Just a quick question, I am going through some books on complex analysis and I'm wondering what an integral like $\int_f f(z,\bar{z}) \sqrt{dz d\bar{z}}$ means. How is one supposed to take that notation to mean something?? I have not come across things that have roots over the $dz's$ and $d\bar{z}'s$. Besides, the integral here is a single integral, but there are two variables involved?
I am assuming I can do that as perhaps I can parametrise $f$ in terms of some parameter $t$.
Thanks
Best Answer
This is somewhat of a guess... hopefully you can find the definition somewhere in the text. But I think that this notation means you are integrating with respect to path length. Write out the differentials in terms of real and imaginary parts, and $ \sqrt{dz ~ d\bar{z}} $ becomes $ \sqrt{dx^2 + dy^2} $. So if you plug $ f(z,\bar{z}) = 1 $ into your expression, this integral will give you the length of that path.
To make it totally concrete, if you parametrize your path with respect to some parameter $t$, then the differential becomes $\sqrt{(\frac{dx}{dt})^2 + (\frac{dy}{dt})^2}~dt$