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?


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