First Shifting Property of Laplace Transforms: Formula, Proof

The first shifting property of Laplace transforms is used to find the Laplace of a function multiplied by an exponential function. Here we discuss the first shifting property along with its proof and solved examples.

State First Shifting Property

Statement: If L{f(t)} = F(s), then for s>a we have

L{eat f(t)} = F(s-a)

Proof:

As L{f(t)} = F(s), by definition

F(s) = $\int_0^\infty$ e^-st f(t) dt.

Now, F(s-a)

= $\int_0^\infty$ e^-(s-a)t f(t) dt

= $\int_0^\infty$ e^-st {e^at f(t)} dt

= L{e^atf(t)}

Thus, we have shown that F(s-a) = L{e^atf(t)}, and this is the formula for the first property of Laplace transforms.

Read: Laplace Transform: Definition, Table, Formulas, Properties

Solved Examples

Question1: Find the Laplace of e^2tsint, that is, find L{e^2tsint}.

Solution:

By the first shifting property,

L{e^2tsint} = F(s-2) where F(s)= L{sint}.

Now, F(s)= L{sint} = $\dfrac{1}{s^2+1}$ as the Laplace of sinat is a/(s²+a²).

⇒ F(s-2) = $\dfrac{1}{(s-2)^2+1}$

⇒ F(s-2) = $\dfrac{1}{s^2-4s+5}$

So from above, L{e^2tsint} = $\dfrac{1}{s^2-4s+5}$.

More Laplace: Laplace Transform of Derivatives

Laplace Transform of Integrals

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