Extracting coefficients from a sum of exponential functions

Question

I'm dealing with a sum of exponential functions with different exponents

Expr[x_, y_] = a1 Exp[(x - 2)^2 + y^2] + a2 Exp[(x - 0.3)^2 + (y + 0.6)^2] + a3 Exp[(x + 0.5) 2 + (y + 0.88)^2]

and I would like to extract the coeffecients to get a result like this

{a1,a2,a3}

I'm new to mathematica, so I'm not sure about which function fits best to deal with that kind of problem. It seems like CoeffiecientList is only useful for polynomial expressions. In addition I would like to expand the solution to a sum of a thousand exponential function. I'm not sure about how to use Coefficient or Cases in the right way, since the exponent of the exponential summands changes.

Best regards

Alex

Answer 1

First we want to make certain that Mathematica doesn't reorder your coefficients. To do that we turn your sum into a list

Expr[x_, y_]:=a1*Exp[(x-2)^2+y^2]+a2*Exp[(x-0.3)^2+(y+0.6)^2]+a3*Exp[(x+0.5)^2+(y+0.88)^2];
List@@Expr[x,y]

which gives you

{a1*E^((-2+x)^2+y^2),a2*E^((-0.3+x)^2+(0.6+y)^2),a3*E^((0.5+x)^2+(0.88+y)^2)}

Next we want to extract the coefficients

List@@Expr[x,y]/.p_*Exp[_]->p

which gives you

{a1, a2, a3}