# Laplace Transform of Exponential times Function

## Theorem

Let $\map f t: \R \to \R$ or $\R \to \C$ be a function of exponential order $a$ for some constant $a \in \R$.

Let $\laptrans {\map f t} = \map F s$ be the Laplace transform of $f$.

Let $e^t$ be the exponential function.

Then:

$\laptrans {e^{a t} \map f t} = \map F {s - a}$

everywhere that $\laptrans f$ exists, for $\map \Re s > a$

## Proof

 $\ds \laptrans {e^{a t} \map f t}$ $=$ $\ds \int_0^{\to +\infty} e^{-s t} e^{a t} \map f t \rd t$ Definition of Laplace Transform $\ds$ $=$ $\ds \int_0^{\to +\infty} e^{-s t + a t} \map f t \rd t$ Exponent Combination Laws $\ds$ $=$ $\ds \int_0^{\to +\infty} e^{-\paren {s - a} t} \map f t \rd t$ $\ds$ $=$ $\ds \map F {s - a}$ Definition of Laplace Transform

$\blacksquare$

## Also known as

This property of the Laplace transform operator is sometimes seen referred to as:

the first translation property

or:

the first shifting property.

## Examples

### Example $1$

$\laptrans {e^{-t} \cos 2 t} = \dfrac {s + 1} {s^2 + 2 s + 5}$

### Example $2$

$\laptrans {t^2 e^{3 t} } = \dfrac 2 {\paren {s - 3}^3}$

### Example $3$

$\laptrans {e^{-2 t} \sin 4 t} = \dfrac 4 {s^2 + 4 s + 20}$

### Example $4$

$\laptrans {e^{4 t} \cosh 5 t} = \dfrac {s - 4} {s^2 - 8 s - 9}$

### Example $5$

$\laptrans {e^{-3 t} \paren {3 \cos 6 t - 5 \sin 6 t} } = \dfrac {3 s - 24} {s^2 + 4 s + 40}$