The Laplace Transform

  • Eugen Skudrzyk


The Fourier transform exists only if the function that is to be transformed is absolutely integrable. Tlis means that s (t) or S (ω) must decrease faster than
$$ s(t) \to 1/t $$
$$ S(\omega ) \to 1/\omega $$
as t or ω approach infinity If methods of analysis that are based on Fourier transforms are to lead to practical results, the initial values of all variables (t = − ∞) must be zero; and they must be zero, too, in the final state of the system. Thus, the standard Fourier analysis is not capable of any initial values or any final values of the functions other than zero. There are two possibilities for dealing with this situation. It can be assumed that all time functions are zero initially and increase to the given values in a very short time interval and finally decrease abruptly to zero when the time exceeds a certain value. Thus, the time functions are truncated (made zero), fortT and for t ≥ + T. This procedure will be investigated in Chapter VII. The second procedure is to assume that the function s (t) is generated at t = 0:
$$> s(t) = 0,t < 0 m $$
and to introduce an infinitely small amount of damping
$$ s(t) \to \mathop {\lim }\limits_{\delta \to 0} s(t){e^{ - \delta t}},t >0 $$
as we have already done in considering the step function and the switched on sinusoidal vibration.


Half Plane Time Function Imaginary Axis Fourier Spectrum Laplace Transform 
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Copyright information

© Springer-Verlag/Wien 1971

Authors and Affiliations

  • Eugen Skudrzyk
    • 1
  1. 1.Ordnance Research Laboratory and Physics DepartmentThe Pennsylvania State UniversityUniversity ParkUSA

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