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Laplace Transform Definition

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aryajur

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There are 2 definitions of Laplace transform specified - the unilateral one in which the lower limit is 0- and the bilateral one in which the lower limit is -∞. Wikipedia says that generally when we say Laplace transform we mean the unilateral one, and the bilateral one is a general case and the unilateral one is the special case for that.
My question is that the properties of Laplace transformation, especially the differentiation, how is it derived for the bilateral one? Its written that for bilateral definition the differentiation property is:

L[f'(t)] = s F

But I cannot derive it. Any help would be appreciated.
 

f(t) <==> F(s) = ∫f(t)*exp{-st} dt

g(t) = f '(t) <==> G(S) = ∫g(t)*exp{-st} dt = ∫f '(t) *exp{-st} dt = (apply the Leibniz formula) = f(t)*exp{-st} - ∫f(t)*d/dt [exp{-st}] dt

Remember that: d/dt [exp{-st}] = -s *exp{-st}

The term f(t)*exp{-st} valueted between the integral extremes gives the initial condition. Don't considering this term You have:

G(s) = - ∫f(t)*d/dt[exp{-st}] dt = - ( -s *∫f(t) *exp{-st} dt ) = s F(s) ==> L [ f '(t)] = s F(s)


Oss. Leibniz formula: d/dt [x(t)*y(t)] = x '(t)*y(t) + x(t)*y '(t) ==> x '(t)*y(t) = d/dt [x(t)*y(t)] - x(t)*y '(t)

Taken the integral of first and second member You have: ∫x '(t)*y(t) dt = x(t)*y(t) - ∫x(t)*y '(t) dt
 

Thanks for the reply, but the problem is that 1st term when you put the integration limits. If you define the laplace transform limits from 0- to infinity then the 1st term reduces to f(0-) but when your laplace transform limits are -infinity to +infinity then your 1st term cannot be evaluated.
So my main question is why did you loose the inital condition term? That is included in the derivative formula when the laplace transform is defined to be unilateral, so it should be evaluated when the laplace transform is defined to be bilateral.
 

When you use Bilateral Laplace transform there is no sense in refering to inicial condition since you start from minus infinite, so to have a initial condition it should be in a time minus infinite ago, wich cleary is not possible. On the other hands, when you analise a unilateral one you start from 0+ in this way you need to know, which were the conditions before you start.

Proving the bilateral laplace transform properties is very simililar to prove Fourier Transform properties.
 

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    aryajur

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