Fourier integral

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Problem: Find the Fourier image of function \( f(x) \), if \( f(x) \equiv 0 \) when \( x \notin\left[x_{1}, x_{4}\right] \), and when \( x \in\left[x_{1}, x_{4}\right] \) the graph of this function consists of links of a polygonal chain, passing through points \( A\left(x_{1}, y_{1}\right), B\left(x_{2}, y_{2}\right) \), \( C\left(x_{3}, y_{3}\right), D\left(x_{4}, y_{4}\right) \). The coordinates of the points are presented in the table: \begin{tabular}{|c|c|c|c|} \hline\( A \) & \( B \) & \( C \) & \( D \) \\ \hline\( (-2,-2) \) & \( (1,0) \) & \( (4,-2) \) & \( (6,-2) \) \\ \hline \end{tabular}

2.6.1.1 Fourier integral

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Problem: Find the Fourier image of function \( f(x) \), if \( f(x) \equiv 0 \) when \( x \notin\left[x_{1}, x_{4}\right] \), and when \( x \in\left[x_{1}, x_{4}\right] \) the graph of this function consists of links of a polygonal chain, passing through points \( A\left(x_{1}, y_{1}\right), B\left(x_{2}, y_{2}\right) \), \( C\left(x_{3}, y_{3}\right), D\left(x_{4}, y_{4}\right) \). The coordinates of the points are presented in the table: \begin{tabular}{|c|c|c|c|} \hline\( A \) & \( B \) & \( C \) & \( D \) \\ \hline\( (-1,-1) \) & \( (0,-1) \) & \( (1,-2) \) & \( (2,-1) \) \\ \hline \end{tabular}

2.6.1.2 Fourier integral

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Problem: Find the Fourier image of function \( f(x) \), if \( f(x) \equiv 0 \) when \( x \notin\left[x_{1}, x_{4}\right] \), and when \( x \in\left[x_{1}, x_{4}\right] \) the graph of this function consists of links of a polygonal chain, passing through points \( A\left(x_{1}, y_{1}\right), B\left(x_{2}, y_{2}\right) \), \( C\left(x_{3}, y_{3}\right), D\left(x_{4}, y_{4}\right) \). The coordinates of the points are presented in the table: \begin{tabular}{|c|c|c|c|} \hline\( A \) & \( B \) & \( C \) & \( D \) \\ \hline\( (0,-1) \) & \( (2,2) \) & \( (4,-1) \) & \( (5,-1) \) \\ \hline \end{tabular}

2.6.1.3 Fourier integral

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Problem: Find the Fourier image \( F[f(x)](v) \) \( =\int_{-\infty}^{+\infty} f(t) e^{-i v t} d t \) of function \( f(t) \), using equalities \( F[ \) rect \( t](v)=\operatorname{sinc} \frac{v}{2} \) and \( F[\Lambda(t)](v)=\operatorname{sinc}^{2} \frac{v}{2} \). The graph of function \( f(t) \) consists a polygonal chain, connecting points \( A, B, C \) and \( D \), as well as from parts of the \( \mathrm{x} \)-axis (to the left of \( A \) and to the right of \( D \) ). \begin{tabular}{|c|c|c|c|} \hline\( A \) & \( B \) & \( C \) & \( D \) \\ \hline\( (-1,1) \) & \( (1,-2) \) & \( (4,-2) \) & \( (6,1) \) \\ \hline \end{tabular}

2.6.1.4 Fourier integral

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Problem: Find the Fourier image of function \( f(x) \), if \( f(x) \equiv 0 \) when \( x \notin\left[x_{1}, x_{4}\right] \), and when \( x \in\left[x_{1}, x_{4}\right] \) the graph of this function consists of links of a polygonal chain, passing through points \( A\left(x_{1}, y_{1}\right), B\left(x_{2}, y_{2}\right) \), \( C\left(x_{3}, y_{3}\right), D\left(x_{4}, y_{4}\right) \). The coordinates of the points are presented in the table: \begin{tabular}{|c|c|c|c|} \hline\( A \) & \( B \) & \( C \) & \( D \) \\ \hline\( (-2 ;-1) \) & \( (-1 ;-1) \) & \( (0 ; 0) \) & \( (1 ;-1) \) \\ \hline \end{tabular}

2.6.1.5 Fourier integral

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Problem: Let \( f(x)=e^{-|x|} \cos \omega x \). Calculate the integral: \[ F(y)=\frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{+\infty} f(x) e^{i y x} d x \text {. } \]

2.6.1.6 Fourier integral

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Problem: Represent the following function as a Fourier integral, extending it evenly to the interval \( (-\infty, 0) \). \[ f(x)=\left\{\begin{array}{ll} 1, & \text { if } 0 \leq x \leq 1 \\ 0, & \text { if } x>1 \end{array} .\right. \]

2.6.1.7 Fourier integral

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