How is the weak derivative of a function $f\in W^{1,2}(\Omega;\mathbb{C})$, $\Omega\subset \mathbb{R}$ open, defined?
Is it the function $f^\prime\in L^2(\Omega;\mathbb{C})$ that satisfies $\langle v,f^\prime\rangle=-\langle v^\prime, f\rangle$ for all bumb function $v\in C_0^\infty(\Omega;\mathbb{C})$?
And does this function has all the properties as in the real valued case?
Does anyone know a reference for that definition?
Note: Since the functions are complex valued we have to take care of the complex conjugate in the dot product.