= Solution
For a <holomorphic function> with $z=re^{i\theta}$, the <chain rule> gives
$$
f_r=e^{i\theta}f'(z),\qquad f_\theta=ire^{i\theta}f'(z)=irf_r.
$$
Writing $f=u+iv$ and equating <real parts> and <imaginary parts>,
$$
u_\theta=-rv_r,\qquad v_\theta=ru_r.
$$
For $r>0$, division by $r$ gives the <Cauchy-Riemann equations> in <polar coordinates>:
$$
\boxed{\frac1r u_\theta=-v_r,\qquad u_r=\frac1r v_\theta.}
$$
The restriction $r>0$ reflects the coordinate singularity at the origin; the Cartesian equations have no such restriction.
Back to article page
