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| iamkennyk |
 Posted: Sun May 04, 2008 6:48 pm Post subject: Imaginary Number raised to an Imaginary Number?? |
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Forum Freshman
Joined: 23 Apr 2008
Posts: 5
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I got a question wrong on an Exam in Applied Electromagnetics (An engineering course)...
True or False-
j^(j) has infinitely many values, all of which are real numbers...?
answer is True!
Why? |
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| serpicojr |
| Posted: Sun May 04, 2008 8:57 pm Post subject: |
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Forum Ph.D.
Joined: 17 Jul 2007
Posts: 871
Location: JRZ
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So what does ii mean? (I'm a mathematician, so I can't help but use i, sorry.) It means ei ln i. Okay, so what does ln i mean? It's any complex number z such that ez = i, and we can calculate it as ln i = ln|i| + i arg i, where arg i is the angle that i makes with the positive real axis. ln|i| = ln 1 = 0, and arg i = π/2... or 5π/2 or 9π/2 or any number of the form (4n+1)π/2, where n is any integer. So ln i = i(4n+1)π/2 for any integer, and thus:
ii = ei*i(4n+1)π/2 = e-(4n+1)π/2
Neat. |
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| iamkennyk |
| Posted: Mon May 05, 2008 8:55 am Post subject: |
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Forum Freshman
Joined: 23 Apr 2008
Posts: 5
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Very Nice. Makes perfect sense. Problem sure did get me thinkin...
Thank you much. |
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