketchy (2-.6i would have been a sure thing) but it worked anyway.
clear i                    % precaution
xx = 0:.01:20;
yy  = -1:.01:1;
[x y] = meshgrid(xx,yy);
z = x+i*y;
f = @(z) i*besselj(1,z) - (z/2).*(besselj(0,z) - besselj(2,z));
dfdz = @(z) ((i-1)/2)*(besselj(0,z) - besselj(2,z)) ...
         + (z/2).*((3/2)*besselj(1,z) -(1/2)*besselj(3,z));
contour(x,y,abs(f(z)))
grid on
w0 = [3;6;9;12;15;18];   % initial estimates
w = w0;
for k = 1:10
  w = w - f(w)./dfdz(w);   
end    
w           % roots
f(w)        % should be small
w =
   2.0811 - 0.6681i
   5.3355 - 0.1967i
   8.5372 - 0.1193i
  11.7063 - 0.0863i
  14.8637 - 0.0677i
  18.0156 - 0.0557i
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