Fourier transform of $e^{-j|t|}$ ?

Question

Fourier transform of $e^{-j|t|}$ ? I have seen numerous examples of the normal form ($e^{j\omega0t}$), but I cannot seem to understand how to compute the modulus form of it.

Answer 1

I'm assuming you're using $j = \sqrt{-1}$ here.

Writing your function as $H(t) e^{-jt} + (1-H(t)) e^{jt}$, where $H(t)$ is the Heaviside step function, the Fourier transform (in the sense of tempered distributions) is $$ \pi \delta(s-1) + \pi \delta(s+1) + \frac{2j}{s^2-1} $$ where $\delta$ is the Dirac delta.