The second shifting property of Laplace transforms is used to find the Laplace of a function multiplied by a unit step function. Let us discuss the second shifting property, its proof and some solved examples.

Before we proceed, let us recall the definition of a unit step function. It is defined as follows:

u(t-a) = $\begin{cases} 1 & \text{ if } t>a \\ 0 & \text{ if } t < a \end{cases}$

Table of Contents

## Second Shifting Property Statement

Let L{f(t)} = F(s). For a>0 we have

L{f(t-a) u(t-a)} = e^{-as} F(s) |

**Proof:**

As L{f(t)} = F(s), by definition of Laplace transforms we have that

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

Now, L{f(t-a) u(t-a)}

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

= $\int_a^\infty$ e^{-st} f(t-a) dt, by the definition of unit step function.

We now make a change of variable. Let us put

z=t-a.

So dz=dt.

t | z |

a | 0 |

∞ | ∞ |

Therefore, from above we obtain that

L{f(t-a) u(t-a)} = $\int_0^\infty$ e^{-s(z+a)} f(z) dz

= e^{-as} $\int_0^\infty$ e^{-sz} f(z) dz

= e^{-as} F(s).

This proves the second shifting property of Laplace transforms whose formula states that L{f(t-a) u(t-a)} = e^{-as} F(s) for a>0.

**Read Also: **First shifting property of Laplace transforms

Laplace Transform: Definition, Table, Formulas, Properties

## Solved Examples

**Question:** Find the Laplace of (t-2) u(t-2), that is, find L{(t-2) u(t-2)}.

**Solution:**

Put a=2 in the second shifting property. Thus,

L{(t-2) u(t-2)} = e^{-2s} F(s)

where F(s)= L{t} = 1/s^{2}.

So, L{(t-2) u(t-2)} = $\dfrac{e^{-2s}}{s^2}$

So the Laplace of (t-2) u(t-2) is equal to e^{-2s}/s^{2}.

Laplace Transform of Derivatives | Laplace Transform of Integrals

## FAQs

**Q1: What is the second shifting property of Laplace transforms?**

Answer: The second shifting property of Laplace transforms states that if L{f(t)} = F(s) then L{f(t-a) u(t-a)} = e^{-as} F(s) for a>0. Here u(t-a) is the unit step function.

This article is written by Dr. T. Mandal, Ph.D in Mathematics. On Mathstoon.com you will find Maths from very basic level to advanced level. Thanks for visiting.