Is there a name for this series?

Question

I am doing a presentation on nonlinear optics and I ran into a paper that uses a series to describe a wave equation of a system. The paper will be linked below and the series is:

$\sum_{n=1}^{\infty}\frac{(-i)^{n-1}}{n!}k^n$

I mostly am looking for what i should call this other than "series." The equations are presented on pg. 1-2, eqns. 1a, 1b, and 1c (all are the same thing for 3 different waves).

http://www.few.vu.nl/~switte/papers/OPCPA_review_published_early_edition.pdf

EDIT: The full equation is

$\frac{\partial A}{\partial z}+\sum_{n=1}^{\infty}\frac{(-i)^{n-1}}{n!}k^n \frac{\partial^{n} A}{\partial t^{n}}=-i \frac{\chi^{(2)}\omega}{2nc}AA^{*}e^{-i\Delta \textbf{k}\cdot \textbf{z}}$

so it may just be simpler to discuss it in terms of what each portion does rather than by mathematical terminology..

Answer 1

$$\sum\limits_{n=1}^\infty \frac{(-i)^{n-1}}{n!}k^n=\sum\limits_{n=1}^\infty \frac{(-i)(-i)^{n-1}}{(-i)n!}k^n=\frac{1}{-i}\sum\limits_{n=1}^\infty\frac{(-i)^n}{n!}k^n=i(e^{-ik}-1).$$ So I would say yes it is known and there is a name for it -- exponential function.

Answer 2

$$\sum_{n=1}^\infty\frac{(-ik)^n}{(-i)\,n!}=i(e^{-ik}-1)=i(\cos k-1-i\sin k).$$

---

This can be rewritten as

$$2i\cos\frac k2\left(\cos\frac k2-i\sin\frac k2\right)=2i\cos\frac k2e^{-ik/2}$$ but there is little benefit.

Answer 3

Think the name of the function is "Exponential generating function (EGF)" see Wikipedia.

$\sum\limits_{n=1}^\infty ik_n \frac{(-i)^n}{n!}=EGF(ik_n,-i)-ik(0)$