delay theorem


Theorem.  If  f(t)0  for  t<0  and  {f(t)}:=F(s),  one has

{f(t-t0)}=e-t0sF(s).

Proof.  Since  f(t-t0)0  for  t<t0,  the definition of Laplace transformMathworldPlanetmath at first gives

{f(t-t0)}=t0e-stf(t-t0)𝑑t.

The substitution (http://planetmath.org/SubstitutionForIntegration)  t-t0:=u  yields

{f(t-t0)}=0e-s(u+t0)f(u)𝑑u=e-t0s0e-suf(u)𝑑u=e-t0sF(s).

Corollary.  For any f(t) and the Heaviside step function H(t), one has

{f(t-a)H(t-a)}=e-asF(s).
Title delay theorem
Canonical name DelayTheorem
Date of creation 2013-03-22 18:02:46
Last modified on 2013-03-22 18:02:46
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 10
Author pahio (2872)
Entry type Theorem
Classification msc 44A10
Synonym delay theorem of Laplace transform
Related topic HeavisideStepFunction
Related topic TelegraphEquation