Question

# Sketch the given function and represent it as indicated. If you have a CAS, graph approximate curves obtained by replacing ∞ with finite limits; also look for Gibbs phenomena. f(x)=x+1 if 0<x<1 and 0 otherwise; by the Fourier sine transform

Solution

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We shall use the following formula:

\begin{align*} \hat{f_s}(x)&=\sqrt{ \frac{2}{\pi} } \int_{0}^{\infty} f(x)\sin wx \,dx \\[7pt] &=\sqrt{ \frac{2}{\pi} } \left( \int_{0}^{1} (x+1)\sin wx \,dx \right) \\[7pt] &=\sqrt{ \frac{2}{\pi} } \left( -\frac{1}{w}\cos \left(wx\right)-\frac{1}{w}x\cos \left(wx\right)+\frac{1}{w^2}\sin \left(wx\right) \right)\Bigg|_0^1 \\[7pt] &=\sqrt{ \frac{2}{\pi} } \left( -\frac{2}{w}\cos \left(w\right)+\frac{1}{w}+\frac{1}{w^2}\sin \left(w\right) \right) \\[7pt] &= \boxed{ \color{#500050} \sqrt{ \frac{2}{\pi} } \frac{\sin w -2w\cos w + w}{w^2}} \end{align*}

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