I'm trying to compute/estimate a rather complicated looking integral. Here
is the code I'm trying to run, with the necessary constants defined:
var('t1,t2,u,w,k')
T = 1
m = 100
E = 1
v = 0
y=1
O = 1
integral(integral(integral(
integral(integral(
e^(-t1^2/T^2)*e^(-t2^2/T^2)*e^(I*O*t1)*
e^(-I*O*t2)*e^(-I*E*y^2*(1 - v)*t1^2/2)*
e^(-I*E*y^2*(1 - v)*t2^2/2)*e^(-I*k*y*(1 - u)*t1)*
e^(I*k*y*(1 - v)*t2)*
e^((1 + I)*(sqrt(E)*y*w*t1 + w*k/sqrt(E)))*
e^((1 - I)*(sqrt(E)*y*u*t2 + u*k/sqrt(E)))*
e^(-w^2/2)*e^(-u^2/2)*w^(-1/2 + I*m^2/(2*E))*
u^(-1/2 - I*m^2/(2*E)), (u, 0, Infinity)), (w, 0,
Infinity)), (t2, -Infinity, Infinity)), (t1, -Infinity,
Infinity)), (k, -Infinity, Infinity))
I haven't been able to get a result from this code, it seems to run
forever. I was hoping to be able to estimate the integral with some
numerical methods, however I was having trouble getting a numerical
integral set up properly. My first question is, can someone help me set up
multivariable numerical integrals properly. I was trying something like
numerical_integral(x*y,(x,0,1),(y,0,1))
or
numerical_integral(numerical_integral(x*y,(x,0,1)),(y,0,1))
but neither seem to be the correct format, as they both give errors.
My second question is, can anyone give some advice on how to approximate
such an integral, where it's multivariable and the bounds are at infinity?
I don't really know where to start.
Thanks!
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