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?

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    $\begingroup$ It 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. $\endgroup$ – Ross Millikan Jan 23 '11 at 16:08
  • $\begingroup$ One 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. $\endgroup$ – Hans Lundmark Jan 23 '11 at 16:15
  • $\begingroup$ Thank 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 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. $\endgroup$ – Harry Weston Jan 24 '11 at 15:09
  • $\begingroup$ In the first place don't assume that we all have this book handy. $\endgroup$ – Christian Blatter May 12 '18 at 15:31

If $z = x+iy$ and $w = \exp(x'+iy')$, then the set of all powers $w^z$ is given by $$ \{ w^z \} = \{ \exp(z\log w) \} = \{\exp(xx'-yy'-2\pi ky +i(xy'+x'y+2\pi k x) : k \in \Bbb{Z} \}$$ You can easily check that this set lies on the logarithmic (equiangular) spiral given in polar coordinates $(r,\theta)$ by $$ r(t) = \exp(\alpha_m t + \beta_m), \quad \theta(t) = t $$ where $$ \alpha_m = \frac{y}{m-x},\quad \beta_m = xx' - yy' - \alpha_m(xy'+x'y) $$ and $m \in \Bbb{Z}$ is an arbitrary integer $\neq x$. Moreover, you can also easily check that the intersection of any two such spirals for integer $m,n$ with $|m-n| = 1$ is precisely the above set of all complex powers $w^z$. The surprising thing is that this set in itself can suggest a shape (see this picture) that is more complicated than a logarithmic spiral, despite (or because of) the fact that it lies on infinitely many of them.


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