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Old Mar 9th 2009, 09:47 AM   #1
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Post Laplace Transformation

Hello,

I am facing problem understanding the following equation I have just faced. It will be very kind of you if you can clarify it to me.

The equation is......

L [f(at)] = 1/a*F(s/a)

I can not find the "F(s/a)" where does it come from? And what does it mean?
I have started Laplace Transform lately and to understand it completely. If you can help me I would be very glad.

Thank you.
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Old Mar 9th 2009, 06:07 PM   #2
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When you take the Laplace transform of a function $\displaystyle f(t) $ of the varible $\displaystyle t $, you get a function $\displaystyle F(s) $ of a different variable $\displaystyle s $.

The rule
$\displaystyle {\cal L} f(at) = \frac {1}{|a|} F(s/a) $
tells you how to find the Laplace transform of $\displaystyle f(at) $ when you only know the Laplace transform of $\displaystyle f(t) $.

For example, given that $\displaystyle {\cal L} \sin(t) = \frac {1}{s^2 + 1} $,
find $\displaystyle {\cal L} \sin(3t) $.

$\displaystyle {\cal L} \sin(3t) = \frac {1}{|3|} \frac {1}{ (s/3)^2 + 1} $
$\displaystyle = \frac {1} {3} \frac {1}{ \frac {s^2}{ 9} + 1} $
$\displaystyle = \frac {3}{9} \frac {1}{ \frac {s^2}{ 9} + 1} $
$\displaystyle = \frac {3}{s^2 + 9} = \frac {3}{s^2 + 3^2} $
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Old Mar 9th 2009, 08:56 PM   #3
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Thanks a lot sir, I was thinking that "F" could be another kind of function. Thank you for your easy solution.

I will be glad if you please send me the derivation. I will try it myself but it will be good for me if you send it.

Thank you.
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