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!
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:
$\displaystyle \int_{-\infty}^{\infty} \frac{1}{(1+x^2)^2}\; dx=\int_{-\infty}^{\infty} |F(\nu)|^2\; d\nu$
where:
$\displaystyle F(\nu)=\int_{-\infty}^{\infty} \frac{1}{1+x^2}e^{2\pi i x \nu}\; dx$
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:
$\displaystyle F(\nu)=\int_{-\infty}^{\infty} \frac{1}{1+x^2}e^{2\pi i x \nu}\; dx=\pi e^{-|2 \pi \nu|}$
Your detail may vary depending on your definition of the FT.
CB