Fourier series: Coefficients of Fourier series
Calculation of Fourier coefficients
Calculate the Fourier series of the #2\pi#-periodic function #f# determined by \[f(x)=\left\{\begin{array}{l cl} 0& \phantom{x}\text{if} & -\pi \le x\lt 0 \\ 4 x & \phantom{x}\text{if}& 0\leq x\lt\pi\end{array}\right.\]
A plot of the function \( f \) over three periods is given in the figure below.
Enter simplified expressions for the coefficients \(a_0\), \(a_k\), and \(b_k\) \( (k=1,2,\dots)\), where the Fourier series is given by \[s(x)=\frac{a_0}{2}+\sum_{k=1}^{\infty}\left(a_k\cdot\cos\left(k\cdot x\right)+b_k\sin\left(k\cdot x\right)\right)\]
A plot of the function \( f \) over three periods is given in the figure below.
Enter simplified expressions for the coefficients \(a_0\), \(a_k\), and \(b_k\) \( (k=1,2,\dots)\), where the Fourier series is given by \[s(x)=\frac{a_0}{2}+\sum_{k=1}^{\infty}\left(a_k\cdot\cos\left(k\cdot x\right)+b_k\sin\left(k\cdot x\right)\right)\]
| \(a_0=\) |
| \(a_k=\) | for \(k=1,2,3,\dots\) |
| \(b_k=\) | for \(k=1,2,3,\dots\) |
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