 Solution

For the preliminary definition, put $S_N(z)={\sum }_{k=1}^N{f}_k(z)$. The series converges uniformly to $f$ on $U$ when, for every $ϵ>0$, some $N_0$ satisfies $|S_N(z)-f(z)|<ϵ$ for all $z\in U$ and all $N\ge N_0$. Equivalently, ${\sup }_{z\in U}|S_N(z)-f(z)|\to 0$.

Let $L(\gamma )$ be the finite length of the piecewise-smooth path. Finite linearity of the contour integral and the integral bound giveThe uniform limit is continuous by the stated allowed result, so its integral is defined. Consequently