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.

Table of Contents

## State First Shifting Property

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

L{e^{at }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^{at}f(t)}

Thus, we have shown that F(s-a) = L{e^{at}f(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^{2t}sint, that is, find L{e^{2t}sint}.

**Solution:**

By the first shifting property,

L{e^{2t}sint} = 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/(s^{2}+a^{2}).

⇒ F(s-2) = $\dfrac{1}{(s-2)^2+1}$

⇒ F(s-2) = $\dfrac{1}{s^2-4s+5}$

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

**More Laplace:** Laplace Transform of Derivatives

Laplace Transform of Integrals

## FAQs

**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{e^{at }f(t)} = F(s-a) when s>a.