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Math Help - Integration question

  1. #1
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    Integration question

    I'm trying to integrate the function exp((-2*pi*i*y^2)/m) dy from 0 to inf.

    I've tried using the substitution u=y/sqrt(m) but just cannot do it.

    Is this integral even possible to evaluate?
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Hint

    Separate real and imaginary parts. With an adequate substitution you'll obtain the Fresnel integrals.
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  3. #3
    MHF Contributor chisigma's Avatar
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    In M.R. Spiegel - Laplace Transforms - Mc Grow-Hill 1965 You can read...

    \displaystyle \mathcal{L} \{e^{-a^{2}\ t^{2}}\} = \int_{0}^{\infty} e^{-a^{2}\ t^{2}}\ e^{-s\ t}\ dt = \frac{\sqrt{\pi}}{2\ a}\ e^{\frac{s^{2}}{4\ a^{2}}}\ erfc (\frac{s}{2a}) (1)

    Setting in (1) a^{2}= 2\ \pi\ \frac{i}{m} and s=0 with little computation You [should] obtain...

    \displaystyle  \int_{0}^{\infty} e^{-2\ \pi\ i\ \frac{t^{2}}{m}}\ dt = \frac{\sqrt{m}}{4}\ (1-i) (2)

    Kind regards

    \chi \sigma
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  4. #4
    MHF Contributor FernandoRevilla's Avatar
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    Let us see if M.R. Spiegel is right

    We have

    I:=\displaystyle\int_0^{+\infty}e^{-2\pi iy^2/m}dy=\displaystyle\int_0^{+\infty}\cos (2\pi y^2/m)dy-i\displaystyle\int_0^{+\infty}\sin (2\pi y^2/m) dy

    using x=\sqrt{2\pi/m}\;y :

    I=\sqrt{m/2\pi}(S(x)-iC(x))

    where S(x)=C(x)=\sqrt{\pi/8} are the Fresnel integrals. So,

    I=\sqrt{m}(1-i)/4
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  5. #5
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    Thanks for this.
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