Exponential equivalent of Taylor Series Let $f$ be a function which extends to an entire function. Then we know that it can be expressed as a series of monoids.
$$f(x)=T_f(x):=\sum_{j=0}^\infty c_jx^j$$
Is there a well known class of functions which may be expressed as a sum of decaying exponentials?
$$f(x)=\mathcal{E}_f(x):=\sum_{j=0}^\infty c_j\exp(-\nu_j x),\quad \nu_j,c_j\in \mathbb{R}_+$$
A guess would be Analytic decaying functions?
 A: Great question! If we substitute $x = \ln t$ into your expression
$$\begin{align}\mathcal{E}_f(\ln t)&=\sum_{j=0}^\infty c_j\exp(-\nu_j \ln t)\\
&= \sum_{j=0}^\infty c_jt^{\nu_j}\end{align}$$
So this is just a different form of Taylor series, where the exponents aren't required to be integers.  Unfortunately, I don't think this is particularly useful - Taylor series are often nice because they form a polynomial approximation to a function, and polynomials are nice.  But, $t^{\nu_j}$ might not behave nicely at all, it might not even be differentiable at the origin.  
Going in a different direction however, what if we instead have an uncountable number of $\nu_j$, by which I mean turn the sum into an integral.  In a Taylor series, we try to rewrite some function in terms of a countable number of values, the $c_j$ in your expression.  Instead, we could see what happens if we use a function instead, say $c(t)$. 
$$C(x) = \int_0^\infty x^t c(t) dt$$
This function might not give $c(t)$ itself on the other side, but it might be worth investigating.  But again we have an unpleasant monomial, let's make it into an exponential using $x = e^{-s}$.
$$\mathcal{L}_c(s) = \int_0^\infty c(t) e^{-st} dt$$
Why the $e^{-s}$ instead of $e^{s}$? Well, $e^{-st}$ decays to zero as $t \to \infty$ for any real $s>0$, so the integral we have is more likely to converge, compared to $e^{st}$. 
We've now got a new function $L(s)$ in terms of $s$ and $c(t)$, but it is any use? Yes! It turns out this is actually the Laplace transform - and it's heavily used in electrical engineering as it can really help solving differential equations (exercise - find $\mathcal{L}_{c'}(s)$ in terms of $\mathcal{L}_c(s)$).  The Laplace transform gives an idea of what the exponential parts of $c(t)$ are like, in the same way as the Taylor series gives you an idea of what the polynomial bits of $f(x)$ are like.
