Laplace transform of cosine and sine
We start from the easily formula
where the curved 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 .
|Title||Laplace transform of cosine and sine|
|Date of creation||2013-03-22 18:18:27|
|Last modified on||2013-03-22 18:18:27|
|Last modified by||pahio (2872)|
|Synonym||Laplace transform of sine and cosine|