I am trying to compute $\oint\limits_C m(x,y) dx + n(x,y) dy$ where $F(x,y)=\langle m(x,y), n(x,y) \rangle = \langle \frac{\cos{(\ln{(xy)})}}{x}, \frac{\cos{(\ln{(xy)})}}{y} \rangle $ and $C$ is the ellipse parameterized by $(x(t),y(t))=(2\cos{(t)}+11, 3\sin{(t)}-15)$ for $0 \leq t \leq 2 \pi$.
I know that $F$ is an irrotational vector field since $\frac{\partial m}{\partial y} - \frac{\partial n}{\partial x}=0$ and since $C$ is a closed curve, by the fundamental theorem of line integrals, this integral should calculate to zero...that is if the curve enclosed no singularities of $F$. I know the singularity at $(0,0)$ doesn't matter for this computation as $C$ doesn't enclose it, but the logarithm will output complex values over the interior of $C$, which leads to hyperbolic cosine outputting complex values over the region.
When I compute this integral directly rather than applying the fundamental theorem of line integrals I do get zero. My question is why? Can we just ignore the fact that we get complex values over the region, is the term singularity only used to designate complete undefinition (like division by zero), rather than undefinition over the reals? My thought is that we get an "explosion" of the vector field near $(0,0)$ but not at the other points discussed. Is this valid logic?