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November 11th, 2009, 09:14 PM   #1
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Complex exponent

How do you evaluate (without using a calculator)? I know it is equal to -1 but I have no idea as to how to arrive at that result.
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November 12th, 2009, 05:30 AM   #2
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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.
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November 12th, 2009, 01:51 PM   #3
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Re: Complex exponent

Quote:
Originally Posted by greg1313
How do you evaluate (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 for x.
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November 17th, 2009, 07:39 PM   #4
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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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