Laplace Transform: Order of transformation

[libelec]

Homework Statement

Find the Laplace transform for $$\[t\left( {\int\limits_0^{t - a} {f(u)du} } \right)H(t - a)\]$$ proving the properties used.

Homework Equations

If f(t) transforms into F(s):

a) $$\[{( - t)^{n}f(t) \to {F^{(n)}}(s)\]$$

b) $$\[\int\limits_0^t {f(u)du} \to \frac{{F(s)}}{s}\]$$

c) $$\[f(t - a)H(t - a) \to {e^{ - as}}F(s)\]$$

The Attempt at a Solution

Here's my question: in what order should I use these properties? There's only one posibility for the translation, and that's that the transform is multiplied by e^-as. But with the derivative of F(s) and the division for s there are two posibilities:

1) That the Laplace transform of $$\[t\left( {\int\limits_0^{t - a} {f(u)du} } \right)H(t - a)\]$$ be $$\[\frac{{ - {e^{ - as}}{F^'}(s)}}{s}\]$$.

2) That the Laplace transform be $$\[ - {e^{ - as}}{\left[ {\frac{{F(s)}}{s}} \right]^'}\]$$.

What's the correct one and why? I couldn't find a reason to use either one but not the other.

EDIT: In property a), the lineal variable t should have an exponent without brackets, since it's not the n-th derivative but the n-th power.

 

[mighty2000]

The correct approach is to first use property c) to transform \[f(u)H(t-a)\] into \[e^{-as}F(s)\]. Then, use property b) to transform the integral \[\int_0^{t-a}e^{-as}F(s)ds\] into \[\frac{e^{-as}}{s}F(s)\]. Finally, use property a) to transform the t in front of the integral into \[\frac{d}{ds}\left(\frac{e^{-as}}{s}F(s)\right)\]. This will give you the Laplace transform of \[t\left(\int_0^{t-a}f(u)du\right)H(t-a)\] as \[\frac{d}{ds}\left(\frac{e^{-as}}{s}F(s)\right)\].

The reason why we use property b) first and then property a) is because the integral \[\int_0^{t-a}f(u)du\] is inside the function t, so we need to transform it first before we can use property a). And the reason why we use property c) is because the function \[f(u)H(t-a)\] has a shifted argument, so we need to use the property that deals with shifted arguments first.

In general, when using Laplace transforms, it is important to pay attention to the order in which the properties are used. This will help ensure that the correct transformation is obtained.