I'm not very into integral to be honest. But I love theorems related to generalisation of Infinity series such as Abel-Plana formula, Poisson summation or analytic continuation of series derived by Ramanujan.
I would like to study such topic more systematicly but I have no idea how to find good sourse of information about that. Most of articels or textbooks, I was trying to find, are focused on some specific type of series like some very complicated series related to mixing logaritgms with harmonic or hyper-harmonic numbers etc.
For examples I will write some theorems I'm into, to show what I desire.
$\displaystyle \sum\limits_{ k \in \mathbb {Z}}f (k)=\sum\limits_{ k \in \mathbb {Z}} \hat {f} (k) \Leftarrow \frac {f (0)}{2}+ \sum_{k=1}^{\infty} f (k)=\sum_{n=-\infty}^{\infty} \mathcal{L} \{ f \} (2 \pi i n) $
$ \displaystyle \sum_{k=x}^{\infty}f (k)=-F(x)+\frac {1}{2}f (x)+\int_{0}^{\infty} \frac{f (x+it)-f (x-it)}{e^{2\pi t}-1} dt. $
$ \displaystyle \sum_{k=1}^{\infty}f (k)=\int_0^{\infty}\frac {\mathcal {L} \{f \}(t)}{e^t-1}dt$
$\displaystyle \sum_{k=x}^{\infty} f(k)=-2\pi \int_{-\infty}^{\infty}\frac {F (x-1/2+it)}{(e^{\pi t}+e^{-\pi t})^2}dt$
