If I have a Gaussian function $f(x)=c e^{-(x-a)^2}$.

I can calculate the $x$ value where it peaks using:

$$a = \frac{\int_{-\infty}^{\infty} x f(x) dx}{\int_{-\infty}^{\infty} f(x) dx} $$

But suppose I have a two peak Gaussian function $f(x)=c_1 e^{-(x-a_1)^2} + c_2 e^{-(x-a_2)^2}$

Is there a similar integral expression that will let me find the two peaks $a_1$ and $a_2$ ? The above formula would only give me some weighted average of the two peaks.

My initial thought is that there would be a 2x2 matrix given by some integrals of $f$ and the values $a_1$ and $a_2$ would be eigenvalues. i.e. if there were integrals that could give $a_1+a_2$ and $a_1 a_2$ then from these we could work out the individual values.


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