So basically I want to integrate over two complex variables, so my integration will look something like this
$\int uv\cdot e^{-uv}dudv$
where u and v are complex coordinates, in this case two dimensional (u=x+iy) and v=(x-iy). Im wondering how to solve this integral. I have seen a number of different suggestions. Can someone point me in the right direction or in the direction of some appropriate literature?
It is rare that one encounters an indefinite integral over two variables, and I myself have never seen such an idea defined explicitly. However, the notions of differentiation and indefinite/definite integration over a single complex variable certainly exist, and are considered a part of what is broadly referred to as complex analysis.
As for your question, perhaps we can simply interpret your integral as $$ \int\left[\int uv\,e^{-uv}\,du \right]dv $$ In other words, we would be solving the partial differential equation $$ \frac{\partial^2 f(u,v)}{\partial u \,\partial v} = uv e^{-uv} $$