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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 mathematics, the Riemann–Lebesgue lemma, named after Bernhard Riemann and Henri Lebesgue, states that the Fourier transform or Laplace transform of an L 1 function vanishes at infinity. It is of importance in harmonic analysis and asymptotic analysis .
Inverting the Laplace transform using contour integral methods then gives u ( x ) = 1 2 e − x sin ( 2 x ) θ ( x ) {\displaystyle u(x)={\frac {1}{2}}e^{-x}\sin(2x)\theta (x)} . Alternatively, one can complete the square and use a table of Laplace transforms ("exponentially decaying sine wave") or recall from memory to proceed:
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.
The first integral formula corresponds to the Laplace transform (or sometimes the formal Laplace–Borel transformation) of generating functions, denoted by [] (), defined in. [7] Other integral representations for the gamma function in the second of the previous formulas can of course also be used to construct similar integral transformations ...
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.