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May 25th, 2015, 05:49 PM   #1
Tau
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Help with Euler's identity; i = 0??

- e^(i2π) = 1
- e^(i2π) = e^0
- i2π = 0
- i = 0 (?)

Where is the mistake?
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May 25th, 2015, 06:38 PM   #2
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You are equating a principal value with a non-principal value.
$$\newcommand{\e}{\mathrm e}
\newcommand{\i}{\mathrm i}
\e^{2\pi\i} = 1 = \e^{0 + 2n\pi\i} \\
2\pi\i = 2n\pi\i \\
n = 1
$$
Also, the complex logarithm is multivalued and $\log 1 = 2n\pi\i$.
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May 25th, 2015, 07:01 PM   #3
Tau
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I get it. So is it correct to also distinguish them qualitatively? To say that with the complex exponent we have a rotation that settles to 1 while for n=0 we don't have a rotation to settle to 1?

Last edited by skipjack; May 25th, 2015 at 08:42 PM.
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May 25th, 2015, 07:11 PM   #4
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I think so, yes. It is rare that we are interested in any but the principal values anyway.
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May 25th, 2015, 08:43 PM   #5
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What does "a rotation that settles to 1" mean?
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May 26th, 2015, 03:34 AM   #6
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Quote:
Originally Posted by skipjack View Post
What does "a rotation that settles to 1" mean?
The visual interpretation of the identity on the complex plane.
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