# Finding multiple peaks of gaussians?

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.