The Laplace transform of u(t-1) is equal to e^{-s}/s, that is, L{u(t-1)} = e^{-s}/s. Note that u(t-1) is the shifted unit step function by 1 and it is defined as follows.

u(t-1) = 0 if t<1

u(t-1) = 1 if t≥1.

Let us now learn how to find the Laplace transform of u(t-1).

Table of Contents

## Laplace of u(t-1)

We will find the Laplace of u(t-1) by definition. The definition of Laplace transforms says that the Laplace of a function f(t) is given by the following integral formula

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

So the Laplace of u(t-1) is

L{u(t-1)} = $\int_0^\infty$ e^{-st} u(t-1) dt.

By the definition of u(t-1) given above, this integral will be equal to

L{u(t-1)}= $\int_1^\infty$ e^{-st} dt

= $\Big[ \dfrac{e^{-st}}{-s}\Big]_1^\infty$

= lim_{t→∞} $\dfrac{e^{-st}}{-s}$ + e^{-s}/s

= 0 + e^{-s}/s as we know that lim_{t→∞} e^{-st} = 0.

= e^{-s}/s.

So the Laplace transform of u(t-1) is equal to e^{-s}/s, that is, L{u(t-1)} = e^{-s}/s, and this is obtained by the definition of Laplace transforms.

**More Laplace Transforms:**

Laplace of unit step function, L{u(t)}

## FAQs

**Q1: What is the Laplace transform of u(t-1)?**

Answer: The Laplace transform of u(t-1) is e^{-s}/s, that is, L{u(t-1)} = e^{-s}/s.

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.