- #1
Incand
- 334
- 47
Homework Statement
Use the Fourier transform to compute
[tex]\int_{-\infty}^\infty \frac{(x^2+2)^2}{(x^4+4)^2}dx[/tex]
Homework Equations
The Plancherel Theorem
##||f||^2=\frac{1}{2\pi}||\hat f ||^2##
for all ##f \in L^2##.
We also have a table with the Fourier transform of some function, the ones of possible use may be
##(x^2+a^2)^{-1} \to (\pi/a)a^{-\xi^2/(2a)}##
and
##f(x)g(x) \to \frac{1}{2\pi} (\hat f * \hat g)(\xi )##
The Attempt at a Solution
Not really sure where to start here, every other similar exercise use Plancherel's theorem so I assume I should use that one here. The problem is then that I would need to computer the Fourier transform of
##\frac{x^2+2}{x^4+4}## which doesn't seem to easy at all. I tried writing this as
##\frac{x^2+2}{(x^2+2i)(x^2-i2)}## and somehow do partial fractions but doesn't seem to get me anywhere. Another similar approach that perhaps could work suggested by our professor was
##\int_{-\infty}^\infty \frac{(x^2+2)^2}{(x^4+4)^2} = \int_{-\infty}^\infty \left| \frac{(x^2+2)}{(x^2+2i)^2} \right|^2## but this doesn't seem to work either.
Any ideas on how to procede?
Edit: I should mention that it's possible to find the Fourier transform in some tables, although not the one in our book. If I use that one however I would need to proove it first.