# Exists a Laplace transform property to calculate functions of the type below int_a^x f(x−y)g(y)dy, when a>0?

Laplace transform property for convolutions of the type
${\int }_{0}^{x}f\left(x-y\right)g\left(y\right)\phantom{\rule{thinmathspace}{0ex}}dy$
is very well known. There exists a Laplace transform property to calculate functions of the type below
${\int }_{a}^{x}f\left(x-y\right)g\left(y\right)\phantom{\rule{thinmathspace}{0ex}}dy,$
when a>0?
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iletsa2ym
You should look at the bilateral Laplace transform :

$=\left({\int }_{0}^{\mathrm{\infty }}f\left(t\right){e}^{-st}\phantom{\rule{thinmathspace}{0ex}}dt\right)\left({\int }_{a}^{\mathrm{\infty }}g\left({t}_{2}\right){e}^{-s{t}_{2}}d{t}_{2}\right)={\int }_{a}^{\mathrm{\infty }}h\left(\tau \right){e}^{-s\tau }\phantom{\rule{thinmathspace}{0ex}}d\tau$
where
$h\left(\tau \right)=\left(f\left(t\right){1}_{t>0}\right)\ast \left(g\left(t\right){1}_{t>a}\right)\left(\tau \right)={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f\left(\tau -t\right){1}_{\tau -t>0}g\left(t\right){1}_{t>a}\phantom{\rule{thinmathspace}{0ex}}dt={\int }_{a}^{\tau }f\left(\tau -t\right)g\left(t\right)\phantom{\rule{thinmathspace}{0ex}}dt$