Example Proof using Complex Numbers

[Chantry]

Homework Statement

http://www-thphys.physics.ox.ac.uk/people/JamesBinney/complex.pdf
Example 1.2 (Page 6)

Homework Equations

De Moivre's Theorem, Euler's Formula, and other simple complex number theory formulas

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.

 

[rock.freak667]

That symbol just means the imaginary part.

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

So the imaginary part of e^i(2n+1)θ, written as Im(e^i(2n+1)θ)=sin(2n+1)θ

 

[Chantry]

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.