How do I properly write this integration in Matlab?

Question

I'm trying to evaluate the following integral in matlab: http://i.imgur.com/Iuc4VT5.png

Here is my code:

alpha = 2;
F1 = @(u,v) 2*u.*v.*exp(-u.^2)./(1+2*z.*u.*v);  
F2 = @(v) v;
F3 = @(z) exp(-z)./sqrt(z);
I1 = dblquad(F1,0,1e5,2,1e5);
I2 = quad(F2,2,1e5);
quad(F3*exp(-(I2-I1)),0,1e5);

I'm getting the errors shown below. These errors don't show much, but I'm guessing it's because of the way I wrote F1 . I defined F1 as a function of u and v for the double integral, but there is also a variable z which is the variable of the outer integral. I did that because there is no way I can separate z from the inner integrals. Is there any better way to write this integration?

Error in ==> @(u,v)2*u.*v.*exp(-u.^2)./(1+2*z.*u.*v)


Error in ==> dblquad>innerintegral at 73
fcl = intfcn(xmin, y(1), varargin{:}); %evaluate only to get the class below

Error in ==> quad at 76
y = f(x, varargin{:});

Error in ==> dblquad at 53
Q = quadf(@innerintegral, ymin, ymax, tol, trace, intfcn, ...

I'm choosing 1e5 to represent infinity.

Answer 1

After having successfully answered a follow-up question by the same poster, I realize a relevant part of this answer is wrong. I'd delete the answer, but I can't, since it is accepted. Therefore this disclaimer...

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The simple answer is: Your definition of F1 contains a reference to z , but it is not specified as an argument of that function.

However, it won't help to specify z as an additional argument, because then I1 is no longer a constant, but itself a function of z .

I'm not an expert on numerical integration, but as far as I can see this means that you cannot numerically integrate your expression, at least not using a combination of quad and dblquad . The argument to the outer exponential function is just not a constant, and a numerical integration cannot return a function.

It might be possible to rearrange the integral to bring it into a form that can be numerically integrated, but I can't tell you how.

Another problem is that "representing" infinity by 10^5 is not necessarily a useful approximation – it all depends on the behavior of the function that is being integrated. A possible trick might be to do a variable substitution such that each variable that goes to infinity is written as a function of another variable with a limited range.

My recommendation: Try to get as far as possible evaluating this integral analytically , and only use numerics when you are sure there is no analytical approach. And try to get help on that on math.stackexchange.com, because it is not a programming problem.