Laplace Transform of Function of t minus a

Theorem
Let $f$ be a function such that $\laptrans f$ exists.

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

Let $a \in \C$ or $\R$ be constant. Let $g$ be the function defined as:


 * $\map g t = \begin{cases} \map f {t - a} & : t > a \\ 0 & : t \le a \end{cases}$

Then:


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

Also known as
This property of the Laplace transform operator is sometimes seen referred to as:
 * the second translation property

or:
 * the second shifting property.