It is a standard fact from elementary complex analysis that a holomorphic function $f:\mathbb{C}\to \mathbb{C}$ is a conformal mapping. Now, suppose I have a map $f':\mathbb{R}^2\to \mathbb{R}^2$ which is a conformal mapping of the plane onto itself. Write $$f'(x,y) = (f\_1(x,y),f\_2(x,y)).$$ Is $f\_1 + if\_2$ holomorphic?



When are conformal maps holomorphic?
If we define **$f' : \mathbb{R}^2 \rightarrow \mathbb{R}^2$** by **$f'(x,y) = (x, -y)$**, then it is conformal, but the corresponding map $f\_1 + i f\_2$ is not holomorphic.


