Abstract
Laplace transforms associate a function of a complex variable with a function defined on a half-axis in “time.” The formal properties of the Laplace transform lead to its use in solution of initial value problems for differential equations. The definition of convolution introduces input-output models and linear system responses. Impedance analysis for linear electrical circuits actually arises from the Laplace transform analysis of the circuit element governing equations.
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- 1.
If the integral is considered as a Lebesgue integral, the restriction is that g: g(t) = f(t)e −α t is an element of L 1(0, ∞). In the case of a Riemann integral, the essential restriction is to exponential growth of f (at worst).
- 2.
f here is evidently ambiguous to the extent that F remains invariant. In short, \(\mathcal{L}^{-1}\{F\}\) should be regarded as a class of such functions.
- 3.
Strictly speaking, \(\mathcal{L}\) acts on functions to produce a function; however, the slight abuse of notation makes the formal properties clearer.
- 4.
The expansion is a consequence of Laurent expansions and Liouville’s theorem.
- 5.
With some care about the class of functions allowed as inputs, and continuity hypotheses, representations of this form must hold.
- 6.
The fact that this is the case for a given circuit arrangement may be verified through a state-variable formulation of the model.
References
E.S. Kuh, R.A. Rohrer, The state variable approach to network analysis. Proc. IEEE 53 (7) (1965)
Further Reading
C.M. Close, The Analysis of Linear Circuits (Harcourt, Brace and World, New York, 1966)
P.M. DeRusso, R.J. Roy, C.M. Close, State Variables for Engineers (Wiley, New York, 1965)
C. Doetsch, Guide to the Applications of Laplace Transforms (D. Van Nostrand, Princeton, 1961)
N.V. Widder, The Laplace Transform (Princeton University Press, Princeton, 1946)
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Davis, J.H. (2016). Laplace Transforms. In: Methods of Applied Mathematics with a Software Overview. Applied and Numerical Harmonic Analysis. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-43370-7_6
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DOI: https://doi.org/10.1007/978-3-319-43370-7_6
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