Approximation of the cosine by gaussian functions Choosing the parameters $\alpha$, $\omega$, and $\phi$ carefully, the function
$$f(x) = \alpha \cos(\omega x + \phi)$$
can be almost perfectly approximated (at least visually) by the sum of two gaussian functions
$$g(x) = e^{-x^2} -e^{-(x-1)^2}$$
in the interval $x \in [-0.24,1.24]$:

Roughly guessed values are
$\alpha \approx 0.7304$
$\omega  \approx 2/3\ \pi$
$\phi  \approx 0.5$
I wonder which closed formulas - only containing natural numbers and standard constants like $\pi, e$, or the golden ratio $\varphi$ - might yield the optimal parameters. Maybe the formulas for $\omega$ and $\phi$ in these approximations are already exact?
[Note, that $\alpha \approx 0.7304$ is nothing but the maximal value of $g(x)$. So it can be obtained by finding $x_0$ with $g'(x_0)=0$ and $g(x_0)>0$ – which is approx. $-0.24$ – and calculating $g(x_0)$.] 
 A: The answer will depend on on your definition of approximation. In general setting, you take a (semi)norm $\|\cdot\|$ in the space of functions over your interval $[a,b]=[-0.24,1.24]$ and then minimize the norm of the difference 
$$\left\|e^{-x^2} -e^{-(x-1)^2} - \alpha\cos(\omega x+\phi)\right\|$$
with respect to parameters $\alpha$, $\omega$, and $\phi$.
Depending on your choice of the norm, this minimisation problem will be easier/harder to solve. I'd say that the easiest one is the $L^2$ norm, followed by $L^\infty$ norm and $L^1$ norm.
For obvious reasons you will never obtain an exact approximation - trigonometric functions are linearly independent with gaussian functions.
On a side note, you can even relax the problem to use a distance instead of a norm, but it will only add complexity to the problem.
A: There is probably no answer.
Because the "optimal" constants are solutions of transcendental equations, and probably have no closed-form expression in terms of the "standard" constants.
Then you can get closer and closer to the desired values with silly formulas such as
$$\frac{7304\pi}{10000\pi}$$ giving arbitrary precision approximations without ever reaching the exact value.
