plancherel relation

**morganfor** · May 30th 2011, 03:33 PM

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!

**CaptainBlack** · May 30th 2011, 09:37 PM

Originally Posted by morganfor

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

**morganfor** · May 31st 2011, 06:16 PM

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?

**CaptainBlack** · May 31st 2011, 07:17 PM

Originally Posted by morganfor

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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