Complex integration over a general ellipse

Question

I'm having trouble evaluating the complex integral over an ellipse :

$\int_C{\dfrac{1}{z^{4} + 1}} dz$

where C is the ellipse given by $x^{2} - xy + y^{2} + x + y = 0$. How should I go about it?

Answer 1

The simplest way to work this out is to figure out which poles of the integrand lie within the given ellipse, and then compute the residues at those poles. The poles lie at

$$(x,y) = \left (\pm \frac1{\sqrt{2}},\pm\frac1{\sqrt{2}} \right )$$

A point $(x,y)$ lies within the ellipse when $x^2-x y+y^2+x+y<0$. Plug in each of these four poles and show that only the pole at $ \left (- \frac1{\sqrt{2}},-\frac1{\sqrt{2}} \right )$ is inside, i.e. $z=-e^{i \pi/4}$. Thus the integral is

$$\frac{i 2 \pi}{-4 e^{i 3 \pi/4}} = \frac{\pi}{2} e^{i 3 \pi/4}$$