Vrftjkry

up vote 1 down vote favorite 1 First off: I know that "almost everywhere" has a legitimate meaning in measure theory, but I'm not referring to that in this case, really. I'm talking about the case where there are only a small (finite) number of points where the function is not analytic inside a piecewise-smooth simple closed curve. I am wondering why we can't use Cauchy's Integral Formula or Cauchy's theorem in this case. Some examples of what I'm referring to: a) $int_gamma f(z) dz = int_gamma frac{e^{z^2}}{z(z-2)}dz$ where $gamma(t) = 3+2e^{it}$ , $0 leq t leq 2pi$ b) $int_gamma g(z) dz=int_gamma frac{z^2 - 1}{z^2 + 1}$ where $gamma(t) = 1-i+2e^{it}$ , $0 leq t leq 2pi$ In the first instance, $f$ is analytic on $mathbb{C} setminus {2}$ but unfortunately, the curve $gamma$ , which is ...