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Math Help - Branch cut when exponentiating by an imaginary number?

  1. #1
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    Branch cut when exponentiating by an imaginary number?

    Hello, I'm new to the forums, so hopefully this is the correct place for me to post.

    What is the branch cut when the exponent is an imaginary number?

    IE, am I incorrectly crossing a branch cut when performing the following set of operations:



    Is the problem somewhere else?
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  2. #2
    Junior Member
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    Jan 2009
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    The problem is obviously in the last step.

    When you rasing some number by a complex you should use the following definition:
    z^i = exp( i Log z)

    So in your case we have z=1=exp (2\pi i n), raising to the power i we get:
    exp( i Log(exp(2 \pi i n)))

    Now remember the definitino of Log in complex plane:
    Log(z)= log |z| + i arg(z), and this function is multivalued so you need to pick a branch cut for the argument.
    In your case obviosuly the equality is wrong cause one side of the equality you pick n=0 and on the other you picked another different number, you can't do this.

    Edit: Does some know why the tex doesn't get compiled?
    Last edited by InvisibleMan; April 18th 2011 at 07:24 AM.
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  3. #3
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    Quote Originally Posted by InvisibleMan View Post
    The problem is obviously in the last step.

    When you rasing some number by a complex you should use the following definition:
    z^i = exp( i Log z)

    So in your case we have z=1=exp (2\pi i n), raising to the power i we get:
    exp( i Log(exp(2 \pi i n)))

    Now remember the definitino of Log in complex plane:
    Log(z)= log |z| + i arg(z), and this function is multivalued so you need to pick a branch cut for the argument.
    In your case obviosuly the equality is wrong cause one side of the equality you pick n=0 and on the other you picked another different number, you can't do this.

    Edit: Does some know why the tex doesn't get compiled?
    Thanks, yeah, you're completely correct about the log. I realized it had to be something like that but couldn't put 2 and 2 together!

    Tex is broken so I suggest you use http://www.quicklatex.com/ to make images that you can imbed. Your (slightly modified tex) can be rendered as:



    Which is generated by
    Code:
    \begin{align*}
    z^i = \exp( i \log (z)) \\
    z=1=\exp (2\pi i n) \\
    \exp( i \log(\exp(2 \pi i n))) \\
    \log(z)= \log |z| + i \arg(z)
    \end{align*}
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