Seeing the actual problem helped.
It sounds like this is supposed to be a region-1-only problem with region 2 out of the picture. Assuming that’s the case, you can find k^2, and k itself. Of course k is complex, as is gamma1 = sqrt(nu.^2-k^2).
It’s possible to use the ‘integral’ function to do the integration, but that function has some problems because the branch point involving 1/gamma1 is close to the axis of integration and causes sharp peaks. It works to use the simple trapz integration, and there is more control over how it’s done.
Once you have k, here is a sample calculation
In the problem, they use (x1,x2) for the source point coordinates and (y1,y2) for the observation point coordinates. This is really confusing. It should have been (x1,y1) for source and (x2,y2) for obs. Well, we are stuck with the system.
nu = -50:.00001:50;
gamma1 = sqrt(nu.^2-k^2);
x1 = 1;
x2 = 1;
y1 = 3;
y2 = 2;
f = (1/(4*pi))*exp(i*nu*(x1-y1)).*exp(-gamma1*abs(x2-y2))./gamma1;
T = trapz(nu,f)
B = (i/4)*besselh(0,1,k*sqrt((x1-y1)^2+(x2-y2)^2))
The plot allows you to see the extent of the integrand f, and to pick upper and lower limits on nu to make sure that f is tiny there. You can see large peaks in f, and to successfully integrate across those takes small spacing in nu, such as the .00001 here. You can reduce the spacing until the answer T does not change significantly.
If you compare the resulting integral T to the bessel expression B, they agree to about 12 significant figures in this case, which is not bad.