# Reduction of the optimization of a function of complex variables

Suppose you have some function $$f:\Omega\subseteq\mathbb{C}^n\rightarrow \mathbb{R}$$ which you want to minimize. Of course, this is not a complex variable problem. Indeed, it is a multivariable problem under the identification $$\mathbb{C}^n\cong\mathbb{R}^{2n}$$. Assume this function is smooth. Now, this function is explicitely expressed in terms of the complex coordinates $$z_1,\dots,z_n$$ and its conjugate variables. Can I differentiate with respect to these as if they where real? I also obtain results related by complex conjugation when I differentiate with respect to a complex variable and its complex conjugate. Is this a general feature? In that case, can I solve the problem by just worrying about the complex variables and not about their conjugates?