Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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?

share|cite|improve this question
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. – Ross Millikan Jan 23 '11 at 16:08
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. – Hans Lundmark Jan 23 '11 at 16:15
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, 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. – Harry Weston Jan 24 '11 at 15:09

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.