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Math Help - Integral

  1. #1
    MHF Contributor chiph588@'s Avatar
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    Integral

    Given  \displaystyle a,b\in\mathbb{R} , compute  \displaystyle \int_0^\infty e^{-ax^2}\cos(bx)dx .
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  2. #2
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    Quote Originally Posted by chiph588@ View Post
    Given  \displaystyle a,b\in\mathbb{R} , compute  \displaystyle \int_0^\infty e^{-ax^2}\cos(bx)dx .
    i think we also need a > 0 for the convergence. anyway, i think the answer is:

    Spoiler:
    \displaystyle \frac{1}{2}\sqrt{\frac{\pi}{a}} \exp \left( \frac{-b^2}{4a} \right).

    why? first do the substitution x \to \sqrt{a}x and put \frac{b}{\sqrt{a}}=c. then your integral becomes f(c)=\frac{1}{\sqrt{a}} \int_0^{\infty} e^{-x^2} \cos (cx) \ dx.

    differentiating with respect to c (check the conditions that are needed) will give us: f'(c)=\frac{1}{\sqrt{a}} \int_0^{\infty} -xe^{-x^2} \sin(cx) \ dx.

    now let u=\sin(cx) and dv=-xe^{-x^2} dx and apply by parts to get: f'(c)=\frac{-c}{2\sqrt{a}} f(c).

    solving this differential equation with the fact that f(0)=\frac{1}{2} \sqrt{\frac{\pi}{a}} will give us the result.
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  3. #3
    MHF Contributor chisigma's Avatar
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    Is...

    \displaystyle \int_{0}^{\infty} e^{-a\ x^{2}}\ \cos b x\ dx = \frac{1}{2} \ \mathcal{F} \{e^{-a\ x^{2}}\} = \sqrt{\frac{\pi}{4\ a}}\ e^{-\frac{b^{2}}{4a}}

    Kind regards

    \chi \sigma
    Last edited by chisigma; September 7th 2010 at 10:38 AM. Reason: some little confusion... sorry!...
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  4. #4
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    Quote Originally Posted by NonCommAlg View Post
    Spoiler:
    \displaystyle \frac{1}{2}\sqrt{\frac{\pi}{a}} \exp \left( \frac{-b^2}{4a} \right).

    why? first do the substitution x \to \sqrt{a}x and put \frac{b}{\sqrt{a}}=c. then your integral becomes f(c)=\frac{1}{\sqrt{a}} \int_0^{\infty} e^{-x^2} \cos (cx) \ dx.

    differentiating with respect to c (check the conditions that are needed) will give us: f'(c)=\frac{1}{\sqrt{a}} \int_0^{\infty} -xe^{-x^2} \sin(cx) \ dx.

    now let u=\sin(cx) and dv=-xe^{-x^2} dx and apply by parts to get: f'(c)=\frac{-c}{2\sqrt{a}} f(c).

    solving this differential equation with the fact that f(0)=\frac{1}{2} \sqrt{\frac{\pi}{a}} will give us the result.
    Dear me, that was very clever!
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