compute $\displaystyle{\int_{\delta K_{0}} \frac{1}{z^{2}+2aiz-1}}$ where $\delta K_{0}$ is a circle of radius one around 0. And $a>1$.
the linear factors give: $\displaystyle{\frac{1}{z^{2}+aiz-1} = \frac{1}{(z-(-ai+\sqrt{a^{2}-1})(z-(-ai-\sqrt{a^{2}-1})}}$
The two residues calculated from this are: $\displaystyle{\frac{1}{2\sqrt{a^{2}-1}}} $ and $\displaystyle{\frac{1}{-2\sqrt{a^{2}-1}}}$
So the sum of them is 0. But that is not true! Since it is a circle of the form : $z=e^{i\phi}$ with $\phi\in [0,2\pi]$ Either both linear factors must lie within or none?
Does anybody see where I went wrong? Please do tell.