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I'm a computer engineering student. We often use the convolution integral but in our engineering textbooks the problems are often toy examples that are best solved via graphical trickery. I would like a deeper understanding of the mathematics behind this integration technique. I tutor mathematics and have gone a bit further than my own course offerings to gain a deeper understanding of the techniques we've used and would like to do that here.

I've looked into a few books (ie Theory and applications of the convolution integral-Srivastava) and the only ones I've found assume I've taken a few semesters of real analysis which I haven't and sadly won't be able to. I've been told that differential equations books might cover the topic in a more rigorous way but none that I own do cover it.

jake mckenzie
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  • Do you have a concrete example of something you don't fully understand ? – reuns Dec 10 '17 at 23:59
  • How do I pick my algebraic boundaries without graphical approaches? I was having a very hard time grasping what was going on with the operator learning how engineers use the method. For instance. To convolve E^(-t)U(t) with U(t)-U(t-1) you can split it up into two integrations: one from 0 to t and another from t-1 to t. As the functions get less uniform how do I pick my integration boundaries with algebra. I assume there must be a differential equations book with problems like this but I have not found it. – jake mckenzie Dec 12 '17 at 07:27
  • In your example $f(t) = e^{-t} 1_{t >0}, g(t) = 1_{t > 0}- 1_{t > 1} = 1_{t \in (0,1)}$ then $f \ast g(t) = \int_{-\infty}^\infty f(u) g(t-u)du = \int_{-\infty}^\infty e^{-u} 1_{u >0} 1_{t-u\in(0,1)}du$ $ = \int_0^\infty e^{-u} 1_{t-u\in(0,1)}du= \int_0^\infty e^{-u} 1_{u-t\in(-1,0)}du$ $=\int_0^\infty e^{-u} 1_{u\in(t-1,t)}du =\int_{t-1}^{t} e^{-u}du = e^{-t}-e^{-(t-1)}$ – reuns Dec 12 '17 at 07:33

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