The Laplace transform of unit step function is equal to 1/s. As the unit step function is denoted by u(t), its Laplace is given by L{u(t)} = 1/s. Before we learn how to find it, let us recall the definition of the unit step function.

u(t) = 0 if t<0

u(t) = 1 if t≥0.

This means that a unit step function u(t) is a function of t (represents time), where u is 0 when time is negative and 1 when time is positive. The formula of the Laplace of the unit step function u(t) is as follows.

L{u(t)} = 1/s. |

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## Laplace of u(t)

By the definition of Laplace transforms, the Laplace of a function f(t) is given by

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

So the Laplace of the unit step function u(t) is

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

Now by the above definition of u(t), we deduce that

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

= $\Big[ \dfrac{e^{-st}}{-s}\Big]_0^\infty$

= lim_{t→∞} $\dfrac{e^{-st}}{-s}$ + e^{0}/s

= 0 + 1/s as the limit lim_{t→∞} e^{-st} = 0.

= 1/s.

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

**More Laplace Transforms:**

## Laplace of u(t-a)

More generally, the Laplace transform of the shifted unit step function, denoted by u(t-a), is equal to e^{-as}/s, that is,

L{ u(t-a)} = e^{-as}/s.

## FAQs

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

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