In complex analysis, there is a formula involving the residues of complex functions that one can employ to find the value of certain infinite series.
In general, sums can be evaluated by means of this theorem according to the following result: $$\lim_{N \to +\infty} \sum_{k=-N}^{N} f(k) = - \ \{ \text{sum of the residues of } \pi f(z) \cot(\pi z) \text{ at the poles of }f(z) \} . \qquad \qquad (*)$$
The function $f: \mathbb{C} \to \mathbb{C}$ must satisfy the condition that $$|f(z)| < \frac{M}{z^{k}} $$ - where $k>1$ and $M$ are constants independent of $N$ - along the path $C_{N}$. This path goes counterclockwise along the rectangle $(N + (1/2))(1-i)$, $(N+(1/2))(1+i)$, $(N+(1/2))(-1-i)$ and $(N+(1/2))(-1-i)$ and it encloses the poles of $f$. See p. 3-5 of this document for more details.
Now, according to this paper by Lüdkovsky and Oystaeyen (see also the following MSE question), one can obtain residues of functions in quaternionic analysis as well. I am curious as to whether it may be possible to formulate an analogue of $(*)$ for a function $g$ of a quaternionic variable.
Questions:
- Does an analogue of $(*)$ exist for functions $g$ of a quaternionic variable? So functions like $g: \mathbb{H} \to \mathbb{H}$ ?
- If so, what conditions must $g$ satisfy?
- Can this quaternionic analogue of the residue formula for series be applied to evaluate infinite series that cannot be evaluated by means of the residue sum formula from complex analysis?