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Math Help - plancherel relation

  1. #1
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    plancherel relation

    how does one use the plancherel relation to evaluate the integral

    integral from negative infinity to infinity of dx/(1+x^2)^2

    i think i'm mostly confused about how to obtain u(x) from that integral and apply it to the relation.

    thanks!
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by morganfor View Post
    how does one use the plancherel relation to evaluate the integral

    integral from negative infinity to infinity of dx/(1+x^2)^2

    i think i'm mostly confused about how to obtain u(x) from that integral and apply it to the relation.

    thanks!
    Plancherel's relation for this problem tells us that:

    \int_{-\infty}^{\infty} \frac{1}{(1+x^2)^2}\; dx=\int_{-\infty}^{\infty} |F(\nu)|^2\; d\nu

    where:

    F(\nu)=\int_{-\infty}^{\infty} \frac{1}{1+x^2}e^{2\pi i x \nu}\; dx

    The last expression is a standard Fourier transform and can be looked up and then the right hand side of the first equation above is an easy integral.

    For the form of the FT that I give above we have:

    F(\nu)=\int_{-\infty}^{\infty} \frac{1}{1+x^2}e^{2\pi i x \nu}\; dx=\pi e^{-|2 \pi \nu|}

    Your detail may vary depending on your definition of the FT.

    CB
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  3. #3
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    Thanks! That all makes sense

    I'm just stuck now on how to integrate the norm of pi*e^-2|pi*v| to evaluate the integral in the question. Don't you get a double integral of pi*e^-2|pi*v| from negative infinity to positive infinity?
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  4. #4
    Grand Panjandrum
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    Quote Originally Posted by morganfor View Post
    Thanks! That all makes sense

    I'm just stuck now on how to integrate the norm of pi*e^-2|pi*v| to evaluate the integral in the question. Don't you get a double integral of pi*e^-2|pi*v| from negative infinity to positive infinity?
    \begin{aligned}\int_{-\infty}^{\infty}\pi e^{-2|\pi \nu|}\;d\nu&=2\int_{0}^{\infty}\pi e^{-2|\pi \nu|}\;d\nu\\&=2\int_{0}^{\infty}\pi e^{-2\pi \nu}\;d\nu\end{aligned}

    CB
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