(These are very closely related - see Laplace transform.) It can also be thought of a generalization of Elmore delay, which matches the first moment in the time domain (or computes a one-pole approximation in the frequency domain - they are equivalent).
They are thus more easily analyzed, using powerful frequency domain methods such as Laplace transforms, to determine DC response, AC response, and transient response.
The wide and general applicability of the Laplace transform and its inverse is illustrated by an application in astronomy which provides some information on the spatial distribution of matter of an astronomical source of radiofrequency thermal radiation too distant to resolve as more than a point, given its flux density spectrum, rather than relating the time domain with the spectrum (frequency domain).
Before Kalman, time series analysis and classical control theory studied the frequency domain, using harmonic analysis, especially Laplace and Fourier transforms.
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Cornelis Simon Meijer (August 17, 1904, Pieterburen – April 12, 1974) was a Dutch mathematician at the university of Groningen who introduced the Meijer G-function, a very general function that includes most of the elementary and higher mathematical functions as special cases; he also introduced generalizations of the Laplace transform that are referred to as Meijer transforms.
It can be seen that the Z(k,z) functions are the kernels of the Fourier transform or Laplace transform of the Z(z) function and so k may be a discrete variable for periodic boundary conditions, or it may be a continuous variable for non-periodic boundary conditions.
The Laplace–Stieltjes transform, named for Pierre-Simon Laplace and Thomas Joannes Stieltjes, is an integral transform similar to the Laplace transform.
An even larger, multivolume table is the Integrals and Series by Prudnikov, Brychkov, and Marichev (with volumes 1–3 listing integrals and series of elementary and special functions, volume 4–5 are tables of Laplace transforms).
The driving point impedance is a mathematical representation of the input impedance of a filter in the frequency domain using one of a number of notations such as Laplace transform (s-domain) or Fourier transform (jω-domain).