Cauchy-Riemann conditions in polar coordinates
The Cauchy-Riemann conditions are central to checking when a complex function f(z) is analytic. While they are commonly presented in Cartesian coordinates (x, y) for z = x + iy (here), it is often efficient to adopt polar coordinates (r, \theta) with z = re^{i\theta}, for functions exhibiting radial or angular symmetry.
Given f(z) = u(r, \theta) + iv(r, \theta), the Cauchy-Riemann equations in polar coordinates are:
\begin{aligned} \frac{\partial u}{\partial r} &= \frac{1}{r} \frac{\partial v}{\partial \theta} \\ \frac{\partial v}{\partial r} &= -\frac{1}{r} \frac{\partial u}{\partial \theta} \end{aligned}
Through the chain rule, one connects the polar derivatives to the Cartesian ones using:
\begin{aligned} u_r &= u_x \cos\theta + u_y \sin\theta \\ u_\theta &= -u_x r\sin\theta + u_y r\cos\theta \end{aligned}
Taking u_x = v_y and u_y = -v_x from the Cartesian Cauchy-Riemann conditions, substitution yields the polar versions above, confirming the equivalence under coordinate change.
A feature in the polar picture is that the complex derivative can be approached either radially or angularly. For example,
f^\prime(z) = e^{-i\theta} \left( \frac{\partial u}{\partial r} + i\frac{\partial v}{\partial r} \right)
computed radially, matches the angular approach:
f^\prime(z) = e^{-i\theta} \left( \frac{1}{r}\frac{\partial v}{\partial \theta} - \frac{i}{r}\frac{\partial u}{\partial \theta} \right)
if and only if the polar Cauchy-Riemann conditions hold.
Example: Analyticity of \operatorname{Log} (z)
Consider the principal value of the complex logarithm:
\operatorname{Log} (z) = \ln (r) + i\theta
where z = re^{i\theta} and \theta \in (-\pi, \pi]. Here, u(r, \theta) = \ln r and v(r, \theta) = \theta. The required derivatives are:
\begin{aligned} & \frac{\partial u}{\partial r} = \frac{1}{r} \\ & \frac{\partial u}{\partial \theta} = 0 \\ & \frac{\partial v}{\partial r} = 0 \\ & \frac{\partial v}{\partial \theta} = 1 \end{aligned}
Plugging into the polar Cauchy-Riemann conditions, both are satisfied for r > 0, confirming analyticity except on the branch cut (-\infty, 0] where \theta jumps discontinuously.
For more insights into this topic, you can find the details here.