Math Problem Statement
Find the fourier transform of f(x) given by f(x) =[1,[x]<2 0 [x]<2] and hence evaluate ∫0∞ sinx/x dx and ∫0∞ (sinx/x)²dx
Solution
Let's approach this systematically.
1. Fourier Transform of
The given function is defined as:
\begin{cases} 1, & |x| < 2, \\ 0, & |x| \geq 2. \end{cases}$$ The Fourier transform $$F(k)$$ of $$f(x)$$ is defined by: $$F(k) = \int_{-\infty}^{\infty} f(x) e^{-ikx} \, dx.$$ Substitute the definition of $$f(x)$$: $$F(k) = \int_{-2}^{2} e^{-ikx} \, dx.$$ Now, compute this integral: $$F(k) = \left[ \frac{e^{-ikx}}{-ik} \right]_{-2}^{2}.$$ Substitute the limits: $$F(k) = \frac{e^{-ik(2)}}{-ik} - \frac{e^{-ik(-2)}}{-ik}.$$ Simplify: $$F(k) = \frac{1}{-ik} \left( e^{-2ik} - e^{2ik} \right).$$ Using the property of sine, $$e^{ia} - e^{-ia} = 2i \sin(a)$$, this becomes: $$F(k) = \frac{1}{-ik} \cdot (-2i \sin(2k)).$$ Simplify further: $$F(k) = \frac{2 \sin(2k)}{k}.$$ Thus, the Fourier transform is: $$F(k) = \frac{2 \sin(2k)}{k}.$$ --- ### **2. Evaluation of $$\int_{0}^\infty \frac{\sin x}{x} \, dx$$** This integral is related to the Fourier transform of a square pulse. From Fourier analysis, we know that: $$\int_{0}^\infty \frac{\sin x}{x} \, dx = \frac{\pi}{2}.$$ --- ### **3. Evaluation of $$\int_{0}^\infty \left(\frac{\sin x}{x}\right)^2 \, dx$$** Using Parseval's theorem, which states that: $$\int_{-\infty}^\infty |f(x)|^2 \, dx = \frac{1}{2\pi} \int_{-\infty}^\infty |F(k)|^2 \, dk,$$ we compute $$\int_{0}^\infty \left(\frac{\sin x}{x}\right)^2 \, dx$$. Here, $$|f(x)|^2 = f(x)$$ since $$f(x)$$ is either 1 or 0, and its Fourier transform is $$F(k) = \frac{2 \sin(2k)}{k}$$. Substituting $$F(k)$$ into Parseval's theorem: $$\int_{-\infty}^\infty f(x)^2 \, dx = \frac{1}{2\pi} \int_{-\infty}^\infty \left|\frac{2 \sin(2k)}{k}\right|^2 \, dk.$$ Evaluate the left-hand side: $$\int_{-\infty}^\infty f(x)^2 \, dx = \int_{-2}^2 1^2 \, dx = 4.$$ Now compute the right-hand side: $$4 = \frac{1}{2\pi} \int_{-\infty}^\infty \frac{4 \sin^2(2k)}{k^2} \, dk.$$ Simplify: $$4 = \frac{2}{\pi} \int_{-\infty}^\infty \frac{\sin^2(2k)}{k^2} \, dk.$$ Using symmetry and a known result for the integral of $$\sin^2(x)/x^2$$: $$\int_{0}^\infty \left(\frac{\sin x}{x}\right)^2 \, dx = \frac{\pi}{4}.$$ --- ### **Final Results** 1. Fourier transform: $$F(k) = \frac{2 \sin(2k)}{k}.$$ 2. Integral evaluations: $$\int_{0}^\infty \frac{\sin x}{x} \, dx = \frac{\pi}{2},$$ $$\int_{0}^\infty \left(\frac{\sin x}{x}\right)^2 \, dx = \frac{\pi}{4}.$$ --- Let me know if you'd like to see detailed steps for any part or have further questions! 😊 --- ### Related Questions: 1. What is the significance of the sinc function in Fourier analysis? 2. How does Parseval’s theorem relate energy in time and frequency domains? 3. What are the properties of Fourier transforms of rectangular functions? 4. Can you derive $$\int_{0}^\infty \frac{\sin(ax)}{x} \, dx$$ for general $$a$$? 5. How does the function $$\sin(x)/x$$ relate to signal processing? --- ### Tip: When solving Fourier problems, always check if symmetry or known Fourier pairs simplify the computation!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Fourier Transform
Definite Integrals
Signal Processing
Parseval's Theorem
Formulas
Fourier Transform: F(k) = ∫_{-∞}^∞ f(x)e^{-ikx} dx
Integral relation: ∫_{0}^∞ (sin(x)/x) dx = π/2
Parseval's Theorem: ∫_{-∞}^∞ |f(x)|² dx = (1/2π) ∫_{-∞}^∞ |F(k)|² dk
Theorems
Parseval's Theorem
Fourier Transform properties
Suitable Grade Level
Undergraduate Mathematics or Advanced High School (Grades 11-12)
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