In Penrose's book "The Road to Reality" page 97 figure 5.9 he shows the values of the complex logarithm in a diagram as the intersection of two logarithmic spirals. Can you please explain how these particular spirals are derived from the definition of the logarithm?
Logarithm of a complex number as intersections of two logarithmic spirals
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complex-analysis
logarithms
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1It would have been nice to post a reference. This is regarding Section 5.4 and Figure 5.9. I find his description unclear, too. The spirals in the figure are $w^z$ for various choices of $\log w$, but I don't get the emphasis on the intersections. – 2011-01-23
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0One of the spirals should be $\exp(z \operatorname{Log} w) \cdot \exp(2\pi i z t)$ for $t$ real, which gives the values of $w^z$ when $t=n$ is an integer, but I don't see immediately what the other spiral is. – 2011-01-23
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0Thank you for the responses. I must rather shamefacedly admit that I had already looked at an answer to this on the 'Road to Reality' Internet Forum at http://roadtoreality.info/viewtopic.php?f=19&t=100, but the caveats expressed by the site owner made me doubt it and look for a simpler answer. Now I have studied that reference carefully I can see that it is correct and complete: but no way could I have worked it out on my own. – 2011-01-24
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0In the first place don't assume that we all have this book handy. – 2018-05-12