Fouriertransform : Determine the inverse Fourier transform

Question

Determine the inverse Fourier transform to

$$F(ω) = \frac{1}{(ω^2 + 4)(ω^2 + 9)}$$

using

a) Partial fractions, and then using the Fourier Transform table.

b) Convolution.

I don't know to to solve it :( please help.

Answer 1

HINT 1:

For partial fraction expansion, we have

$$\frac{1}{(\omega^2+4)(\omega^2+9)}=\frac15\left(\frac{1}{\omega^2+4}-\frac{1}{\omega^2+9}\right)$$

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HINT 2:

For using the convolution theorem, note that

$$\mathscr{F}(f(\omega)g(\omega))=\mathscr{F}(f(\omega))*\mathscr{F}(g(\omega))$$

where here, $f(\omega)=\frac{1}{\omega^2+4}$ and $g(\omega)=\frac{1}{\omega^2+9}$.

Here we define the Fourier Transform as

$$\mathscr{F}(f(\omega))=\int_{-\infty}^{\infty}f(\omega)e^{i2\pi \omega t}\,d\omega$$