Question

can anyone say one or more samples for this Laplace tarnsform (with step by step solving) :
L{(e^ax) f(x)} = F(s-a)
and what means s-a exactly ? i cant understand that .

Answer 1

\[L \left\{ e ^{ax}f(x) \right\} = F(s-a)\]

Answer 2

When dealign with Laplace transforms, it's useful to think of a *transformation* from the time-domain (t represents a time) to the *frequency-domain* (s represents a complex angular frequency, r e^(it))... in this case, multiplying by \(e^{ax}\) translates to a shift of the frequency \(s\).

Answer 3

\[\begin{align*}
\mathcal L \big\{ e ^{5x}f(x) \big\}\\
&=\int\limits_0^\infty e^{5x}f(x)e^{-sx}\mathrm dx\\
&=\int\limits_0^\infty f(x)e^{-(s-5)x}\mathrm dx\\
\text{let }s-5=p\\
&=\int\limits_0^\infty f(x)e^{-px}\mathrm dx\\
&=F(p)\\
p=s-5\\
&= F(s-5)\end{align*}\]