I'm trying to verify that $F_{m}$ it is more continuous, but I'm not getting it. I've tried using Newton's Binomial, but it did not..
Let $\lambda$ be a rectifiable curve and suppose $\phi$ is a function defined and continuos on {$\lambda$}. For each $m\geq 1$ let $F_{m}(z)=\int_\lambda \! \phi(w)(w-z)^{-m} \, \mathrm{d}w$. Then each $F_{m}$ is analytic on $\mathbb{C}$ $-$ ${\lambda}$ and $F_{m}′(z)=mF_{m+1}(z)$. (Conway, Lemma 5.1, pg. 83 (second edition))