A lot of times in math one considers not just one object but a family of objects, or a space of objects.
In set theory one often deals with collections of numbers that form a set. In functional analysis one can consider function spaces. In group theory one often cares about understanding the discrete collection of symmetries of an object.
I'm wondering if that's done at all with regards to analytic continuation. That is, does one ever consider a family of real functions and continue each and every one of them to the complex plane?
Is there anything one can learn by doing this as opposed to analytically continuing just one candidate function.
To be more specific, if you have $f(x)=1/x$ of a real variable you could analytically continue this to a function of a complex variable $f(z)=1/z$. But what if you have an infinite collection, like $f_k(x)=k/x$ for $k\in \Bbb R.$ This can be done by partitioning the real plane into a disjoint union. Then all the analytic continuations would differ by a constant multiple.
I am highly skeptical that there is any point in continuing each and every curve in the space because I think any one curve will capture the essence, the information that is needed. But I want to hear from more knowledgeable people.
In practice does one analytically continue just one function in the family, or analytically continue the space of all the curves in the family?
Dirichlet L functions are an incredibly important tool in Analytic Number theory and they were first used by Dirichlet in his proof of the Prime Number Theorem in Arithmetic progressions. A key step of his proof is analytically continuing the function
$$\prod_{\chi\mod k}L(\chi,s)$$
to $\Re(s)\geq1$, where the product is taken over all Dirichlet characters $\chi$ with period $k$. I will note that his proof relies on showing that this function must have a pole at $\chi=1$ and thus can't be analytically be continued and instead must be meromorphically continued, but multiplying by $(s-1)$ yields a nice analytic continuation of a family of functions (any choice of $k$).