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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).
The Laplace transform is defined by, U ( s ) = L { u ( x ) } = ∫ 0 ∞ e − s x u ( x ) d x . {\displaystyle U(s)={\mathcal {L}}\left\{u(x)\right\}=\int _{0}^{\infty }e^{-sx}u(x)\,dx.} Upon taking term-by-term Laplace transforms, and utilising the rules for derivatives and integrals, the integro-differential equation is converted into the ...
In mathematics, the Laplace transform is a powerful integral transform used to switch a function from the time domain to the s-domain. The Laplace transform can be used in some cases to solve linear differential equations with given initial conditions. First consider the following property of the Laplace transform:
Gabor transform; Hankel transform; Hartley transform; Hermite transform; Hilbert transform. Hilbert–Schmidt integral operator; Jacobi transform; Laguerre transform; Laplace transform. Inverse Laplace transform; Two-sided Laplace transform; Inverse two-sided Laplace transform; Laplace–Carson transform; Laplace–Stieltjes transform; Legendre ...
An alternative fractional derivative was introduced by Caputo in 1967, [7] and produces a derivative that has different properties: it produces zero from constant functions and, more importantly, the initial value terms of the Laplace Transform are expressed by means of the values of that function and of its derivative of integer order rather ...
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
There is the Caputo fractional derivative defined as: = () () () < < which has the advantage that is zero when f(t) is constant and its Laplace Transform is expressed by means of the initial values of the function and its derivative.