To find the Fourier sine transform of $f(x) = e^{-|x|}$, we follow these steps:

Step 1: Define the Fourier Sine Transform

The Fourier sine transform $F_s(k)$ of a function $f(x)$ is defined as: $F_s(k) = \sqrt{\frac{2}{\pi}} \int_0^\infty f(x) \sin(kx) \, dx$

Given $f(x) = e^{-|x|}$, we need to split the function $f(x)$ into two parts since it is defined piecewise:

e^{-x}, & x \geq 0 \\ e^x, & x < 0 \end{cases}$$ ### Step 2: Simplify the Integral Because the Fourier sine transform is defined for $$x \geq 0$$, we only consider the part of $$f(x)$$ where $$x \geq 0$$: $$f(x) = e^{-x}$$ ### Step 3: Set Up the Integral The Fourier sine transform of $$f(x) = e^{-x}$$ is then: $$F_s(k) = \sqrt{\frac{2}{\pi}} \int_0^\infty e^{-x} \sin(kx) \, dx$$ ### Step 4: Evaluate the Integral To evaluate the integral $$\int_0^\infty e^{-x} \sin(kx) \, dx$$, we use integration by parts or a known integral result. This integral can be solved using the formula: $$\int_0^\infty e^{-ax} \sin(bx) \, dx = \frac{b}{a^2 + b^2}$$ where $$a = 1$$ and $$b = k$$. ### Step 5: Apply the Formula Using the above formula: $$\int_0^\infty e^{-x} \sin(kx) \, dx = \frac{k}{1^2 + k^2} = \frac{k}{1 + k^2}$$ ### Step 6: Multiply by the Prefactor Now, multiply by the prefactor $$\sqrt{\frac{2}{\pi}}$$: $$F_s(k) = \sqrt{\frac{2}{\pi}} \cdot \frac{k}{1 + k^2}$$ ### Final Result The Fourier sine transform of $$f(x) = e^{-|x|}$$ is: $$F_s(k) = \sqrt{\frac{2}{\pi}} \cdot \frac{k}{1 + k^2}$$ ### Summary - We defined the Fourier sine transform. - We considered the function $$f(x)$$ for $$x \geq 0$$. - We set up and evaluated the integral using a known formula. - We multiplied by the necessary prefactor to obtain the final result. Would you like any more details or explanations on specific parts of this solution? ### Relative Questions 1. What is the definition of the Fourier sine transform? 2. How do you handle piecewise functions in Fourier transforms? 3. What is the known integral result for $$\int_0^\infty e^{-ax} \sin(bx) \, dx$$? 4. Can you show the steps for integration by parts used in evaluating the integral? 5. How does the prefactor $$\sqrt{\frac{2}{\pi}}$$ affect the Fourier transform? 6. What is the Fourier cosine transform, and how does it differ from the sine transform? 7. Why do we only consider $$x \geq 0$$ in the sine transform? 8. How would the result change if we considered the Fourier transform instead of the sine transform? ### Tip When dealing with Fourier transforms, always check the conditions of the integral and simplify the function within the given domain for accurate results.