
Complex Analysis Complex Analysis Math Forum 
 LinkBack  Thread Tools  Display Modes 
May 29th, 2017, 12:30 PM  #1 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,158 Thanks: 90  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: 1,574 Thanks: 380 
$e^{i \theta}$ 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...irrationalaro Last edited by Maschke; May 29th, 2017 at 01:00 PM. 
May 29th, 2017, 05:21 PM  #3 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,158 Thanks: 90 
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  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
irrational number multiplied by any real number  mick7  Number Theory  13  July 13th, 2015 10:08 PM 
Find a irrational number  shunya  Elementary Math  2  March 18th, 2014 12:56 PM 
Irrational Number  nfsmwbe  Number Theory  13  April 17th, 2012 07:17 AM 
What are the irrational number  MyNameIsVu  Number Theory  3  June 16th, 2009 08:13 PM 
Is pi irrational number?  habipermis  Algebra  5  December 28th, 2008 02:54 PM 