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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.

jds
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  • Assuming $\lambda_1 < \lambda_2$, the function will be dominated by $b_1e^{\lambda_2x}$, making $a + b_1 e^{\lambda_1x}$ only significantly affecting the match for small $x$ data points. – DanielV Mar 29 '24 at 13:48
  • I also do not see how this is better than just one exponent. Let's say $\lambda_1 > \lambda_2$. This can be written as $a+b_1e^{\lambda_1x}(1+e^{-\lambda_3x})$ which tends to $a+b_1e^{\lambda_1x}$ as $x$ increases. – Vasili Mar 29 '24 at 13:59
  • When the exponents are negative, this sort of function is good for matching combinations of short and long term dynamics. For example, consider two stage radioactive decay, with two different half lives. The amount of final product would have this form. – Alex K Mar 29 '24 at 14:04

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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.

Alex K
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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.

Letac Gérard
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