To find the Fourier transform of $f(x) = e^{-k|x|}$, where $k > 0$, let's proceed step by step.

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Fourier Transform Definition

The Fourier transform of $f(x)$ is defined as: $F(\omega) = \int_{-\infty}^{\infty} f(x)e^{-i\omega x}dx$

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Given Function

The function is $f(x) = e^{-k|x|}$, and because of the absolute value, we split $f(x)$ into two parts:

\begin{cases} e^{-kx}, & x \geq 0 \\ e^{kx}, & x < 0 \end{cases}$$ Thus, the Fourier transform becomes: $$F(\omega) = \int_{-\infty}^{0} e^{kx} e^{-i\omega x} dx + \int_{0}^{\infty} e^{-kx} e^{-i\omega x} dx$$ --- ### **Evaluating the Integrals** 1. **For $$x \geq 0$$:** $$\int_{0}^{\infty} e^{-kx} e^{-i\omega x} dx = \int_{0}^{\infty} e^{-(k + i\omega)x} dx$$ This integral converges for $$k > 0$$ and can be evaluated as: $$\int_{0}^{\infty} e^{-(k + i\omega)x} dx = \frac{1}{k + i\omega}$$ 2. **For $$x < 0$$:** $$\int_{-\infty}^{0} e^{kx} e^{-i\omega x} dx = \int_{-\infty}^{0} e^{(k - i\omega)x} dx$$ This integral converges for $$k > 0$$ and can be evaluated as: $$\int_{-\infty}^{0} e^{(k - i\omega)x} dx = \frac{1}{-(k - i\omega)} = \frac{1}{k - i\omega}$$ --- ### **Combining the Results** The Fourier transform is: $$F(\omega) = \frac{1}{k + i\omega} + \frac{1}{k - i\omega}$$ Combine the terms using a common denominator: $$F(\omega) = \frac{k - i\omega + k + i\omega}{(k + i\omega)(k - i\omega)} = \frac{2k}{k^2 + \omega^2}$$ --- ### **Final Result** The Fourier transform of $$f(x) = e^{-k|x|}$$ is: $$F(\omega) = \frac{2k}{k^2 + \omega^2}$$ --- Do you want further details or explanations? Let me know! --- ### Related Questions: 1. How does the Fourier transform behave for other exponentially decaying functions? 2. What happens to the transform when $$k \to 0$$ or $$k \to \infty$$? 3. Can you compute the inverse Fourier transform of $$\frac{2k}{k^2 + \omega^2}$$? 4. How does $$k$$ affect the width of the Fourier transform in frequency space? 5. How is the Fourier transform of even functions generally simplified? --- **Tip**: For symmetric functions like $$e^{-k|x|}$$, leveraging symmetry properties can often simplify the integration process!