Differentiation of Functions of a Complex Variable 3: The Cauchy-Riemann Conditions in Polar Coordinates

 Let $x=r\cos\theta$ and $y=r\sin\theta$. A complex-valued function of a complex variable $f(z)=u(x,y)+iv(x,y)$ can be viewed as $f(z)=u(r,\theta)+iv(r,\theta)$ in terms of polar coordinates $(r,\theta)$. Using the chain rule, we obtain
\begin{align*}
\frac{\partial u}{\partial r}&=\frac{\partial u}{\partial x}\frac{\partial x}{\partial r}+\frac{\partial u}{\partial y}\frac{\partial y}{\partial r}\\
&=u_x\cos\theta+u_y\sin\theta\\
\frac{\partial u}{\partial\theta}&=\frac{\partial u}{\partial x}\frac{\partial x}{\partial\theta}+\frac{\partial u}{\partial y}\frac{\partial y}{\partial\theta}\\
&=-u_xr\sin\theta+u_yr\cos\theta
\end{align*}
Similarly, we also obtain
\begin{align*}
v_r&=v_x\cos\theta+v_y\sin\theta\\
v_\theta&=-v_xr\sin\theta+v_yr\cos\theta
\end{align*}
Suppose that $f(z)$ satisfies the Cauchy-Riemann conditions. Then
\begin{align*}
v_r&=-u_y\cos\theta+u_x\sin\theta\\
v_\theta&=u_yr\sin\theta+u_xr\cos\theta
\end{align*}
Hence, we get
\begin{equation}
\label{eq:cr2}
ru_r=v_\theta,\ u_\theta=-rv_r
\end{equation}
This is the Cauchy-Riemann conditions in polar coordinates. Assume that $f(z)=u(r,\theta)+iv(r,\theta)$ satisfies the Cauchy-Riemann conditions in polar coordinates and that the partial derivatives of $u(r,\theta)$ and $v(r,\theta)$ are continuous. Then, in terms of polar coordinates, $f'(z)$ is given by
\begin{equation}
f'(z)=e^{-i\theta}(u_r+iv_r)
\end{equation}

Example.
\begin{align*}
f(z)&=\frac{1}{z}\\
&=\frac{1}{r}(\cos\theta-i\sin\theta)
\end{align*}
So, $u(r,\theta)=\frac{1}{r}\cos\theta$ and $v(r,\theta)=-\frac{1}{r}\sin\theta$. The Cauchy-Riemann conditions in polar coordinates are satisfied as
$$ru_r=-\frac{1}{r}\cos\theta=v_\theta,\ u_\theta=-\frac{1}{r}\sin\theta=-rv_r$$
and the partial derivatives of $u(r,\theta)$ and $v(r,\theta)$ are continuous. Hence, $f'(z)$ exists and
\begin{align*}
f'(z)&=e^{-i\theta}\left(-\frac{1}{r^2}\cos\theta+i\frac{1}{r^2}\sin\theta\right)\\
&=\frac{1}{z^2}
\end{align*}