Trigonometric Laurent Series and Complex Integration

[thelema418]

My task is to solve the integral $\frac{1}{\cos 2z}$ on the contour $z=|1|$ 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 $\pm \frac{\pi}{4}$. I realize that one of the problems with a Taylor series expansion of sec z is that it is only convergent for $\pm \frac{\pi}{2}$.

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 $u=2z$, $\frac{1}{\cos{u}}$. I changed the cosine to a Taylor Series, and then recognized the geometric series, so that:

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

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

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

 

[vela]

My task is to solve the integral $\frac{1}{\cos 2z}$ on the contour $z=|1|$ 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 $\pm \frac{\pi}{4}$. I realize that one of the problems with a Taylor series expansion of sec z is that it is only convergent for $\pm \frac{\pi}{2}$.

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 $u=2z$, $\frac{1}{\cos{u}}$. I changed the cosine to a Taylor Series, and then recognized the geometric series, so that:

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

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 $\cos 2z = \cos 2[(z\pm\pi/4)\mp\pi/4]$.

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

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