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)

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 {eat f(t)} dt

= L{eatf(t)}

Thus, we have shown that F(s-a) = L{eatf(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 e2tsint, that is, find L{e2tsint}.

By the first shifting property,

L{e2tsint} = 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{e2tsint} = $\dfrac{1}{s^2-4s+5}$.

More Laplace: Laplace Transform of Derivatives

Laplace Transform of Integrals


Q1: What is the first shifting property of Laplace transforms?

Answer: The first shifting property of Laplace transforms states that if L{f(t)} = F(s) then L{eat f(t)} = F(s-a) when s>a.

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