Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Laplace transform of cosine and sine∗ pahio† 2013-03-22 1:32:01 We start from the easily derivable formula eαt x 1 s−α (s > α), (1) where the curved arrow points from the Laplace-transformed function to the original function. Replacing α by −α we can write the second formula e−αt x 1 s+α (s > −α). (2) Adding (1) and (2) and dividing by 2 we obtain (remembering the linearity of the Laplace transform) 1 1 eαt +e−αt 1 x + , 2 2 s−α s+α i.e. L{cosh αt} = s . s2 −α2 (3) Similarly, subtracting (1) and (2) and dividing by 2 give L{sinh αt} = a . s2 −α2 (4) The formulae (3) and (4) are valid for s > |α|. There are the hyperbolic identities 1 sinh it = sin t i cosh it = cos t, ∗ hLaplaceTransformOfCosineAndSinei created: h2013-03-2i by: hpahioi version: h40929i Privacy setting: h1i hDerivationi h44A10i † This text is available under the Creative Commons Attribution/Share-Alike License 3.0. You can reuse this document or portions thereof only if you do so under terms that are compatible with the CC-BY-SA license. 1 which enable the transition from hyperbolic to trigonometric functions. If we choose α := ia in (3), we may calculate cos at = cosh iat x s s = 2 , s2 −(ia)2 s + a2 the formula (4) analogously gives sin at = 1 a 1 ia = 2 . sinh iat x · 2 2 i i s −(ia) s + a2 Accordingly, we have derived the Laplace transforms L{cos at} = s , s2 +a2 (5) L{sin at} = a , s2 +a2 (6) which are true for s > 0. 2