Also, I tried running the integral with finite bounds, and I get a giac
error. Here is the code:
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, 10)), (w, 0,
10)), (t2, -10, 10)), (t1, -10,
10)), (k, -10, 10))
Here is the error
RuntimeError: An error occurred running a Giac command:
INPUT:
sage20
OUTPUT:
:1: syntax error line 1 col 31 at " in sage20:=int(sage16,sage17"Done
",sage18w,sage190):;
:1: syntax error line 1 col 31 at " in sage20:=int(sage16,sage17"Done",
sage18w,sage190):;
"Done"
On Wednesday, February 26, 2020 at 10:21:21 PM UTC-5, saad khalid wrote:
>
> 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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