On the clip , https://youtu.be/E7NNc-AM7vQ?t=385 (at the current time), the speaker checkes that $P_n(z)$ = $\int_{n}^{n+1} t^{z-1}e^{-t}dt$ is analytic function. To do so, he referred
"Use Thm5 (on the screen) : $\int H(\omega, z) d\omega $" (In case of $P_n(z)$, using variable t instead of $\omega$ )
The question is what is Thm5?... (Although I have tried to google, I cannot find the theorem which is analgolous to the Thm5 on the screen....)
One way to prove analyticity is to use Morera's Theorem.
Let $g(z)=\int_a^{b} H(t,z)dt$ where $H$ is continuous in the first variable and analytic in the second variable. We can show that $H$ is jointly measurable. Also $g$ is continuous by DCT. Consider any open ball $B(z,r)$ in the domain and let $\gamma$ be a closed path in it. Then $\int_{\gamma} g(z)dz=\int_a^{b} \int_{\gamma} H(t,z)dzdt$ by Fubini's Theorem and $\int_{\gamma} H(t,z)dz=0$ for each $t$. Hence $\int_{\gamma} g(z)dz=0$ for every closed path $\gamma$ which implies that $g$ is analytic in $B(z,r)$.