I have a problem with calculating the winding number $n\left ( \gamma ,\frac{1}{3} \right )$ of the curve $\gamma :\left [ 0,2\pi \right ]\rightarrow \mathbb{C}, t \mapsto \sin(2t)+i\sin(3t)$.

According to the formula $\frac{1}{2\pi i}\int_{0}^{2\pi }\frac{\gamma{}' {(t)}}{\gamma (t)-\frac{1}{3}}dt$ i get a strange integral to calculate: $\frac{1}{2\pi i}\int_{0}^{2\pi }\frac{2\cos(2t)+3i\cos(3t)}{\sin(2t)+i\sin(3t)-\frac{1}{3}}dt$...

I don't know how to move on...there should be another way how to do it with plotting the curve around $\frac{1}{3}$, but i don't know how to do it and can't find software for this. Can anybody help me, please? Thank you in advance!


Here is a picture of the curve:

Parametric Plot

Notice that as you trace around the curve, it doesn't wind around the point $z=\frac{1}{3}$, so the winding number is $0$.

  • $\begingroup$ Thank you very much, Jared! $\endgroup$ – Lullaby Apr 26 '13 at 8:45
  • 1
    $\begingroup$ but what If I can not draw the curve?there must be some other analytic approach to solve this problem. $\endgroup$ – Marso May 2 '13 at 7:00

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