Translation PropertyThe translation formula states that Y(s) is the Laplace transform of y(t), then

where a is a constant. Here is an example. The Laplace transform of they(t)=t is Y(s)=1/s^2. Hence

Example:

1) L{e2t t}

Since, f(t) = t, F(s) = L{t} = 1/s2, s > 0.Then, L{e2t t} = F(s 2) = 1(s ! 2)2 , s > 2.

Derivatives of the Laplace TransformLet Y(s) be the Laplace Transform of y(t). Then

EXAMPLE:

Suppose we wish to compute the Laplacetransform of tsin(t). The Laplace transform of sin(t) is 1/(s^2+1).Hence, we have

s-Shift:This property is the Laplace transform corresponds to the frequency shift property of the Fourier transform. In fact, the derivation of thes-shift property is virtually identical to that of the frequency shift property.

L{eatx(t)}====0eatx(t)estdt0x(t)e(a+s)tdt0x(t)e(a++j)tdtX(s+a)

Thes-shift property also alters the region of convergence of the Laplace transform. If the region of convergence forX(s)is>min, then the region of convergence forL{eatx(t)}is>minRe(a).

TIME SHIFTING PROPERTY OF THE LAPLACE TRANSFORM:Delaying x(t) by t0 amount of time is called the time shifting property of Laplace transformation. The target is achieved by multiplying its transform X(s) by e-stoX(t) X(s)For t0>=0X(t-t0) X(s)e-stoNote that x(t) starts at t=0 and x(t-t0) starts at t=t0.Therefore the more accurate statement of the time scaling property is X(t)u(t) X(s)X(t-t0) u(t-t0) X(s)e-sto t0>=0

TimeDelay

The time delay property is not much harder to prove, but there are some subtleties involved in understanding how to apply it. We'll start with the statement of the property, followed by the proof, and then followed by some examples. The time shift property states

We again prove by going back to the original definition of the Laplace Transform

Because

we can change the lower limit of the integral from 0-to a-and drop the step function (because it is always equal to one)

We can make a change of variable

The last integral is just the definition of the Laplace Transform, so we have the time delay property

To properly apply the time delay property it is important that both the function and the step that multiplies it are both shifted by the same amount. As an example, consider the function f(t)=t(t). If we delay by 2 seconds it we get (t-2)(t-2), not (t-2)t(t) or t(t-2)