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I know the formula for a convolution integral but how would you actually carry out one when you have two piece-wise defined functions? If you had

$$ f(x) = \left\{ \begin{array}{ll} e^{y} & \quad {-\infty}<x < 0 \\ e^{-y} & \quad 0<x < {\infty} \\ \end{array} \right. $$

and

$$ g(x) = \left\{ \begin{array}{ll} 1 & \quad |x| \leq 1 \\ 0 & \quad otherwise \end{array} \right. $$ ?

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Hint : $f\ast g(x)=\int _\mathbb{R}f(x-y)g(y)dy=\int _{-1}^1 f(x-y)dy=\int _{-1}^xe^{y-x}dy+\int _x^1 e^{x-y}dy= \ldots $

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