TITLE: Complex differentiation
QUESTION [7 upvotes]: Is differentiation in the complex plane the same as that in the reals? In particular do the normal differentiation rules apply in the complex case such that I can just treat a complex map as a real map? 
Thanks.

REPLY [2 votes]: Here is a "rule" that fails: if $f$ is differentiable, then $\overline f$ is differentiable, and $\overline{f'}=(\overline f) '$.
This rule is true if the variable is real (for complex valued functions). It is also true if the complex plane is treated as $\mathbb{R}^2$ and differentiation as real differentiation on $\mathbb{R}^2$. So you might say that failure of this rule is what distinguishes complex analysis from real analysis.