I want to integrate a complex function over a boundary of a region:
$$-\frac{i}{v}\oint_\Gamma \exp(-2\pi i(ux+vy))\,dx,$$
where $\Gamma$ is the closed boundary of a region and $y = f(x)$.
I know that MMA can extract the boundary of a region. For instance:
R = RegionBoundary[Disk[]];
returns the circumference of the unit circle. Therefore if I do:
Integrate[1, {x, y} ∈ R]
gives its length: $2\pi$.
No problem thus far as I understanding that the integral is a contour integral. I mean, it is integrating over the boundary of the region R.
Now, if I do the following:
Assuming[v >= 0 && u >= 0, -I/v*Integrate[Exp[-2 π I (u*x + v*y)], {x, y} ∈ R] // FunctionExpand]
it should return:
2 π BesselJ[1, 2 π Sqrt[u^2 + v^2]]/Sqrt[u^2 + v^2]
but it doesn't.
Am I not understanding correctly how to extract the boundary of a region to use it in a integral along that contour?, or am I not doing the integration well?
Thanks for your time.
FunctionExpand
outside of integral, I get an answer in terms ofBesselJ
. $\endgroup$2 \[Pi] BesselJ[0, 2 \[Pi] Sqrt[u^2 + v^2]]
? $\endgroup$