Let $f(z)=\frac{1}{[(z-\frac{1}{2}-i)(z-1-\frac{3i}{2})(z-1-\frac{i}{2})(z-\frac{3}{2}-i)]}$ and let $\gamma$ be the polygon $[0,2,2+2i,2i,0]$. Find $\int_{\gamma}^{} f$ .
I'm trying to use the partial fractions decomposition method, but it's getting too long and I'm lost in the accounts. I do not know if the author expects me to do so. Can anybody help me? Conway, pg. 96, prob., 7.
On
Your $\gamma $ encloses the poles of $f.$ Suppose we now consider the circle $|z|=R$ for any $R>\sqrt 8.$ Such a circle encloses $\gamma,$ hence $f$ is analytic on and between these two contours. Thus by Cauchy's theorem,
$$\int_\gamma f(z)\, dz =\int_{|z|=R} f(z)\, dz.$$
Now think of estimating the last integral as $R\to \infty.$ We will have $|f(z)|$ on the order of $1/R^4,$ while the arc length of the contour is $2\pi R.$ It follows that the last integral $\to 0$ as $R\to \infty.$ Since it is constant for large $R,$ it must equal $0$ for such $R.$ Hence $\int_\gamma f(z)\, dz = 0.$
By the residue theorem: $\int_{\gamma}f(z)dz=2\pi i\sum_i\textrm{res}_{z_i}$. So the problem essentially is to evaluate the residue of each pole. The poles are at: $1/2+i$, $1+3i/2$, $1+i/2$, $3/2+i$. Simply check which are in your boundary and evaluate the residues.