Integral transform useful in probability theory, physics, and engineering
In mathematics, the Laplace transform, named after its discoverer Pierre-Simon Laplace (/ləˈplɑːs/), 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 transform is useful for converting differentiation and integration in the time domain into much easier multiplication and division in the Laplace domain (analogous to how logarithms are useful for simplifying multiplication and division into addition and subtraction). This gives the transform many applications in science and engineering, mostly as a tool for solving linear differential equations[1] and dynamical systems by simplifying ordinary differential equations and integral equations into algebraic polynomial equations, and by simplifying convolution into multiplication.[2][3] Once solved, the inverse Laplace transform reverts to the original domain.
The Laplace transform is defined (for suitable functions f) by the integral:
^Lynn, Paul A. (1986). "The Laplace Transform and the z-transform". Electronic Signals and Systems. London: Macmillan Education UK. pp. 225–272. doi:10.1007/978-1-349-18461-3_6. ISBN 978-0-333-39164-8. Laplace Transform and the z-transform are closely related to the Fourier Transform. Laplace Transform is somewhat more general in scope than the Fourier Transform, and is widely used by engineers for describing continuous circuits and systems, including automatic control systems.
In mathematics, the Laplacetransform, named after its discoverer Pierre-Simon Laplace (/ləˈplɑːs/), is an integral transform that converts a function...
In mathematics, the inverse Laplacetransform of a function F ( s ) {\displaystyle F(s)} is the piecewise-continuous and exponentially-restricted[clarification...
following is a list of Laplacetransforms for many common functions of a single variable. The Laplacetransform is an integral transform that takes a function...
Mellin transform is an integral transform that may be regarded as the multiplicative version of the two-sided Laplacetransform. This integral transform is...
frequency domain. Employing the inverse transform, i.e., the inverse procedure of the original Laplacetransform, one obtains a time-domain solution. In...
distributions. The Laplacetransform of the Heaviside step function is a meromorphic function. Using the unilateral Laplacetransform we have: H ^ ( s )...
mathematics, the Laplacetransform is a powerful integral transform used to switch a function from the time domain to the s-domain. The Laplacetransform can be...
differential equations can be solved by a direct use of the Laplacetransform. The Laplacetransform for an M-dimensional case is defined as F ( s 1 , s 2 ...
like the Fourier transform. The major difference is that the Laplacetransform has a region of convergence for which the transform is valid. This implies...
f ( t ) {\displaystyle f(t)} in continuous time has (unilateral) Laplacetransform F ( s ) {\displaystyle F(s)} , then a final value theorem establishes...
delta (in this formula, its derivative appears). The single-sided Laplacetransform of R(x) is given as follows, L { R ( x ) } ( s ) = ∫ 0 ∞ e − s x R...
inputs, and how their behavior is modified by feedback, using the Laplacetransform as a basic tool to model such systems. The usual objective of control...
filter is u(t). Apply z-transform and Laplacetransform on these two inputs to obtain the converted output signal. Perform z-transform on step input Z [ u...
_{p})+k_{p}{\frac {\theta _{p}-\theta _{q}}{\theta _{q}}}.\end{aligned}}} The Laplacetransform of the gamma PDF is F ( s ) = ( 1 + θ s ) − k = β α ( s + β ) α ....
dividing the Laplacetransform of the output, Y ( s ) = L { y ( t ) } {\displaystyle Y(s)={\mathcal {L}}\left\{y(t)\right\}} , by the Laplacetransform of the...
improper definite integral can be determined in several ways: the Laplacetransform, double integration, differentiating under the integral sign, contour...
Borel measure on the real line is of this kind. One can define the Laplacetransform of a finite Borel measure μ on the real line by the Lebesgue integral...
derivative of the Weierstrass transform of f. There is a formula relating the Weierstrass transform W and the two-sided Laplacetransform L. If we define g ( x...
Global Information
