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# Lecture 24: The Fourier Transform ## The Fourier Transform ### Definition $$ F(\omega) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} f(t)e^{-i\omega t} dt $$ ### Inverse Fourier Transform $$ f(t) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} F(\omega)e^{i\omega t} d\omega $$ ## Example 1...

# Lecture 24: The Fourier Transform ## The Fourier Transform ### Definition $$ F(\omega) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} f(t)e^{-i\omega t} dt $$ ### Inverse Fourier Transform $$ f(t) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} F(\omega)e^{i\omega t} d\omega $$ ## Example 1 ### Problem Find the Fourier transform of the function $$ f(t) = \begin{cases} 1, & |t| < a \\ 0, & |t| > a \end{cases} $$ ### Solution $$ F(\omega) = \frac{1}{\sqrt{2\pi}} \int_{-a}^{a} e^{-i\omega t} dt $$ $$ = \frac{1}{\sqrt{2\pi}} \left[ \frac{e^{-i\omega t}}{-i\omega} \right]_{-a}^{a} $$ $$ = \frac{1}{\sqrt{2\pi}} \frac{e^{-i\omega a} - e^{i\omega a}}{-i\omega} $$ Using Euler's formula $e^{ix} = \cos x + i \sin x$, we have $$ = \frac{1}{\sqrt{2\pi}} \frac{\cos(-\omega a) + i\sin(-\omega a) - \cos(\omega a) - i\sin(\omega a)}{-i\omega} $$ Since cosine is an even function and sine is an odd function, we get $$ = \frac{1}{\sqrt{2\pi}} \frac{\cos(\omega a) - i\sin(\omega a) - \cos(\omega a) - i\sin(\omega a)}{-i\omega} $$ $$ = \frac{1}{\sqrt{2\pi}} \frac{-2i\sin(\omega a)}{-i\omega} $$ $$ = \sqrt{\frac{2}{\pi}} \frac{\sin(\omega a)}{\omega} $$ ## Example 2 ### Problem Find the Fourier transform of $f(t) = e^{-a|t|}$, where $a > 0$. ### Solution $$ F(\omega) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} e^{-a|t|}e^{-i\omega t} dt $$ We can split the integral into two parts: $$ = \frac{1}{\sqrt{2\pi}} \left( \int_{-\infty}^{0} e^{at}e^{-i\omega t} dt + \int_{0}^{\infty} e^{-at}e^{-i\omega t} dt \right) $$ $$ = \frac{1}{\sqrt{2\pi}} \left( \int_{-\infty}^{0} e^{(a-i\omega)t} dt + \int_{0}^{\infty} e^{-(a+i\omega)t} dt \right) $$ $$ = \frac{1}{\sqrt{2\pi}} \left( \left[ \frac{e^{(a-i\omega)t}}{a-i\omega} \right]_{-\infty}^{0} + \left[ \frac{e^{-(a+i\omega)t}}{-(a+i\omega)} \right]_{0}^{\infty} \right) $$ $$ = \frac{1}{\sqrt{2\pi}} \left( \frac{1}{a-i\omega} + \frac{1}{a+i\omega} \right) $$ $$ = \frac{1}{\sqrt{2\pi}} \left( \frac{a+i\omega + a - i\omega}{a^2 + \omega^2} \right) $$ $$ = \frac{1}{\sqrt{2\pi}} \frac{2a}{a^2 + \omega^2} $$ $$ = \sqrt{\frac{2}{\pi}} \frac{a}{a^2 + \omega^2} $$