Find the Laplace transform,
F
(
s
)
=
∫
0
∞
1
e
−
s
t
d
t
\displaystyle {F}{\left({s}\right)}={\int_{{0}}^{{\infty}}}{1}{e}^{{-{s}{t}}}{\left.{d}{t}\right.}
F
(
s
)
=
∫
0
∞
1
e
−
s
t
d
t
of the function
f
(
t
)
=
1
\displaystyle {f{{\left({t}\right)}}}={1}
f
(
t
)
=
1
1
1
−
s
,
∣
s
∣
<
1
\displaystyle \frac{{1}}{{{1}-{s}}},\quad{\left|{s}\right|}<{1}
1
−
s
1
,
∣
s
∣
<
1
1
s
,
s
>
0
\displaystyle \frac{{1}}{{s}},\quad{s}>{0}
s
1
,
s
>
0
1
s
−
1
,
s
>
0
\displaystyle \frac{{1}}{{s}}-{1},\quad{s}>{0}
s
1
−
1
,
s
>
0
1
s
−
1
,
∣
s
∣
>
1
\displaystyle \frac{{1}}{{{s}-{1}}},\quad{\left|{s}\right|}>{1}
s
−
1
1
,
∣
s
∣
>
1
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