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The unilateral Laplace transform takes as input a function whose time domain is the non-negative reals, which is why all of the time domain functions in the table below are multiples of the Heaviside step function, u(t). The entries of the table that involve a time delay τ are required to be causal (meaning that τ > 0).
The following table provides Laplace transforms for many common functions of a single variable. [31] [32] For definitions and explanations, see the Explanatory Notes at the end of the table. Because the Laplace transform is a linear operator, The Laplace transform of a sum is the sum of Laplace transforms of each term.
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
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).
Bateman transform; Fourier transform. Short-time Fourier 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 ...
As a function of θ this is a two-sided Laplace transform of h, and cannot be identically zero unless h is zero almost everywhere. [2] The exponential is not zero, so this can only happen if g is zero almost everywhere. By contrast, the statistic (,) is sufficient but not complete.
Pages in category "Laplace transforms" The following 9 pages are in this category, out of 9 total. This list may not reflect recent changes. ...
The state-transition equation is defined as the solution of the linear homogeneous state equation. The linear time-invariant state equation given by = + + (), with state vector x, control vector u, vector w of additive disturbances, and fixed matrices A, B, E can be solved by using either the classical method of solving linear differential equations or the Laplace transform method.