The theory was further developed in the 19th and early 20th centuries by Mathias Lerch, Oliver Heaviside, and Thomas Bromwich. Laplace's use of generating functions was similar to what is now known as the z-transform, and he gave little attention to the continuous variable case which was discussed by Niels Henrik Abel. Laplace wrote extensively about the use of generating functions in Essai philosophique sur les probabilités (1814), and the integral form of the Laplace transform evolved naturally as a result. Here a is any constant such AS 2.The Laplace transform is named after mathematician and astronomer Pierre-Simon, marquis de Laplace, who used a similar transform in his work on probability theory. If function (f) is piecewise continuous and of exponential order, then the Laplace transform F(s) exists for s>a that is s is greater than the a. For What Values of s Does the Laplace Transform Exist? A single/unit impulse input that starts at a time t=0 and rises to the value 1 has a Laplace transform of 1. The Laplace transform for a unit step input starts at a time t=0, rises to the constant value 1 and has a Laplace transform of 1/s. As Laplace transforms to give you a broad perspective of thinking in complex frequency spaces, which can be a bit awkward, and operate using algebraic formulas rather than simply numbers. Fourier transforms only capture the steady state behavior, i.e., Fourier series. Laplace transforms can capture the transient behaviors of systems, such as in the ODE system.
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