MATLAB: How to integrate products of hypergeometric functions and rational functions

confluenthypergeometric

Hello, I am a physics student and I am new to Matlab. I am trying to solve definite integrals involving the product of confluent first order hypergeometric functions and rational functions as follows:
p=0,1,2.... Where p=0,1,2,3…
When using the Matlab "integral" command the computation of these integrals takes a long time. The question is whether there is any routine in Matlab that does quick calculations of integrals that decay very quickly to zero.
Thank you very much for reading, excuse my English I am using a translator.

Best Answer

Hello Nicolas,
I don't know how you are implementing the nth derivative of 1/(lambda^2 + q^2) but it is obviously very time consuming. The method below calculates the nth derivative algebraically and takes about as much time as for the zeroth derivative, about 0.1 sec on my PC. That's with moderate values for m, n, alpha, beta, lambda.
lambda = 1.2;
alpha = 2.3;
beta = 3.4;
m = 2;
n = 3;
F = @(q) lorentzdiff(5,q,lambda).*hypergeom(m,3/2,-alpha*q.^2).*hypergeom(n,3/2,-beta*q.^2);
J = integral(F,0,inf)   
% ------------------
function yn = lorentzdiff(n,x,a)
% nth derivative wrt 'a' of the Lorentian 1/(x^2+a^2),
% as a function of x for fixed scalar 'a'
%
% function yn = lorentzdiff(n,x,a)
yn = (-1)^(n)*(factorial(n)./x).*imag((1./(a-i*x)).^(n+1));
ind = abs(x) < 1e-6*a;
yn(ind) = (-1)^n*factorial(n+1)/a^(n+2);
end
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