I was able to prove the formula $10.32.9$ for $\nu \not \in \mathbb{Z}$ of this link https://dlmf.nist.gov/10.32. I want to prove that this formula is still valid when $\nu=n \in \mathbb{Z}$. From the theory of Bessel functions it's known that $K_n(z)=\lim_{\nu \to n} K_{\nu}(z)$. And so I wish to prove that
$$K_n(z)=\lim_{\nu \to n} \int_{0}^{\infty} e^{-z\cosh(t)}\cosh(\nu t)dt = \int_{0}^{\infty} e^{-z\cosh(t)}\cosh(n t)dt.$$
My idea is try to use the Lebesgue dominated convergence theorem to pass the limit inside the integral sign but i'm failing in show that the integrand is bounded by an integrable function as $\nu \to n$, so if possible, I need I little help in this step. Thank you for the help and every hint will be appreciated.
If is known another method to show that this representation is true for $\nu=n \in \mathbb{Z}$, it will be helpful too, I'm searching in some books about special functions, but none of them gives the detail, they only talk about the case where $\nu \not \in \mathbb{Z}$ and says that is obvious that the property is still true for all integers.
Update: I found one book that says that the formula remains true by a process called analytic continuation, my question now is, how can I know when a function can be analytically continued and how can I know for sure that this can be done for all integers in the above case?