Noting that there are lots of exp(-x^2) in the integrand, I would perform the
numerical approximation using Monte-Carlo techniques with Gaussian
pseudu-random numbers.
________________________________
From: [email protected] <[email protected]> on behalf
of saad khalid <[email protected]>
Sent: Wednesday, February 26, 2020 9:41:19 PM
To: sage-support
Subject: [sage-support] Re: Approximating integral with infinite bounds
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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