Conditions for complex integrability

Question

I was wondering if the Lebesgue conditions for Riemann integrability also hold for a complex path integral, and what that would mean exactly. I am assuming that $f(z)$ would have to be bounded and continuous almost everywhere along a path $\gamma(t)$, but I am wondering what are the conditions on $\gamma(t)$ itself. I've found some references say that $\gamma(t)$ has to be "smooth" on an interval $[a,b]$ (which means $C^{\infty}?$), but another reference only requiring the path to be $C^{1}$, for example.

Answer 1

The path integral $$\int_\gamma f(z)\,dz$$ makes sense if $f$ is continuous and $\gamma$ is continuous with finite length, or bounded variation if you will. You can define it as the limit of sums $$\sum_{k=1}^n f\big(\gamma(t^*_k)\big)\big(\gamma(t_k)-\gamma(t_{k-1})\big)$$ with $t^*_k\in[t_{k-1},t_k]$, as the partition is refined.

An alternate take: You could think of it as a Stieltjes integral $$\int f(\gamma(t))\,d\gamma(t),$$ with the corresponding integration criteria borrowed from Stieltjes theory.