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Example Proof using Complex Numbers

  1. Aug 21, 2011 #1
    1. The problem statement, all variables and given/known data
    http://www-thphys.physics.ox.ac.uk/people/JamesBinney/complex.pdf
    Example 1.2 (Page 6)

    2. Relevant equations
    De Moivre's Theorem, Euler's Formula, and other simple complex number theory formulas

    3. The attempt at a solution
    I'm having troubles understanding the format, which makes me thing the author is assuming prior knowledge in another area of math.

    What I don't understand is where he gets the mSYMBOL format from. I don't know what that symbol is, so I couldn't google it. I get all of the simplifying, except for when the conversion happens to and from the mSYMBOL. It looks like he's simply converting the sin(2n + 1) to the complex exponential function, but how can you do that without i?

    I know sin(n) = 1/(2i) * (e^(in)-e^(-in)), but that's not even close to the result they got.

    If that's the case, then my question is, how is this transformation happening?

    Again, I understand the simplifying of the series, just not the transformation to and from the complex exponential.

    Hopefully I explained that well enough. Any help would be appreciated.
     
    Last edited: Aug 21, 2011
  2. jcsd
  3. Aug 21, 2011 #2

    rock.freak667

    User Avatar
    Homework Helper

    That symbol just means the imaginary part.

    For example for Euler's formula e=cosθ+ isinθ so that Imaginary part of e, written as Im(e) = sinθ.

    So the imaginary part of ei(2n+1)θ, written as Im(ei(2n+1)θ)=sin(2n+1)θ
     
  4. Aug 21, 2011 #3
    Thanks for the help :).

    I now understand where they get the sin(x) + rsin(x) in the numerator, and where the 1 + r^2 comes from in the denominator. However, how do they get the 2rcos(2x) in the denominator?

    EDIT: Never mind, I figured it out.

    I forgot about cos(x) = 1/2(e^ix + e^-ix).

    Thanks again.
     
    Last edited: Aug 21, 2011
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