# Conformal Mapping - Can't Prove Analyticity

by chill_factor
Tags: analyticity, conformal, mapping, prove
 HW Helper Thanks P: 1,025 The Cauchy-Riemann equations are conditions for two functions $\mathbb{R}^2 \to \mathbb{R}$ to be the real and imaginary parts of a differentiable function $\mathbb{C} \to \mathbb{C}$. But to prove analyticity you didn't need to show that the real and imaginary parts of $f(z) = z + K/z$ satisfy the Cauchy-Riemann equations, because you already knew that $f'(z)$ exists for all $z \neq 0$ and is equal to $1 - K/z^2$. Also, doesn't $f'(\sqrt K) = f'(-\sqrt K) = 0$? Your function is then conformal on the open set $\mathbb{C} \setminus \{\sqrt K, 0, -\sqrt K\}$.