How to find inverse of Bessel function of first kind?

I already have a distribution of intensities which contains real and imaginary numbers. How can I find inverse of Bessel function of first kind?

=(distribution I found) , calculate ν.

Answer 1

[Rohit Kulkarni]

Hi Jacky!

Here's an example MATLAB code that demonstrates how to find the inverse of the Bessel function of the first kind using the Newton-Raphson method:

% Distribution of intensities (example values)
intensities = [0.5+0.2i, 0.8-0.3i, 1.2+0.5i, 1.5-0.1i];

% Function to calculate the Bessel function of the first kind
bessel_fn = @(x) besselj(0, x);

% Function to calculate the derivative of the Bessel function of the first kind
bessel_deriv = @(x) -besselj(1, x);

% Tolerance for convergence
tolerance = 1e-6;

% Maximum number of iterations
max_iterations = 100;

% Loop over each intensity value
for intensity = intensities
    % Initial guess for the inverse Bessel function value
    guess = 1.0;
    
    % Newton-Raphson iteration
    for iteration = 1:max_iterations
        % Calculate the value of the Bessel function at the current guess
        bessel_value = bessel_fn(guess);
        
        % Calculate the residual
        residual = intensity - bessel_value;
        
        % Check for convergence
        if abs(residual) < tolerance
            break;
        end
        
        % Calculate the derivative of the Bessel function at the current guess
        bessel_deriv_value = bessel_deriv(guess);
        
        % Update the guess using the Newton-Raphson method
        guess = guess - residual / bessel_deriv_value;
    end
    
    % Display the inverse Bessel function value
    fprintf('Inverse Bessel function value for intensity %f: %f\n', intensity, guess);
end

In this code, we define the distribution of intensities as an array intensities. The Bessel function of the first kind is calculated using the besselj function, and its derivative is approximated by taking the negative of the Bessel function of the next order (besselj(1, x)).

The Newton-Raphson iteration is performed until the residual becomes smaller than the specified tolerance (tolerance). The inverse Bessel function values are then displayed for each intensity.

Hope this helps.

Thanks