Load up Matlab and paste in the following code:

%% Find the zeros of the Bessel Function
alpha  = 2.4;
beta   = 8.65;

%% Other Parameters
aa     = 1;   % what is the length
order  = 0;   % What order of Bessel

rr     = linspace (0,aa,10000);    % define values of rr from 0 to aa
JPa    = besselj(order,alpha*rr);  % Use Matlab built in function for 
JPb    = besselj(order,beta*rr);   %  Bessel functions


%% Intgrate over the r J_p(alpha r / aa) J_p(beta r /aa ) from 0 to aa
Meaty  = rr .* JPa .* JPb;    % Get the middle inside
Value  = trapz(rr, Meaty);    % Do the integral
titleStr = sprintf ('Integral Value = %f', Value);

%% Plot results
plot (rr, JPa, '-k', 'linewidth', 2)
hold on
plot (rr, JPb, '-b', 'linewidth', 2)
hold off

xlabel ('rr');
ylabel ('Bessel');
title  (titleStr);  %Output the results of the intgral
legend ('alpha', 'beta')

Answer the following questions:

• What is the value of J0(alpha aa / aa) = J0(alpha) = J0(2.4)?
• What is the value of J0(beta aa / aa) = J0(beta) = J0(8.65)?
• What is the result of the following integral?

• In the code, alpha = 2.4 and beta = 8.65. This is the first and third zero for the p=0 Bessel function. Can you please find the second and the fourth?
• Can you please fill in the values for the integral for this table? In the box put in the value of the integral?

beta down alpha ->	2.4	??	8.65	??
2.4
??
8.65
??

• Review the notes from class. Develop at least three questions: Class Notes: https://cosmicpathways.org/wp-content/uploads/2021/04/besselorthogonalityproof.pdf
• Write a paragraph discussing how the analytical solution for this integral, shown below, matches the results in the table.

• Write a paragraph discussing how you would find the table for the J1 case.