# Laplace Transform of e^at: Formula, Proof

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 eat Formula

The formula of the Laplace transform of eat is 1/(s-a). Mathematically, we write it as

L{eat} = 1/(s-a).

## eat Laplace Transform

The Laplace transform of eat is equal to 1 divided by (s-a). That is,

L{eat} = 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)=eat.

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

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

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

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

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

= 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:

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

Laplace transform of sin(at): The Laplace transform of sin(at) is a/(s2+a2).

Laplace transform of cos(at): The Laplace transform of cos(at) is s/(s2+a2).

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 eat+e-at.

Solution:

By the linearity property of Laplace transform, we have

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

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

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

= 2s/(s2-a2)

So the Laplace transform of the sum of eat and e-at is 2s/(s2-a2).

## FAQs

Q1: What is the Laplace transform of eat?

Answer: The Laplace transform of eat is 1/(s-a).

Q2: What is the Laplace transform of et?

Answer: The Laplace transform of et 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).

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