MHB Erin's question via email about a Fourier Transform

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The discussion focuses on finding the Fourier Transform of a piecewise function defined for specific intervals. The calculation involves integrating the function over its defined ranges and applying integration techniques to derive the final expression. The resulting Fourier Transform is expressed as F(ω) = (1 - cos(2ω))/ω², highlighting its dependence on the frequency variable ω. Additionally, a comment suggests that understanding Fourier transforms can be enhanced through distribution theory, referencing a specific book for further reading. The conversation emphasizes both the mathematical process and theoretical insights related to Fourier analysis.
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Find the Fourier Transform of $\displaystyle \begin{align*} f \left( t \right) = \begin{cases} 1 - \frac{t}{2} \textrm{ if } 0 \leq t \leq 2 \\ 1 + \frac{t}{2} \textrm{ if } -2 \leq t < 0 \\ 0 \textrm{ elsewhere } \end{cases} \end{align*}$

$\displaystyle \begin{align*} F \left( \omega \right) &= \mathcal{F} \left\{ f \left( t \right) \right\} \\
&= \int_{-\infty}^{\infty}{ f\left( t \right) \mathrm{e}^{-\mathrm{j}\,\omega \, t}\,\mathrm{d}t } \\
&= \int_{-\infty}^{-2}{ 0\,\mathrm{d}t } + \int_{-2}^0{ \left( 1 + \frac{t}{2} \right) \mathrm{e}^{-\mathrm{j}\,\omega\,t}\,\mathrm{d}t } + \int_0^2{ \left( 1 - \frac{t}{2} \right) \mathrm{e}^{-\mathrm{j}\,\omega\,t}\,\mathrm{d}t } + \int_2^{\infty}{0\,\mathrm{d}t} \\
&= 0 + \left[ \left( 1 + \frac{t}{2} \right) \left( \frac{\mathrm{e}^{-\mathrm{j}\,\omega\,t}}{-j\,\omega} \right) \right]_{-2}^0 - \int_{-2}^0{ \frac{\mathrm{e}^{-\mathrm{j}\,\omega\,t}}{-2\,\mathrm{j}\,\omega}\,\mathrm{d}t } + \left[ \left( 1 - \frac{t}{2} \right) \left( \frac{\mathrm{e}^{-\mathrm{j}\,\omega\,t}}{-\mathrm{j}\,\omega} \right) \right]_0^2 - \int_0^2{ \frac{\mathrm{e}^{-\mathrm{j}\,\omega\,t}}{2\,\mathrm{j}\,\omega}\,\mathrm{d}t } + 0 \\
&= \frac{1}{-\mathrm{j}\,\omega} - \left[ \frac{\mathrm{e}^{-\mathrm{j}\,\omega\,t}}{2\,\mathrm{j}^2\,\omega^2} \right] _{-2}^0 - \left( \frac{1}{-\mathrm{j}\,\omega} \right) - \left[ \frac{\mathrm{e}^{-\mathrm{j}\,\omega\,t}}{-2\,\mathrm{j}^2\,\omega^2} \right] _0^2 \\
&= \left[ \frac{\mathrm{e}^{-\mathrm{j}\,\omega\,t}}{2\,\omega^2} \right] _{-2}^0 - \left[ \frac{\mathrm{e}^{-\mathrm{j}\,\omega\,t}}{2\,\omega^2} \right] _0^2 \\ &= \frac{1}{2\,\omega^2} \left\{ \left[ \mathrm{e}^{-\mathrm{j}\,\omega\,t } \right] _{-2}^0 - \left[ \mathrm{e}^{-\mathrm{j}\,\omega\,t} \right] _0^2 \right\} \\
&= \frac{1}{2\,\omega^2} \left[ \left( 1 - \mathrm{e}^{2\,\mathrm{j}\,\omega } \right) - \left( \mathrm{e}^{-2\,\mathrm{j}\,\omega\,t} - 1 \right) \right] \\ &= \frac{1}{2\,\omega^2} \left[ 2 - \left( \mathrm{e}^{2\,\mathrm{j}\,\omega} + \mathrm{e}^{-2\,\mathrm{j}\,\omega} \right) \right] \\ &= \frac{1}{\omega^2} \left[ 1 - \left( \frac{\mathrm{e}^{2\,\mathrm{j}\,\omega} + \mathrm{e}^{-2\,\mathrm{j}\,\omega}}{2} \right) \right] \\ &= \frac{1 - \cos{ \left( 2\,\omega \right) }}{\omega ^2} \end{align*}$
 
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