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.

Statement of First Shifting Property

The first shifting property of Laplace states that if L{f(t)} = F(s), then for s>a we have L{e^at f(t)} = F(s-a).

Proof of First Shifting Property

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

Questions and Answers

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

FAQs