Discovering the Inverse Fourier Transform

[Prove It]

Miko asks:

What is the Inverse Fourier Transform of \(\displaystyle \displaystyle \begin{align*} F(\omega) = \frac{-5(i\omega + 4)}{16+(i\omega + 4)^2} \end{align*}\)?

 

[Prove It]

Miko asks:

What is the Inverse Fourier Transform of \(\displaystyle \displaystyle \begin{align*} F(\omega) = \frac{-5(i\omega + 4)}{16+(i\omega + 4)^2} \end{align*}\)?

Miko has sent me a scan with a great attempt of this question, but has tried to use a shift when a shift is not necessary.

From a table of Fourier Transforms, we can see that \(\displaystyle \displaystyle \begin{align*} \mathcal{F}^{-1} \left\{ \frac{a + i(\omega + k)}{k^2 + (a + i\omega)} \right\} = e^{-at}e^{ikt}H(t) \end{align*}\) and \(\displaystyle \displaystyle \begin{align*} \mathcal{F}^{-1} \left\{ \frac{k}{k^2 + (a + i\omega)^2} \right\} = e^{-at}\sin{(kt)} H(t) \end{align*}\). The function given has an identical denominator (with \(\displaystyle \displaystyle \begin{align*} a = 4 \end{align*}\) and \(\displaystyle \displaystyle \begin{align*} k = 4 \end{align*}\)) and so it would suggest that the function given is a combination of the functions given in the tables. So doing some algebraic manipulation...

\(\displaystyle \displaystyle \begin{align*} F(\omega) &= \frac{-5(i\omega + 4)}{16 + (i\omega + 4)^2} \\ &= \frac{-5i\omega - 20}{16 + (i\omega + 4)^2} \\ &= \frac{-5i(\omega + 4) + 20i - 20}{16 + (i\omega + 4)^2} \\ &= -5 \left[ \frac{4 + i(\omega + 4)}{16 + (i\omega + 4)^2} \right] + 5i \left[ \frac{4}{16 + (i\omega + 4)^2} \right] \end{align*}\)

which is now in the forms given in the tables. Therefore

\(\displaystyle \displaystyle \begin{align*} \mathcal{F}^{-1} \left\{ \frac{-5(i\omega + 4)}{16 + (i\omega + 4)^2} \right\} &= \mathcal{F}^{-1} \left\{ -5 \left[ \frac{4 + i(\omega + 4)}{16 + (i\omega + 4)^2} \right] + 5i \left[ \frac{4}{16 + (i\omega + 4)^2} \right] \right\} \\ &= -5e^{-4t}e^{4it}H(t) + 5i\,e^{-4t}\sin{(4t)}H(t) \end{align*}\)For Miko and anyone else viewing this and having similar questions, I invite you to read and post in the http://www.mathhelpboards.com/f16/ subforum.