Doing a nested numerical integral in Matlab

How do I do the following nested integral in Matlab? The goal is to evaluate the following:

where the function ϕ is defined as

where . The notation seems a bit weird to me since the argument of ϕ in the second equation is only part of the argument where it appears in the first equation. I imagine that just means μ is re-defined in terms of itself. Assuming that's correct, how do I do this nested integral?

Answer 1

[Torsten]

mu = -20:0.01:20;
f = @(mu) integral2(@(u,sigma) sin(mu*sqrt(1-sigma.^2).*cos(pi/2*u)).*(sin(mu*sigma/2)).^2,0,1,0,1);
P = arrayfun(@(mu)f(mu),mu);
plot(mu,P)

[L'O.G.]

arrayfun is slow, is there a faster option?

[Torsten]

Yes, a for-loop over the elements of mu.

But I don't think that "arrayfun" is the time-consuming part - it's "integral2".