Laplace Transform

\(\mathscr L{f(t)} = F(s)\)
\(f(t)\)	\(F(s)\)	Condition on s
a	\(\frac{a}{x}\)	\(s\ >\ 0\)
\(t^n,\ n = 0,1,2...\)	\(\frac{n!}{s^{n+1}}\)	\(s\ >\ 0\)
\(e^{at}\)	\(\frac{1}{s-a}\)	\(s\ >\ a\)
\(sin\ at\)	\(\frac{a}{s^2 + a^2}\)	\(s\ >\ 0\)
\(cos\ at\)	\(\frac{s}{s^2 + a^2}\)	\(s\ >\ 0\)
\(sinh\ at\)	\(\frac{a}{s^2 - a^2}\)	\(s\ >\ \mid a \mid\)
\(cosh\ at\)	\(\frac{s}{s^2 - a^2}\)	\(s\ >\ \mid a \mid\)

First Shifting Proferty

If \(\mathscr L {f(t)} = F(s)\), then \(\mathscr L(e^{at}f(t)) = F(s-a)\)

Multiplication by \(t^n\) Property

If \(\mathscr L {f(t)} = F(s)\), then \(\mathscr L(t^n\ f(t)) = \frac{d^nF}{ds^n},\ n = 1,2,3...\)

Laplace Transform of Unit Step Functions, \(\mathscr L{H(t-a)}\)

\(\mathscr L(H(t-a)) = \frac{e^{-as}}{s},\ a\ >\ 0\)

Laplace Transform of \(\mathscr L{f(t-a)H(t-a)}\)

Second shifting property:

\(\mathscr L{f(t-a)H(t-a)} = e^{-as}\ \mathscr L {f(t)} = e^{-as}F(s)\)

Laplace Transform of Dirac Delta Function \(\mathscr L(\delta(t-a))\)

\(\mathscr L (\delta (t)) = 1\)

\(\mathscr L (\delta (t-a)) = e^{-as},\ a\ >\ 0\)

\(\mathscr L (f(t) \delta (t-a)) = e^{-as}f(a),\ a\ >\ 0\)

Inverse Laplace Transform

\(\mathscr L^{-1}(F(s-a)) = e^{at} f(t)\)	\(\mathscr L^{-1}\frac({e^{-as}}{s}) =H(t-a)\)
\(\mathscr L^{-1}(a^{-as}F(s)) = f(t-a)H(t-a)\)	\(\mathscr L^{-1}(e^{-as}) =\delta(t-a)\)
\(\mathscr L^{-1}(e^{-as}f(a)) =f(t)\delta(t-a)\)

Convolution Theorem

|—| |\(\mathscr L F(s)G(s) = \int_{0}^{t} f(u)g(t-u)\ du\)|