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APM346-2015F => APM346--Home Assignments => HA7 => Topic started by: Victor Ivrii on November 01, 2015, 05:10:24 PM

Title: HA7-P5
Post by: Victor Ivrii on November 01, 2015, 05:10:24 PM

Problem 5

http://www.math.toronto.edu/courses/apm346h1/20159/PDE-textbook/Chapter5/S5.2.P.html#problem-5.2.P.5 (http://www.math.toronto.edu/courses/apm346h1/20159/PDE-textbook/Chapter5/S5.2.P.html#problem-5.2.P.5)

Title: Re: HA7-P5
Post by: Xi Yue Wang on November 03, 2015, 08:11:49 PM

For part (a), given that $|x| \leq a$, we get $$ \hat{f}(\omega) = \frac{1}{2\pi}\int_{-a}^{a} e^{-i\omega x } dx\\= \frac{1}{2\pi}\frac {e^{-i\omega a}-e^{i\omega a}}{(-i\omega)}\\=\frac{\sin(\omega a)}{\pi\omega}$$


For part (b), we take $g(x) = xf(x)$, $\hat{g}(\omega) = i\hat{f'}(\omega)$, where $f(x)$ is defined in part (a).$$\hat{f'}(\omega) = \frac{a\cos(\omega a)}{\pi\omega} - \frac{\sin(\omega a)}{\pi\omega^2}\\\hat{g}(\omega) = i( \frac{a\cos(\omega a)}{\pi\omega} - \frac{\sin(\omega a)}{\pi\omega^2})$$

For part (c), by Inverse Fourier Transformation, $$\int_{-\infty}^{\infty} \hat{f}(\omega)e^{i\omega x} d\omega=\int_{-\infty}^{\infty} \frac{\sin(\omega a)}{\pi\omega}e^{i\omega x} d\omega= f(x) = 1$$ where $|x|\leq a$
We take $x=0, a=1$, then we get $$\frac{1}{\pi}\int_{-\infty}^{\infty} \frac{\sin(\omega)}{\omega} d\omega = 1 $$
Hence, $$\int_{-\infty}^{\infty} \frac{\sin(\omega)}{\omega} d\omega = \pi$$