# Problem with calculating a winding number

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!

## 1 Answer

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

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