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\end{align*}  Now first look at the integral over $\gamma_1$.  \begin{align*}  \int_{\gamma_1}\frac{dz}{z^{4}+1} &= \int_{\gamma_1}\frac{dz}{(z-(\frac{\sqrt2}{2}+i\frac{\sqrt2}{2})} \int_{\gamma_1}\frac{dz}{(z-(\frac{\sqrt2}{2}+i\frac{\sqrt2}{2}))}  \end{align*}