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6.6:  The Convolution Integral

Defn: The convolution of $f$ and $g$ is the function $f * g$ defined by

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\centerline{$(f * g)(t) = \int_0^t f(t - s)g(s)ds = \int_0^t f(x)g(t - x)dx  $}

Note $*$ is

1.)  commutative: $f * g = g * f$

2.) associative:  $(f*g)*h = f*(g*h)$

3.)  distributive w.r.t $+$:  $f*(g_1 + g_2) = f*g_1 + f* g_2$

4.)  $f*0 = 0*f = 0$

Example:  $cos(t) * 1 =$

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Example:  $sin(t) * sin(t) \not\geq 0$

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Thm: ${\cal L}((f*g)(t)) = {\cal L}(f(t)){\cal L}(g(t))$
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Proof:
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${\cal L}(f(t)){\cal L}(g(t)) 
= \int_0^\infty e^{-sy}f(y)dy \int_0^\infty e^{-sx}g(x)dx$
\s
$= \int_0^\infty [\int_0^\infty e^{-sy}f(y)dy] e^{-sx}g(x)  dx$
\s

$= \int_0^\infty [\int_0^\infty e^{-sy}f(y) e^{-sx}g(x) dy] dx$
\s


$= \int_0^\infty [\int_0^\infty e^{-s(y + x)}f(y) g(x) dy] dx$
\s

$= \int_0^\infty [\int_0^\infty e^{-s(y + x)}f(y) g(x) dx] dy$
\s


Let $t = x + y$, $dt = dx$
\s

$= \int_0^\infty [\int_y^\infty e^{-s(y + t - y)}f(y) g(t-y) dt] dy$
\s

$= \int_0^\infty [\int_y^\infty e^{-st}f(y) g(t-y) dt] dy$
\s

$= \int_0^\infty [\int_0^t e^{-st}f(y) g(t-y) dy] dt$
\s

$= \int_0^\infty e^{-st} [\int_0^t f(y) g(t-y) dy] dt$
\s

$= \int_0^\infty e^{-st} (f*g)(t) dt$
\s

$= {\cal L}(f*g)$

Example:  ${\cal L}^{-1}({1 \over s (s - a)}) = $


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