- I've been asked to evaluate $\int_C Im(z) dz$ for a variety of contours, which I've had no issue in doing.
- For the sake of clarity, these contours included the upper and lower halves of the circle with radius 2, centre zero,
- and the line segments from $-2$ to $-2 - i$ to $2 - i$ to $2$.
- After parametrizing and integrating, I acquired values of $-2 \pi i$ for the individual halves, and $-4$ for the line integrals.
My question comes in what we can conclude about the $Im(z)$ function. I'm assuming it's meant to be to do with either the analyticity or path dependence/independence of the function, but I'm really not sure.
Any guidance would be fantastic!!
Consider Cauchy's theorem: the integral of an analytic function about a closed loop is zero. The obvious corollary is that the integral of an analytic function between two points is path independent. The converse is that the integral of a non analytic function between two points may be path dependent.