Consider some kernel function $f(n,s)$ with $\sum_n^\infty f(n,s)=f(s).$ Say an extension of $f$ is desired via analytic continuation. I'm curious about examples when that analytic continuationof $f(s),$ is expressible in the following form:
$$f(s)=\int_{n=a}^\infty f(n,s)~dn+ \sum_{n=b}^\infty f(n,s)$$
I can give an example:
$$ f(s)=\int_0^\infty \exp(-n^s)~dn + \sum_{n=1}^\infty \exp(-n^s) $$
$$ = \Gamma\big(1+\frac{1}{s}\big)+\sum_{k=0}^\infty \frac{(-1)^k}{k!}\zeta(-sk) $$
Does this example just coincidentally give the correct result?
I know that this is verified by the contour integration approach here.
Just wondering why both these attempts agree.
