We start from the easily derivable formula
where the curved arrow points from the Laplace-transformed function to the original function. Replacing by we can write the second formula
Adding (1) and (2) and dividing by 2 we obtain (remembering the linearity of the Laplace transform)
Similarly, subtracting (1) and (2) and dividing by 2 give
The formulae (3) and (4) are valid for .
There are the hyperbolic identities
which enable the transition from hyperbolic to trigonometric functions. If we choose in (3), we may calculate
the formula (4) analogously gives
Accordingly, we have derived the Laplace transforms
which are true for .