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 $\map \Re s > a$

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.