Continuum sums -- a big problem and some interesting observations

[Ansus]

a sort of sum-analogue of the error function. With the Fourier-based sum, it does not settle to a limit as \( x \rightarrow \infty \), but instead goes to a small 1-cyclic wobble about the limit of the discrete sum.

I believe in such cases the periodic component can be easily excluded (i.e. by requireing monotonous derivatives of higher orders). This is related to positive real axis. So to obtain the "natural solution" to the continous sum, just purify it of the periodic component on the positive real axis. Of course in any case the periodic component on the complex plane still will remain.

You can also use checks by other methods such as Mueller's formula (this formula is well-applicable to the function you presented as an example).

If
\( \lim_{x\to{+\infty}}f(x)=0 \)

then

\( \sum _x f(x)=\sum_{n=0}^\infty\left(f(n)-f(n+x)\right)+ C \)