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 Complex Analysis Complex Analysis Math Forum

 May 29th, 2017, 12:30 PM #1 Banned Camp   Joined: Mar 2015 From: New Jersey Posts: 1,720 Thanks: 126 irrational power of complex number What is $\displaystyle z^{a}$ when a is irrational? Formally, $\displaystyle z^{a}=r^{a}e^{i(a\theta \pm an2\pi) }$, n=0,1,2,.... but an is never an integer, so it looks like an infinite number of roots: points on the circle of radius $\displaystyle r^{a}$. May 29th, 2017, 12:52 PM #2 Senior Member   Joined: Aug 2012 Posts: 2,393 Thanks: 749 $e^{i \theta}$ is a point on the unit circle. If $\theta$ is a rational multiple of $2 \pi$, the points $e^{n i \theta}$ are a finite set. That is, they repeat after a while. You can see this because $(e^{\frac{n}{m} 2 \pi i})^m = 1$. But if $\theta$ is an irrational multiple of $2 \pi$, the $n$-th powers never repeat. Amazingly, they are dense on the circle. They get arbitrarily close to every point on the circle. https://en.wikipedia.org/wiki/Irrational_rotation https://math.stackexchange.com/quest...irrational-aro Thanks from zylo Last edited by Maschke; May 29th, 2017 at 01:00 PM. May 29th, 2017, 05:21 PM #3 Banned Camp   Joined: Mar 2015 From: New Jersey Posts: 1,720 Thanks: 126 If $\displaystyle \alpha$ is angle of arbitary point on cricle, can I find n and m so that I can come arbitrarily close to $\displaystyle (\alpha + m2\pi)$, ie, can i find m and n st $\displaystyle |(a\theta+na2\pi)-(\alpha + m2\pi)|<\epsilon$ Tags complex, irrational, number, power Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post mick7 Number Theory 13 July 13th, 2015 10:08 PM shunya Elementary Math 2 March 18th, 2014 12:56 PM nfsmwbe Number Theory 13 April 17th, 2012 07:17 AM MyNameIsVu Number Theory 3 June 16th, 2009 08:13 PM habipermis Algebra 5 December 28th, 2008 02:54 PM

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