$ ( a ) Â \dfrac { 1 } { s } + 2 $

$ ( b ) \dfrac { 1 } { s } \: – \: 2 $

$ ( c ) Â \dfrac { e ^ { 2 s } } { s } $

$ ( d ) \dfrac {e ^ { – 2 s } } { s } $

This**Â article aims**Â to find the **Laplace transform**Â of a **given function**. The **article uses the concept**Â of how to find the **Laplace transform**Â of the step function. The reader should know the basics of **Laplace transform.**

In mathematics, **Laplace transform**, named after its **discoverer Pierre-Simon Laplace**, is an integral transformation that converts function of a real variable (usually $ t $, in the time domain) to a part of a complex variable $ s $ (in the complex frequency domain, also known as $ s $-domain or s-plane).

The transformation has many applications in **science and engineering**Â because it is a tool for solving differential equations. **In particular**, it converts ordinary differential equations to **algebraic equations and convolution to multiplication. **

For any given function $ f $, the Laplace transform is given as

\[F ( s ) = \int Â _ { 0 } ^ { \infty } f ( t ) e ^ { – s Â t } dt\]

**Expert Answer**

**We know that **

\[ L ( u ( t ) ) Â = Â \dfrac { 1 } { s } \]

By $ t $ **shifting theorem**

\[ L ( u ( t – 2 ) ) = e ^ { – 2 s } L ( u ( t ) ) Â = \dfrac Â { e ^ { – 2 s } } { s } Â \]

**Option $ d $ is correct**.

**Numerical Result**

The **Laplace transform**Â of $ u( t – 2 ) $ is $ \dfrac { e ^ { – 2 s } } { s } $.

**Option $ d $ is correct.**

**Example**

**What is the Laplace transform of $ u ( t – 4 ) $?**

**$ ( a ) Â \dfrac { 1 } { s } + 4 $**

**$ ( b ) \dfrac { 1 } { s } \: – \: 4 $**

**$ ( c ) Â \dfrac { e ^ { 4 s } } { s } $**

**$ ( d ) \dfrac {e ^ { – 4 s } } { s } $**

**Solution**

\[ L ( u ( t ) ) Â = Â \dfrac { 1 } { s } \]

By $ t $ **shifting theorem**

\[ L ( u ( t – 4 ) ) = e ^ { – 4 s } L ( u ( t ) ) Â = \dfrac Â { e ^ { – 4 s } } { s } Â \]

\[ L ( Â u ( Â t – 4 ) ) = \dfrac Â { e ^ { – 4 s } } { s } \]

**Option $ d $ is correct**.

The **Laplace transform** of $ u( t – 4 ) $ is $ \dfrac Â { e ^ { – 4 s } } { s }$.