Laplace transform of cosine and sine

We start from the easily formula

eαt1s-α  (s>α), (1)

where the curved from the Laplace-transformed functionMathworldPlanetmath to the original function.  Replacing α by -α we can write the second formula

e-αt1s+α  (s>-α). (2)

Adding (1) and (2) and dividing by 2 we obtain (remembering the linearity of the Laplace transformDlmfMathworldPlanetmath)



{coshαt}=ss2-α2. (3)

Similarly, subtracting (1) and (2) and dividing by 2 give

{sinhαt}=as2-α2. (4)

The formulae (3) and (4) are valid for  s>|α|.

There are the hyperbolic identities


which enable the transition from hyperbolic to trigonometric functionsDlmfMathworldPlanetmath.  If we choose  α:=ia  in (3), we may calculate


the formula (4) analogously gives


Accordingly, we have derived the Laplace transforms

{cosat}=ss2+a2, (5)
{sinat}=as2+a2, (6)

which are true for  s>0.

Title Laplace transform of cosine and sine
Canonical name LaplaceTransformOfCosineAndSine
Date of creation 2013-03-22 18:18:27
Last modified on 2013-03-22 18:18:27
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 8
Author pahio (2872)
Entry type Derivation
Classification msc 44A10
Synonym Laplace transform of sine and cosine