Polar partial derivatives continuously differentiable implies holomorphic

Question

I need to show that if $f(re^{i\vartheta}) = U(r,\vartheta) + iV(r, \vartheta)$ and $U,V$ are continuously differentiable and satisfy the Cauchy-Riemann equations, then $f$ is holomorphic. I am attempting to show this by demonstrating that $f$ has a linear approximation, similar to $$f(x + iy) = f(x_{0} + iy_{0}) + (x - x_{0})f_{x} + (y - y_{0})f_{y}.$$ However, I'm not sure what form the approximations for (U,V) (which are guaranteed as they are differentiable) should take.

Answer 1

The function $g(x,y) = U(x,y) + iV(x,y)$ is holomorphic, and $f$ is defined via $f(re^{i\theta}) = g(r,\theta)$. For the purpose of polar coordinates, I suppose $(r,\theta)$ should only take values in the rectangle $r > 0, 0 \leq \theta < 2\pi$. At each point in this strip, denote the function $z + i\theta \mapsto re^{i\theta}$ by $Exp$. On the strip, it locally has an inverse which is holomorphic. I'll temporarily denote this locally defined inverse by $\log(re^{i\theta}) = (r,\theta)$. Then if $w = Exp(r,\theta)$, then $g(log(w)) = g(r,\theta) = f(w)$ is holomorphic near $w$.