Mark
Suppose that G\subset \mathbb C is a region, that \gamma is a smooth curve from A to B in G, that f:G\to \mathbb C, and that F'(z)=f(z) for every z\in G. Show by direct computation that
\int_{\gamma} f = F(B)-F(A).
An archive the questions from Mark's Fall 2018 Complex Variables Course.
Antiderivatives
axk
Suppose that \gamma is parameterized by \gamma(t) such that a \leq t \leq b, \gamma(a) = A, and \gamma(b) = B. Then
\int_{\gamma} f = \int_{a}^{b} f(\gamma(t)) \, \gamma'(t) \, dt
\!\!\!\!\!\!\!\!\!\!= \int f(u)\, du
\!\!\!\!\!\!\!\!\!\!\!= [F(\gamma(t))]_{a}^{b}
\,\,\,\,\,\,\,\,\,\,\,\,\!= F(\gamma(b)) - F(\gamma(a))
\,\!= F(B) - F(A).