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behzad24b
 2 years ago
can anyone say one or more samples for this Laplace tarnsform (with step by step solving) :
L{(e^ax) f(x)} = F(sa)
and what means sa exactly ? i cant understand that .
behzad24b
 2 years ago
can anyone say one or more samples for this Laplace tarnsform (with step by step solving) : L{(e^ax) f(x)} = F(sa) and what means sa exactly ? i cant understand that .

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behzad24b
 2 years ago
Best ResponseYou've already chosen the best response.0\[L \left\{ e ^{ax}f(x) \right\} = F(sa)\]

oldrin.bataku
 2 years ago
Best ResponseYou've already chosen the best response.1When dealign with Laplace transforms, it's useful to think of a *transformation* from the timedomain (t represents a time) to the *frequencydomain* (s represents a complex angular frequency, r e^(it))... in this case, multiplying by \(e^{ax}\) translates to a shift of the frequency \(s\).

UnkleRhaukus
 2 years ago
Best ResponseYou've already chosen the best response.1\[\begin{align*} \mathcal L \big\{ e ^{5x}f(x) \big\}\\ &=\int\limits_0^\infty e^{5x}f(x)e^{sx}\mathrm dx\\ &=\int\limits_0^\infty f(x)e^{(s5)x}\mathrm dx\\ \text{let }s5=p\\ &=\int\limits_0^\infty f(x)e^{px}\mathrm dx\\ &=F(p)\\ p=s5\\ &= F(s5)\end{align*}\]
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