Discussion
The complex Fourier transform of \(e^{-|x|}\) is :
Answer: A
The complex Fourier transform is \(\frac{2}{1 + n^{2}}\).
Using \(F(n) = \int_{-\infty}^{\infty} f(x)e^{inx}dx\) and splitting the range at zero, where \(|x|\) changes sign:
\[ \begin{aligned} F(n) &= \int_{-\infty}^{0} e^{(1 + in)x}dx \\ &\quad + \int_{0}^{\infty} e^{-(1 - in)x}dx \\ &= \frac{1}{1 + in} + \frac{1}{1 - in} \\ &= \frac{2}{1 + n^{2}} \end{aligned} \]
The paper uses the convention without the factor \(1/\sqrt{2\pi}\). With it, the result would be \(\sqrt{2/\pi}/(1 + n^{2})\), which is not offered. The other numerators come from dropping one of the two halves (1) or mishandling the constants (4 or \(\pi\)).
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