if $z=x+iy=re^{i\theta}$ and $\omega :=f(z) = u+iv = \rho e^{i\phi},$ How can we represent f with
(a) $u$ and $v$ as functions of $x,y$;
(b) $u$ and $v$ as functions of $r,\theta$;
(c) $\rho$ and $\phi$ as functions of $x,y$
(d) $\rho$ and $\phi$ as functions of $r,\theta$
In each of the cases, assuming the differentiability, I have to write down the neccessary conditions in order for $f$ to be holomorphic.
I am allowed to assume $z\ne0$ and/or $\omega \ne 0$ if neccessary.
Now for part (a) , I wrote $f(z) = f(x+iy) = Re(x+iy) + iImf(x+iy) = u(x,y) + iv(x,y)$
If the functions $u(x,y)$ and $v(x,y)$ satisfy the Cauchy-Riemann equations and have continuous partial derivatives in an open set $U$, then the function $f(z)=u(x,y)+iv(x,y)$, where $z=x+iy$, is analytic in $U$.
and for part (b) I wrote $f(z) = u(r,\theta) + iv(r,\theta) $ and I wrote this:
Now I do not know parts (c) and (d), I would appreciate any help, thanks in advance.