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The Laplace transform's key property is that it converts differentiation and integration in the time domain into multiplication and division by s in the Laplace domain. Thus, the Laplace variable s is also known as an operator variable in the Laplace domain: either the derivative operator or (for s −1 ) the integration operator .
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 two-sided Laplace transform or bilateral Laplace transform is an integral transform equivalent to probability's moment-generating function. 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 .
By a derivation similar to Eq.1, there is an analogous theorem for sequences, such as samples of two continuous functions, where now denotes the discrete-time Fourier transform (DTFT) operator. Consider two sequences u [ n ] {\displaystyle u[n]} and v [ n ] {\displaystyle v[n]} with transforms U {\displaystyle U} and V {\displaystyle V} :
A similar derivation can be done using the unilateral Laplace transform (one-sided Laplace transform). The convolution operation also describes the output (in terms of the input) of an important class of operations known as linear time-invariant (LTI). See LTI system theory for a derivation of convolution as the result of LTI constraints.
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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:
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