Discussion
If \(w = \log_{e} z = \log_{e}(x + iy)\), where \(i = \sqrt{-1}\), then which one of the following statements is correct?
Answer: A
The correct statement is \(w\) is analytic everywhere except at \(z = 0\).
Write \(w = u + iv\) with \(u = \frac{1}{2}\log_{e}(x^{2} + y^{2})\) and \(v = \tan^{-1}(y/x)\). Then
\[ \begin{aligned} u_{x} &= \frac{x}{x^{2} + y^{2}} = v_{y} \\ u_{y} &= \frac{y}{x^{2} + y^{2}} = -v_{x} \end{aligned} \]
so the Cauchy-Riemann equations hold wherever \(x^{2} + y^{2} \ne 0\), and \(dw/dz = 1/z\) exists for every \(z \ne 0\). (Taking a single-valued branch also needs a cut from the origin, but the textbook answer, and the key, single out \(z = 0\).)
So \(w\) is not non-analytic everywhere. Its conjugate pair is \(\frac{1}{2}\log_{e}(x^{2} + y^{2})\) and \(\tan^{-1}(y/x)\), not \(\log_{e}(x^{2} - y^{2})\) and not \(\tan^{-1}(x/y)\).
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