Properties of sum of two exponential functions

Question

I have some data that can be fit reasonably well with an exponential function. However, a colleague mentioned that it would be better to use the sum of two exponentials:

$$ f(x; a, b_1, b_2, \lambda_1, \lambda_2) = a + b_1 e^{\lambda_1 x} + b_1 e^{\lambda_2 x} $$

There was no mathematical justification given, and I'm curious: assuming that this is used to model what is roughly exponential (so $\lambda_1$ and $\lambda_2$ have the same sign, for example), what properties does this kind of function have? Does this function have a name? I assume there must be a fully general case, which is an infinite sum of exponentials? I'm having trouble searching online for this, because Google thinks I'm referring to the sum of exponential random variables.

Answer 1

Such functions are the solutions to second order differential equations of the form $y''+ay'+by=0$ -- though for many $a$ and $b$, the values of $\lambda$ will be complex:$$ \lambda=\frac{-a \pm \sqrt{a^2-4b}}2 $$

Compared to a single exponential, having this mix lets you do a couple of things. First, it lets you have a mix of time scales. Like, if you have heat transfer between two objects and the rest of the world -- you might have a short time scale on which heat moves between the two objects, and a long time scale on which heat moves between them and their surroundings.

It lets you fit systems that can't start changing immediately, but have some inertia -- though for many such systems, you end up with complex $\lambda$.

I'm not sure about the name for this family of functions. But the underlying dynamics would be called a second order system. Specifically, for $\lambda$ being negative real, an "overdamped" second order system.

Answer 2

Functions of the type $$x\mapsto f(x) =\sum_{i=1}^nP_i(x)e^{\lambda_i x}$$ where the $P_i$ are polynomials are called exponential polynomials. When all coefficients are real and if $\deg P_i=d_i$ this is a nice exercise using Rolle theorem to prove by induction on $N=\sum_{i=1}^n(d_i+1)$ that $f$ has at most $N-1$ real roots, with a suitable interpretation in case of multiple roots.