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- https://math.libretexts.org/Bookshelves/Differential_Equations/Elementary_Differential_Equations_with_Boundary_Value_Problems_(Trench)/08%3A_Laplace_Transforms/8.06%3A_ConvolutionThis section deals with the convolution theorem, an important theoretical property of the Laplace transform.
- https://math.libretexts.org/Courses/Cosumnes_River_College/Math_420%3A_Differential_Equations_(Breitenbach)/09%3A_Laplace_Transforms/9.06%3A_Convolution\[\begin{aligned}{\cal L}|\sin t|&={1\over 1-e^{-s\pi}}\int_0^\pi e^{-st}|\sin t|\,dt \\[5pt] &={1\over 1-e^{-s\pi}}\int_0^\pi e^{-st}\sin t\,dt\\[5pt] &={1\over 1-e^{-s\pi}}\left(\int_0^\pi e^{-st}\s...\[\begin{aligned}{\cal L}|\sin t|&={1\over 1-e^{-s\pi}}\int_0^\pi e^{-st}|\sin t|\,dt \\[5pt] &={1\over 1-e^{-s\pi}}\int_0^\pi e^{-st}\sin t\,dt\\[5pt] &={1\over 1-e^{-s\pi}}\left(\int_0^\pi e^{-st}\sin t\,dt+\int_\pi^\infty e^{-st}(0)\,dt\right)\\[5pt] &={1\over 1-e^{-s\pi}}{\cal L}\left(\sin t-\sin t {\cal U}(t-\pi)\right)\\[5pt] &={1\over 1-e^{-s\pi}}{\cal L}\left(\sin t+\sin (t-\pi){\cal U}(t-\pi)\right)\\[5pt] &={1\over 1-e^{-s\pi}}\left({1\over s^2+1}+{e^{-\pi s}\over s^2+1}\right)\\[5pt]…
- https://math.libretexts.org/Bookshelves/Differential_Equations/Introduction_to_Partial_Differential_Equations_(Herman)/09%3A_Transform_Techniques_in_Physics/9.06%3A_The_Convolution_OperationIn the list of properties of the Fourier transform, we defined the convolution of two functions, f(x) and g(x) to be the integral (f∗g)(x). In some sense one is looking at a sum of the overlaps of on...In the list of properties of the Fourier transform, we defined the convolution of two functions, f(x) and g(x) to be the integral (f∗g)(x). In some sense one is looking at a sum of the overlaps of one of the functions and all of the shifted versions of the other function. The German word for convolution is faltung, which means "folding" and in old texts this is referred to as the Faltung Theorem. In this section we will look into the convolution operation and its Fourier transform.
- https://math.libretexts.org/Courses/Chabot_College/Math_4%3A_Differential_Equations_(Dinh)/06%3A_Laplace_Transforms/6.06%3A_ConvolutionThis section deals with the convolution theorem, an important theoretical property of the Laplace transform.
