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behzad24b
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 .
\[L \left\{ e ^{ax}f(x) \right\} = F(s-a)\]
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 \(s\).
\[\begin{align*} \mathcal L \big\{ e ^{5x}f(x) \big\}\\ &=\int\limits_0^\infty e^{5x}f(x)e^{-sx}\mathrm dx\\ &=\int\limits_0^\infty f(x)e^{-(s-5)x}\mathrm dx\\ \text{let }s-5=p\\ &=\int\limits_0^\infty f(x)e^{-px}\mathrm dx\\ &=F(p)\\ p=s-5\\ &= F(s-5)\end{align*}\]