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The following is a list of Laplace transforms for many common functions of a single variable. [1] The Laplace transform is an integral transform that takes a function of a positive real variable t (often time) to a function of a complex variable s (complex angular frequency ).
In mathematics, the Laplace transform, named after Pierre-Simon Laplace (/ l ə ˈ p l ɑː s /), is an integral transform that converts a function of a real variable (usually , in the time domain) to a function of a complex variable (in the complex-valued frequency domain, also known as s-domain, or s-plane).
Two-sided Laplace transforms are closely related to the Fourier transform, the Mellin transform, the Z-transform and the ordinary or one-sided Laplace transform. If f ( t ) is a real- or complex-valued function of the real variable t defined for all real numbers, then the two-sided Laplace transform is defined by the integral
In an electrical network, ω is a natural angular frequency of a response function f(t) if the Laplace transform F(s) of f(t) includes the term Ke −st, where s = σ + ωi for a real σ, and K ≠ 0 is a constant. [2] Natural frequencies depend on network topology and element values but not their input. [3]
The starred transform is a convenient mathematical abstraction that represents the Laplace transform of an impulse sampled function (), which is the output of an ideal sampler, whose input is a continuous function, ().
In mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace, who first studied its properties.This is often written as = or =, where = = is the Laplace operator, [note 1] is the divergence operator (also symbolized "div"), is the gradient operator (also symbolized "grad"), and (,,) is a twice-differentiable real-valued function.
Similarly, the Green's function for the three-variable Laplace equation can be given as a Fourier integral cosine transform of the difference of vertical heights whose kernel is given in terms of the order-zero modified Bessel function of the second kind as | ′ | = (+ ′ ′ (′)) [(′)].
Laplace transform; List of Laplace transforms; I. Inverse Laplace transform; L. ... This page was last edited on 2 August 2022, at 15:01 (UTC).