I'm having trouble computing the Fourier transform of the following function:

$$y(x)=De^{-\lambda \lvert x\rvert}$$

It mainly has to do with the integration, I think, but I'll try to attempt it and illustrate where it all goes wrong.

$$\bar f(k) = \frac{D}{2\pi}\int_{-\infty}^{\infty}e^{ikx}e^{-\lambda \lvert x \rvert} \ dx$$

$$\bar f(k) = \frac{D}{2\pi}\int_{-\infty}^{\infty}e^{ikx-\lambda \lvert x \rvert} \ dx$$

From here I don't know how to integrate this. I tried a substitution, and quickly realized I couldn't due to the modulus. I was thinking it may be possible to convert the complex exponential into sines and cosines to try and integrate it by parts for each trigonometric component, but I don't think that will be doable since differentiating $e^{-\lambda \lvert x \rvert}$ will not allow for the typical way to integrate an exponential mutipled by a sine or cosine.

  • 3
    $\begingroup$ $\int_{-\infty }^{\infty }{{}}=\int_{-\infty }^{0}{{}}+\int_{0}^{\infty }{{}}$ and think about the modulus of x over these ranges. $\endgroup$ – Paul Sep 28 '18 at 11:05
  • $\begingroup$ Really the same as above, but first show that $\int_{-\infty}^\infty f(x){\rm d}x = \int_{0}^\infty [f(x)+f(-x)]{\rm d}x$. With this formulation $|x|$ is not a problem anymore. $\endgroup$ – Winther Sep 28 '18 at 11:19

\begin{equation} \begin{split} \bar f(k) &\stackrel{(a)}{=} \frac{D}{2\pi}\int_{-\infty}^{\infty}e^{ikx}e^{-\lambda \lvert x \rvert} \ dx\\ &\stackrel{(b)}{=} \frac{D}{2\pi}\int_{-\infty}^{0}e^{ikx}e^{\lambda x } \ dx + \frac{D}{2\pi}\int_{0}^{\infty}e^{ikx}e^{-\lambda x} \ dx \\ &\stackrel{(c)}{=}\frac{D}{2\pi}\big( \frac{1}{\lambda + ik}e^{(\lambda+ik)x}\big]_{-\infty}^0 - \frac{1}{-\lambda + ik}e^{(-\lambda+ik)x}\big]^{\infty}_{0} \big)\\ &\stackrel{(d)}{=}\frac{D}{2\pi}\big( \frac{1}{\lambda + ik} - \frac{1}{-\lambda + ik} \big) \end{split} \end{equation} In $(a)$ we used the definition of Fourier transform.

In $(b)$ we have split the integral and made use of $\vert x \vert = -x$ if $x < 0$ and $\vert x \vert = x$ if $x \geq 0$.

In $(c)$, we just integrated.

In $(d)$, we replaced the limits.

  • $\begingroup$ How did you evaluate the limits of the $e^{(-\lambda + ik)x}$ integral in (c)? $\endgroup$ – sangstar Oct 3 '18 at 16:00

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