Show that in polar coordinates, the Cauchy-Riemann equations take the form \[ \frac{\partial u}{\partial r}=\frac{1}{r} \frac{\partial v}{\partial \theta} \text { and } \frac{\partial v}{\partial r}=-\frac{1}{r} \frac{\partial u}{\partial \theta} .\]
Use (a) to show that the logarithm function, defined as \[ \log z=\log r+i \theta \text { where } z=r e^{i \theta} \text { with }-\pi<\theta<\pi .\] is holomorphic on the region \(r> 0, -\pi < \theta < \pi\).
Show also that this branch cannot be extended continuously across the negative real axis to all of \(\CC\setminus\{0\}\).
Solution
Goal: (a) Derive the polar form of the Cauchy–Riemann equations; (b) use it to show the principal logarithm \(\log z = \log r + i\theta\) (\(z = re^{i\theta}\), \(-\pi < \theta < \pi\)) is holomorphic on the slit plane, and show that this branch has no continuous extension across the negative real axis.
1. Chain rule in polar coordinates: \(u_r = u_x \cos\theta + u_y \sin\theta\) and \(u_\theta = r(-u_x \sin\theta + u_y \cos\theta)\); same for \(v\).
With \(x = r\cos\theta\), \(y = r\sin\theta\), \(dr = \cos\theta\,dx + \sin\theta\,dy\) and \(d\theta = -r\sin\theta\,dx + r\cos\theta\,dy\) by the chain rule.
2. (a): \(u_r = \frac{1}{r} v_\theta\) and \(v_r = -\frac{1}{r} u_\theta\).
The Cauchy–Riemann equations \(u_x = v_y\), \(u_y = -v_x\) imply, by 1, \(u_r = u_x\cos\theta + u_y\sin\theta = v_y\cos\theta - v_x\sin\theta = \frac{1}{r}\qty(-v_x r\sin\theta + v_y r\cos\theta) = \frac{1}{r} v_\theta\). Similarly \(v_r = v_x\cos\theta + v_y\sin\theta = -u_y\cos\theta + u_x\sin\theta = -\frac{1}{r}\qty(-u_x r\sin\theta + u_y r\cos\theta) = -\frac{1}{r} u_\theta\).
3. (b): For \(\log z = \log r + i\theta\) on \(\theset{r > 0, -\pi < \theta < \pi}\), \(u = \log r\) and \(v = \theta\) satisfy the polar Cauchy–Riemann equations of
2.
\(u_r = \frac{1}{r}\), \(u_\theta = 0\), \(v_r = 0\), \(v_\theta = 1\), so \(u_r = \frac{1}{r} = \frac{1}{r}\cdot 1 = \frac{1}{r} v_\theta\) and \(v_r = 0 = -\frac{1}{r} u_\theta\).
4. (b): \(\log z\) is holomorphic on \(\theset{r > 0, -\pi < \theta < \pi}\).
\(u, v\) are \(C^1\) there (on the slit plane, \(\theta\) is a smooth single-valued function of \(z\), and \(\log r\) is smooth), and they satisfy the Cauchy–Riemann equations in polar form by 3; by the standard criterion, \(\log z\) is holomorphic, with derivative \(e^{-i\theta}\qty(u_r + i v_r) = \frac{e^{-i\theta}}{r} = \frac{1}{z}\).
5. (b): The principal branch has no continuous extension to all of \(\CC\setminus\{0\}\).
The principal branch is continuous on its slit-plane domain. To see that it cannot extend across the cut, fix \(z_0 = -r_0\) with \(r_0 > 0\) and approach it from the upper half-plane, \(z = r_0 e^{i\theta}\) with \(\theta \to \pi^-\): \(\log z \to \log r_0 + i\pi\). Approaching from the lower half-plane, \(\theta \to -\pi^+\): \(\log z \to \log r_0 - i\pi\). These two limits differ by \(2\pi i\), so no continuous extension can be defined at \(z_0\).
6. Q.E.D.
2 proves (a), and <1>4–<1>5 prove (b): holomorphicity on the slit plane and failure of continuous extension across its branch cut.