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I need to calculate the integral $$\frac{1}{2\pi}\int_{0}^{2\pi}\log{|a-e^{i\theta}|}d\theta$$

When $|a|> 1$, I can show it equals $\log|a|$. But I failed to handle the situation when $|a|\leqslant 1$. I know it should be $0$.

I tried to use $\log(a-z)$, but when $|a|\leqslant 1$, such $\log$ can not be defined. So anyone can help?

Updated: the second situation should be $|a|<1$.

Lord_Farin
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hxhxhx88
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  • Why can't that function be defined when $,|a|\le 1,$ ? – DonAntonio Jun 26 '13 at 10:54
  • @DonAntonio, the unit circle surrounding $a$ will contain $0$, then $\log$ function has no single branch on it, right? – hxhxhx88 Jun 26 '13 at 11:00
  • @DonAntonio, maybe it should be $|a|<1$ – hxhxhx88 Jun 26 '13 at 11:01
  • Well, two things: first, a branch cut can be chosen and we can try to work with it, and second: is your function the real or the complex logarithm? Note that you wrote in your question the absolute value of $,a-e^{i\theta},$ ... – DonAntonio Jun 26 '13 at 11:03
  • @DonAntonio, when dealing with absolute value ,it is real log, otherwise it is complex log. – hxhxhx88 Jun 26 '13 at 11:23
  • @DonAntonio, I thought the single branch of log can only be chosen on the complex plane excluding a ray starting at the origin. – hxhxhx88 Jun 26 '13 at 11:24
  • That's what I thought, @hxhxhx88... – DonAntonio Jun 26 '13 at 11:24
  • @julien, thanks! But I still have a confusion. In the case $|a|=1$, particularly $a=1$, is the question well-posed? Because when integrating by $\theta$ from $0$ to $2\pi$, at some point the function $\log|a-e^{i\theta}|$ may equals $\log(0)$, which is not defined. – hxhxhx88 Jun 26 '13 at 13:18
  • Was it absolutely necessary to delete my comment linking to another relevant thread? What was the motivation? Just curious. – Julien Jun 26 '13 at 15:46
  • @julien: your comment was deleted by the system. When a question is deemed a duplicate, the system removes comments that state the question is a duplicate since the question will be labelled a duplicate anyway. Unfortunately, the script doesn't realize there was more information there. – robjohn Jun 26 '13 at 21:53
  • @robjohn Thanks a lot for the explanation. Strange script... – Julien Jun 27 '13 at 03:38

1 Answers1

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The Jensen's formula implies $$\frac{1}{2\pi}\int_{0}^{2\pi}\log{|a-e^{i\theta}|}d\theta = \max(\log(|a|),0).$$

user64494
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