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

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!

Answer 1

[David Goodmanson]

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

[bohan shen]

Thank you for your solution! It worked and helped me a lot!

Answer 2

[Torsten]

  x0 = [2; 1];
  fun = @(x)[real(1i*besselj(1,complex(x(1),x(2)))-complex(x(1),x(2)).*(besselj(0,complex(x(1),x(2)))-besselj(2,complex(x(1),x(2))))/2);imag(1i*besselj(1,complex(x(1),x(2)))-complex(x(1),x(2)).*(besselj(0,complex(x(1),x(2)))-besselj(2,complex(x(1),x(2))))/2)];
  sol = fsolve(fun,x0)
  fun(sol)
  
  

Note that your equation appears to have an infinite number of solutions. Depending on the initial guess vector x0, "fsolve" converges to different roots.

For comparison with David's solution:

  X0 = [3 0;6 0; 9 0; 12 0; 15 0; 18 0];
  fun = @(x)[real(1i*besselj(1,complex(x(1),x(2)))-complex(x(1),x(2)).*(besselj(0,complex(x(1),x(2)))-besselj(2,complex(x(1),x(2))))/2);imag(1i*besselj(1,complex(x(1),x(2)))-complex(x(1),x(2)).*(besselj(0,complex(x(1),x(2)))-besselj(2,complex(x(1),x(2))))/2)];
  for i = 1:size(X0,1)
    x0 = X0(i,:);
    sol = fsolve(fun,x0);
    Sol(i) = complex(sol(1),sol(2));
  end  
  Sol

[bohan shen]

Thank you for your solution!