Consider the line $\theta - \ln r = c$ where $-\pi < \theta \leq \pi$ and $c$ is a fixed real constant. How would I find the point $(r, \theta)$ which minimizes $1/r$? Alternatively, what is the minimum $z \in \mathbb{C}$ on the curve $\operatorname{Arg} z - \ln |z| = c$ such that $1/|z|$ is minimized?
Minimize a function in polar coordinates along a curve
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complex-analysis
1 Answers
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$1/r$ is minimized when $r$ is maximized. Since $\ln r = \theta-c$, $r$ is maximized when $\theta$ is maximized. Thus the minimum of $1/r$ occurs at $\theta=\pi$; the corresponding value of $r$ is $\mathrm e^{\pi-c}$.