integral using residual theorem

Question

I have the following problem: $$\int_C\cot z\ \mathrm{d}z,\ \ \ C(t)=2e^{it},\ t\in[0,2\pi]$$ my solution is: $$\mathrm{res_0}\cot z=\frac{\cos(0)}{\cos(0)}=1$$ The circle only surrounds one singularity (0), so the result should be: $$I=2\pi i\frac{1}{2\pi i}\int_C\frac{\mathrm{d}z}{z}=2\pi i$$ But according to the (probably) correct restults in book it should be $4\pi i$. It seems so straightforward, that I don't see where could I've made a mistake.

Answer 1

that I don't see where could I've made a mistake.

You didn't. The result is indeed $2\pi i$, since the contour surrounds only one singularity, and is traversed only once.

If it had been $C(t) = e^{2it}$, then the result would be $4\pi i$.