Trigonometric Fourier series

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Problem: Expand function \( f(x)=\frac{x^{2}-1}{2} \) into a trigonometric Fourier series on the line segment \( [-\pi ; \pi] \). Plot the graphs of function \( f(x) \) and the sum of the Fourier series. Write down the first three partial sums of the Fourier series and approximately plot their graphs.

2.6.2.1 Trigonometric Fourier series

5.08 $

Problem: The periodic function is given: \[ f(x)=\left\{\begin{array}{ll} x, & 0 \leq x \leq 1 \\ 1, & 1

2.6.2.2 Trigonometric Fourier series

6.35 $

Problem: Expand function \( f(x) \) into a trigonometric Fourier series: \[ f(x)=\cos \alpha x, \quad-\pi

2.6.2.3 Trigonometric Fourier series

1.27 $

Problem: Expand functon \( f(x) \) into a trigonometric Fourier series: \[ f(x)=\left\{\begin{array}{cc} 0, & -6

2.6.2.4 Trigonometric Fourier series

2.54 $

Problem: Expand function \( f(x) \) in a Fourier series in the form of a superposition of simple harmonics, given on the segment line \( [-T / 2, T / 2] \). Plot the amplitude and phase spectra. The values of parameters \( T, h, p \) and \( q \) are given in the table: \begin{tabular}{|c|c|c|c|} \hline\( T \) & \( h \) & \( p \) & \( q \) \\ \hline 2 & 2 & -2 & 1 \\ \hline \end{tabular} \( f(x)=\left\{\begin{array}{cc}h-\frac{2 h}{T} x, & -T / 2 \leq x<0 \\ p, & 0 \leq x

2.6.2.5 Trigonometric Fourier series

5.08 $

Problem: Expand function \( f(x) \) in a Fourier series in the form of a superposition of simple harmonics, given on the segment line \( [-T / 2, T / 2] \). Plot: 1. Amplitude and phase spectra; 2. Graphs of partial sums of the Fourier series \( S_{3}(x), S_{10}(x), S_{20}(x), S_{100}(x) \). The values of parameters \( T, h, p \) and \( q \) are given in the table:

2.6.2.6 Trigonometric Fourier series

6.35 $

Problem: Expand function \( f(x) \) into Fourier series in the form of a superposition of simple harmonics, given on the segment line \( [-T / 2, T / 2] \). Plot: 1. Amplitude and phase spectra; 2. Graphs of partial sums of the Fourier series \( S_{3}(x), S_{10}(x), S_{20}(x), S_{100}(x) \). The values of parameters \( T, h, p \) and \( q \) are given in the table: \begin{tabular}{|c|c|c|c|} \hline\( T \) & \( h \) & \( p \) & \( q \) \\ \hline 2 & -2 & 2 & -1 \\ \hline & \( p \), & \begin{tabular}{l} \( / 2 \leq \) \\ \( -T / 4 \) \end{tabular} & \\ \hline & \( 2-\frac{2 l}{T} \) & & \\ \hline \end{tabular}

2.6.2.7 Trigonometric Fourier series

6.35 $

Problem: Expand function \( f(x) \) into the trigonometric Fourier series in cosines. 1) \( f(x)=x^{3} \) in \( [0 ; 1] \), 2) \( f(x)=\sin 2 x \) in \( [0 ; 1] \), 3) \( f(x)=\frac{\pi}{4}-\frac{x}{2} \) in \( [0 ; 1] \).

2.6.2.8 Trigonometric Fourier series

3.81 $

Problem: Calculate coefficient \( a_{3} \) in the expansion of function \( f(x) \) in the trigonometric Fourier series \[ f(x)=\left\{\begin{array}{lr} 0, & -2

2.6.2.9 Trigonometric Fourier series

0.51 $

Problem: Expand function \[ f(x)=\frac{\pi}{4}-\frac{x}{2}, \] given in interval \( (0 ; \pi) \), into Fourier series in sines. Plot the graph of the sum of Fourier series.

2.6.2.10 Trigonometric Fourier series

1.78 $

Problem: Expand function, given in interval \( x \in[-\pi ; \pi] \) into Fourier series. \[ f(x)=\left\{\begin{array}{cc} x^{2}, & 0

2.6.2.11 Trigonometric Fourier series

2.54 $

Problem: Expand the function, given in interval \( x \in[0 ; \pi] \), by cosines and sines of multiple arcs. \[ f(x)=\frac{\pi}{4}-\frac{x}{2} \text {. } \]

2.6.2.12 Trigonometric Fourier series

2.54 $

Problem: Expand function \( f(x) \) into the trigonometric Fourier series in \( [-\pi ; \pi] \), continuing function in \( [-\pi ; 0] \) evenly. \[ f(x)=\left\{\begin{array}{ll} x-\frac{\pi}{2}, & 0 \leq x \leq \frac{\pi}{2} \\ 2 x-\pi, & \frac{\pi}{2} \leq x \leq \pi \end{array}\right. \]

2.6.2.13 Trigonometric Fourier series

2.54 $

Problem: a) Expand function \( y=f(x) \), given on the halfcycle \( (0 ; l) \), into Fourier series in cosines. Plot the graphs of the 2nd and 3rd partial sums. Write Parseval's equality for the resulting series. b) Expand function \( y=f(x) \), given on the halfcycle \( (0 ; l) \), into Fourier series in cosines. Plot the graphs of the 2nd and 3rd partial sums. c) Expand function \( y=f(x) \) into Fourier series, continuing it on the half-cycle \( (0 ; l) \) with a function, equal to zero. Construct the graphs of the second and the fourth partial sums. \[ y=1-x, \quad l=4 \]

2.6.2.15 Trigonometric Fourier series

7.62 $

Problem: Extend function \( f(x)=x^{3} \) into Fourier series in cosines on the segment line \( [0 ; 1] \).

2.6.2.16 Trigonometric Fourier series

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