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Math Help - Asymptotic evaluation of an integral..

  1. #1
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    Asymptotic evaluation of an integral..

    Let be the integral:

    Int(-oo,oo)dx f(x)exp(iux) I would like to know if there is any method to

    evaluate it for u-->oo

    -Stationary phase method:=can't be applied since y=x has no "stationary points" for any real x.

    - by the way if we had f(x)=g(x)+ih(x) could the integral above be evaluated splitting it into the real and complex part?..thanks.
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by lokofer View Post
    Let be the integral:

    Int(-oo,oo)dx f(x)exp(iux) I would like to know if there is any method to

    evaluate it for u-->oo

    If |f(x)|^2 is integrable over (-inf,inf) then your integral goes to 0 almost
    everywere as u goes to infinity, in fact it's absolute value squared is itself
    integrable from -inf to inf.

    (This is a fourier transform).

    RonL
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  3. #3
    Grand Panjandrum
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    Quote Originally Posted by lokofer View Post
    Let be the integral:

    Int(-oo,oo)dx f(x)exp(iux) I would like to know if there is any method to

    evaluate it for u-->oo

    -Stationary phase method:=can't be applied since y=x has no "stationary points" for any real x.

    - by the way if we had f(x)=g(x)+ih(x) could the integral above be evaluated splitting it into the real and complex part?..thanks.
    Int(-oo,oo)dx f(x)exp(iux)=Int(-inf,inf) [g(x)+ih(x)][cos(ux)+isin(ux)] dx

    ................................ ..=Int(-inf,inf) {[g(x)cos(ux)-h(x)sin(x)]+i[g(x)sin(ux)+h(x)cos(ux)]}dx

    ....................................=Int(-inf,inf)[g(x)cos(ux)-h(x)sin(x)]dx +
    .........................................i.Int(-inf,inf)[g(x)sin(ux)+h(x)cos(ux)]dx

    RonL
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