Uniform convergence and contour integration
= Uniform convergence and contour integration
If continuous functions $f_n$ converge uniformly to $f$ on the image of a rectifiable curve $\gamma$, then
$$
\int_\gamma f_n(z)\,dz\longrightarrow\int_\gamma f(z)\,dz.
$$
Indeed, the absolute difference is at most the length of $\gamma$ times $\sup_\gamma|f_n-f|$.