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Trigonometric Laurent Series and Complex Integration

  1. Dec 23, 2012 #1
    My task is to solve the integral [itex]\frac{1}{\cos 2z}[/itex] on the contour [itex]z=|1|[/itex] using a Laurent series.

    The easy part of this is the geometric part. I drew a picture of the problem. I see that there are two singularity points that occur within the contour region at [itex]\pm \frac{\pi}{4}[/itex]. I realize that one of the problems with a Taylor series expansion of sec z is that it is only convergent for [itex]\pm \frac{\pi}{2}[/itex].

    I have tried several methods of determining the Laurent series, but I am having difficulty with this problem. I have tried two methods, and I am not certain what might be a better strategy -- or if I am going down the wrong road completely. In one method (a), I create a Taylor series, and then construct a geometric series. In the other (b), I use a trigonometric identity and derive partial fractions.

    In method (a) I considered [itex]u=2z[/itex], [itex]\frac{1}{\cos{u}}[/itex]. I changed the cosine to a Taylor Series, and then recognized the geometric series, so that:

    [itex]\sum_{j=0}^\infty \left( \sum_{k=0}^\infty \frac{z^{(2k+2)}}{(2k+1)!} \right)^j[/itex]

    In method (b) I got
    [itex] \frac{1}{2} \left( \frac{1}{1-\sqrt{2} \sin z} - \frac{1}{1+\sqrt{2} \sin z}\right)[/itex]

    Is one of these methods better to keep following -- or am I on the wrong track altogether? Also, sorry to be not as thoroughly detailed... I wrote a lot, but the system logged me out during one of my previews and I lost everything.
    Thanks
     
  2. jcsd
  3. Dec 23, 2012 #2

    vela

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    The problem here is that you're expanding about z=0. You want to find the Laurent series about each pole. I'd start by writing ##\cos 2z = \cos 2[(z\pm\pi/4)\mp\pi/4]##.

     
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