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Thread: related integrals

  1. #1
    Super Member Random Variable's Avatar
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    related integrals

    $\displaystyle \int_{0}^{\infty} \cos \Big(x + \frac{1}{x} \Big) \frac{dx}{\sqrt{x}} = 2 \int^{\infty}_{0} \cos \Big(x^{2}+ \frac{1}{x^{2}} \Big) dx = 2 \ \Re \int^{\infty}_{0} e^{-i(x^{2}+\frac{1}{x^{2}})} dx $


    and $\displaystyle \int_{0}^{\infty} \sin \Big(x + \frac{1}{x} \Big) \frac{dx}{\sqrt{x}} = 2 \int^{\infty}_{0} \sin \Big(x^{2}+ \frac{1}{x^{2}} \Big) dx = -2 \ \Im \int^{\infty}_{0} e^{-i(x^{2}+\frac{1}{x^{2}})} dx $


    Could I use $\displaystyle \int^{\infty}_{0} e^{-a(x^{2}+\frac{1}{x^{2}})} dx = \frac{\sqrt{\frac{\pi}{a}}}{2e^{2a}} $ and substitute i for a?
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  2. #2
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    Re: related integrals

    Good question! Any kind of substitution would have to be justified. It boils down to whether the last result extends to complex $\displaystyle a$'s. I'll need to think about this a while.
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  3. #3
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    Re: related integrals

    The answer to your question is yes you can. You can extend the identity to complex values of $\displaystyle a$, in a suitable neighborhood, because of the uniqueness principle in complex analysis.
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  4. #4
    Super Member Random Variable's Avatar
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    Re: related integrals

    I'm not familiar with a uniqueness principle. All I know is that when $\displaystyle a$ is real valued and $\displaystyle a \le 0$, the integral diverges. I don't know for what imaginary or complex values the integral is convergent. Substituting $\displaystyle a= i $ does lead to the correct solution, though.

    EDIT: It should be OK if I can show that $\displaystyle \int^{\infty}_{0} \cos \Big( x^{2}+\frac{1}{x^{2}} \Big) \ dx $ and $\displaystyle \int^{\infty}_{0} \sin \Big( x^{2} + \frac{1}{x^{2}} \Big) \ dx $ actually converge.
    Last edited by Random Variable; Jun 26th 2011 at 06:57 PM.
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  5. #5
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    Re: related integrals

    The question of whether an identity involving a real parameter can extended to complex values comes up a lot in complex analysis. There's more to it than just showing that certain integrals converge. A reference for the uniqueness principle is Gamelin's book on complex analysis. He addresses questions very similiar to the one you're considering.
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