Question regarding Fourier Transforms

Question

Still wrapping my head around FT.

How do you go about finding the Fourier Transform of the following function?

$$y(t) = \frac{1}{a^2 + t^2}$$

where $t$ is the time variable and $a$ is some constant.

Answer 1

From

$$\mathcal{F}(e^{-at}u(t))=\frac{1}{j\omega+a},\, a>0,$$

calculate the FT of $e^{-a|t|}$ using the properties of FT.

You can expand $e^{-a|t|}$ as follows:

$$e^{-a|t|}=e^{-at}u(t)+e^{at}u(-t)$$

After you did that, use the duality property of the FT to find the answer to your question.