I would like to ask probably a small question.
I am reading Brown-Churchill complex variable book. There it is defined that contour is a piecewise smooth arc, where smooth arc is a differentiable arc having nonzero derivative of the arc parametrization. To be precise, if $z(t)$ is the parametrization of the arc, then having $z'(t)\neq 0$ anywhere in the interval of $t$ means smooth.
My question would be: Can we still define contour integration where the contour needs not to be piecewise smooth (according to the above definition?)? Like perhaps, using piecewise differentiable arc only? Where will anything fall off if the smoothness above is removed?
Thank you so much!
Smoothness is not at all required for the definition. If $\gamma: [a,b] \to \mathbb C$ is piece-wise differentiable then we define $\int_{\gamma} f(z)dz$ as $\int_a^{b}f(\gamma (t)) \gamma'(t)dt$ for any continuous function $f$ and this integral does exist.