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 November 11th, 2009, 09:14 PM #1 Global Moderator     Joined: Oct 2008 From: London, Ontario, Canada - The Forest City Posts: 7,963 Thanks: 1148 Math Focus: Elementary mathematics and beyond Complex exponent How do you evaluate $e^{\pi i}$ (without using a calculator)? I know it is equal to -1 but I have no idea as to how to arrive at that result.
 November 12th, 2009, 05:30 AM #2 Global Moderator     Joined: Nov 2006 From: UTC -5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms Re: Complex exponent exp(z) = 1 + z/1 + z^2/2 + z^3/6 + z^4/24 + ... + z^k/k! + .... You can use the first half-dozen or so terms to convince yourself that it's close, or split it into parts and use limits to show the equality. Alternatively, break it into magnitude and angle components: r * exp(i * theta). The angle tells you what direction on the complex plane (0 is to the right, along the positive real axis; move counter-clockwise; 2 pi is a full rotation) and the magnitude says how far.
November 12th, 2009, 01:51 PM   #3
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Re: Complex exponent

Quote:
 Originally Posted by greg1313 How do you evaluate $e^{\pi i}$ (without using a calculator)? I know it is equal to -1 but I have no idea as to how to arrive at that result.
You could use the Euler identity: exp(ix)=cos(x) + isin(x), and plug in ${\pi}$ for x.

 November 17th, 2009, 07:39 PM #4 Global Moderator     Joined: Oct 2008 From: London, Ontario, Canada - The Forest City Posts: 7,963 Thanks: 1148 Math Focus: Elementary mathematics and beyond Re: Complex exponent Thanks, guys. It always amazes me how far those mathematicians got with the technology they had. Says a lot for studying, I guess. I don't know where I'd be without my calculator/computer/internet and so on.

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