I am trying to find the Fourier transform of the expression $e^{-\nu |t-s|}$, which is given in a number of places (such as here, pg. 248) as being $\frac{\nu}{\omega^2+\nu^2}$. However I get a very different answer:
\begin{align} \int_{-\infty}^\infty e^{-\nu |t-s|}e^{-i \omega t}dt &= \int_{-\infty}^\infty e^{-\nu |u|-i \omega (u+s)}du \quad\textrm{with $u=t-s$}\\ &=\int_0^\infty e^{-u(\nu+i \omega)-i\omega s}du+\int_{-\infty}^0 e^{u(\nu-i \omega)-i \omega s}du\\ &=\left[ \frac{-1}{\nu+i \omega}e^{-u(\nu+i\omega)-i\omega s}\right]_0^\infty+\left[\frac{1}{\nu-i\omega}e^{u(\nu-i\omega)-i\omega s}\right]^0_{-\infty} \\ &=e^{-i\omega s}\left( \frac{1}{\nu+i\omega}+\frac{1}{\nu-i\omega} \right) \end{align} Where am I going wrong here?
The reference you cite studies the spectrum of: $$ \frac{D}{\tau} e^{ - |t-s|/\tau } $$
From your proposed solution (taking $D=1$ and $\nu=1/\tau$), this leads to: \begin{align} \int_{-\infty}^\infty \nu\ e^{-\nu |t-s| }e^{-i\omega t} dt &= \nu\ e^{-i\omega s} \left( \frac{1}{\nu + i\omega} + \frac{1}{\nu - i\omega} \right) \\ &= \nu\ e^{-i\omega s} \frac{ \nu - i\omega + \nu + i\omega }{ (\nu + i\omega) (\nu - i\omega) } \\ &= e^{-i\omega s} \frac{ 2\nu^2 }{ \nu^2 + \omega^2 } \end{align}
Which corresponds to the result proposed in your reference: $$ \frac{2D}{\tau^2\omega^2+1} $$
Maybe what bothers you is that this is only the magnitude of the power spectrum; I think this is simply an abusive language from the book (the magnitude of the power spectrum is often what people think about when talking about the "power spectrum" of a 1d signal). The Wiener-Khinchin theorem relates the PSD to the autocorrelation function via: $$ R(t) = \int_{-\infty}^\infty S(\nu)\ e^{i\nu t}\ d\nu $$ so clearly if $R(t)$ should be real, then $S(\nu)$ should be complex.