I had a thought today and I've tried to see if it is a thing. I'm certain it is a thing, I just don't know how to search for it.
We have the Taylor series which is a summation of monomials:
$f(x)=\sum_{n=0}^\infty a_nx^n$
And the Fourier series that is a sum of sines and cosines:
$f(x)=\sum_{n=0}^\infty A_n\sin{nx}+\sum_{n=0}^\infty B_n\cos{nx}$
Is there an equivalent "exponential" series that sums to a function? Something like the following, but it could be of any form where an exponential is being summed.
$?f(x)=\sum_{n=0}^\infty A_{n}e^{a_{n}x}+\sum_{n=0}^\infty B_{n}e^{-b_{n}x}?$
I'm assuming here that the $a_n$s and $b_n$s (or which ever coefficients arise in the formulation) are not complex so this won't reduce to the Fourier series or exponential sum.
I suspect that hyperbolic trig functions might come into it.
(Internet searching has been very little help - searching for "exponential", "series", and "sum" produces predicatable results)
General Dirichlet Series:
$$f(s)=\sum_{n=1}^\infty A_n e^{-\lambda_n s}$$
http://en.wikipedia.org/wiki/General_Dirichlet_series
EDIT: An example of Dirichlet series with real coefficients $A_n$, is given by the celebrated Riemann Zeta function:
$$\zeta(s)=\sum_{n=1} n^{-s}=\sum_{n=1}^\infty A_n e^{-\lambda_n s}$$
where $A_n=1$ and $\lambda_n=\log(n)$