The Laplace transform of the exponential function e to the power at is 1/(s-a). In this article, we will learn how to prove this Laplace transform formula of exponential functions.

## Laplace Transform of e^{at} Formula

The formula of the Laplace transform of e^{at }is 1/(s-a). Mathematically, we write it as

L{e^{at}} = 1/(s-a).

## e^{at }Laplace Transform

The Laplace transform of e^{at }is equal to 1 divided by (s-a). That is,

L{e^{at}} = 1/(s-a).

*Proof:*

Recall, the definition of the Laplace transform of a function f(t).

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

Put f(t)=e^{at}.

∴ L{e^{at}} = $\int_0^\infty$ e^{-st} e^{at} dt

= $\int_0^\infty$ e^{-st+at} dt

= $\int_0^\infty$ e^{-(s-a)t} dt

= lim_{T→∞} $\left[\dfrac{e^{-(s-a)t}}{-(s-a)}\right]_0^T$

= lim_{T→∞} $\left(\dfrac{e^{-(s-a)T}}{-(s-a)}-\dfrac{1}{-(s-a)}\right)$

= 0 – $\dfrac{1}{-(s-a)}$

= $\dfrac{1}{s-a}$

So, the Laplace transform of e^at is 1/(s-a).

Now, replacing a with -a, we obtain the Laplace transform of e^-at which is 1/(s+a).

**Summary:** Denoting the Laplace transform of f(t) by L{f(t)}, we list the Laplace transform formulas for the exponential functions in the below table:

1 | L{e^{at}} | 1/(s-a) |

2 | L{e^{-at}} | 1/(s+a) |

3 | L{e^{t}} | 1/(s-1) |

4 | L{e^{-t}} | 1/(s+1) |

**Read Also:**

**Concept of Laplace Transform: Definition, Table, Formulas, Properties & Examples**

**Laplace transform of sin(at)**: The Laplace transform of sin(at) is a/(s^{2}+a^{2}).

**Laplace transform of cos(at)**: The Laplace transform of cos(at) is s/(s^{2}+a^{2}).

**Laplace transform of constant**: The Laplace transform of c is c/s.

**Inverse Laplace transform of constant**: The inverse Laplace transform of c is cδ(t), where δ(t) is the Dirac delta function.

**Question:** Find the Laplace transform of e^{at}+e^{-at}.

*Solution:*

By the linearity property of Laplace transform, we have

L{e^{at}+e^{-at}} = L{e^{at}} + L{e^{-at}}

= 1/(s-a) + 1/(s+a) from the above table

= (s+a+s-a)/(s-a)(s+a)

= 2s/(s^{2}-a^{2})

So the Laplace transform of the sum of e^{at} and e^{-at} is 2s/(s^{2}-a^{2}).

## FAQs

**Q1: What is the Laplace transform of e**

^{at}?Answer: The Laplace transform of e^{at} is 1/(s-a).

**Q2: What is the Laplace transform of e**

^{t}?Answer: The Laplace transform of e^{t} is 1/(s-1).

**Q3: What is the Laplace transform of e**

^{-at}?Answer: The Laplace transform of e^{-at} is 1/(s+a).

**Q4: What is the Laplace transform of e**

^{-t}?Answer: The Laplace transform of e^{-t} is 1/(s+1).