# Complex Analysis - Contour Integral

Evaluate $\int \frac {1} {z^2+1} dz$ along the contour $\Gamma$. (Gamma is some closed circle centered around i, no specified radius, and is oriented counter clockwise.)

So far, I've used factored the expression into $\int \frac 1 {(z+i)(z-i)}$, then used partial fraction decomposition, and got $\frac 1 {2i}[ \frac{1} {z+i}-\frac{1} {z-i}]$.

Now I can take the integral of both of these separate terms from 0 to 2pi, but I'm having a lot of trouble with parameterization of this curve before I integrate.

I assumed since it's a closed contour centered at i, I would parameterize it by $z(t)=i+e^{it}$? However, when I substitute this into the integral and multiply by it's derivative $z'(t) = ie^{it}$, I don't get the right answer. The book says I should be getting zero for the first integral, but I don't even know how to do this integral.

$$\int_{\Gamma}\frac{1}{z+i}dz =\int_0^{2\pi}(\frac{1}{(i+e^{it})+i})(ie^{it})dt=\int_0^{2\pi}\frac {ie^{it}}{e^{it}+2i}dt$$

Now I have absolutely no clue how to do this integral, but supposedly it is supposed to come out to 0, and the second integral of $$\int_0^{2\pi}\frac{1}{z-i}$$ should be equal to $2\pi i$. They then total together using the 3rd equation from the top, and 2i cancels, and the total integral becomes $\pi$. That should be the total answer.