In Baouenti, Ebenfelt, and Rothschild's Real Submanifolds in Complex Space and Their Mappings, on the very first page, they denote a function $f$ defined on $\mathbb{C}^N$ identified with $\mathbb{R}^{2N}$ as $f(x,y)$ for $x, y \in \mathbb{R}^N$, or "by abuse of notation, $f(z, \overline{z})$" for $z \in \mathbb{C}^N$. What is the usefulness of writing $f$ in this manner? Why the redundancy, and not just $f(z)$?
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3You could write it this way but typically $f(z)$ is typically understood to mean a holomorphic function. Writing $f(z, \bar{z})$ makes it clearer that it need not be holomorphic. – Cameron Williams May 01 '21 at 13:41
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2Not a duplicate, but How exactly does writing $f(z)=f(z,\overline{z})$ work? is related. – Mark S. May 01 '21 at 14:35
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Writing $f$ in this way, when $f$ need not be holomorphic, helps to emphasize the fact that the Wirtinger derivative(s) $\dfrac{\partial f}{\partial\overline z}$ need not be zero, so that the standard $\dfrac{\partial f}{\partial z}$ is not all you're concerned about (which it would be if $f$ were known to be holomorphic).
Mark S.
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