Integration of a piecewise multivariable function

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Divij Gupta
Divij Gupta 2021-7-30
编辑： Paul 2021-8-1
I have a function I need to integrate. The function is a product of 2 functions, one a function in x and the the other a piecewise function in t and x. The piecewise function is something like for rho(x) (where rho is some function) < t, f(x,t) = 0, for rho(x)<t<2rho(x), f(t,x) = rho(x) - t, and for t>2rho, f(t,x) = -rho(x). I want to integrate with respect to x with bounds of Inf and -Inf (100 would do a good enough job as infinity for my purposes so I can just use that). I know the integral is not analytic, but I do know that I should get back a piecewise function in t. I cannot seem to figure it out. Anyone have any tips?
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Divij Gupta
Divij Gupta 2021-7-31
For an example, take f(t,x) to be the function I defined above (let rho(x) = sqrt(1-x^2)) and take the other function (the one you multiply with f(x,t) to form the function to be integrated) to be exp(x^2). How do I find the integral from -Inf to Inf?
Paul
Paul 2021-7-31
编辑：Paul 2021-8-1
Can you clarify the piecewise function f(x,t)? The first condition is:
rho(x) < t
but the second condition is
rho(x) < t < 2*rho(x)
Those conditions overlap. Should the first condition be t < rho(x) ?
Also, with rho(x) = sqrt(1-x^2) it would seem the integrand would become complex for abs(x) > 1. But the integration wrt x is over -inf to inf. Is this problem supposed to cover complex variables?

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