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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. 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 B { f } ( s ) = F ( s ) = ∫ − ∞ ∞ e − s t f ( t ) d t . {\displaystyle {\mathcal {B}}\{f\}(s)=F(s)=\int _{-\infty }^{\infty }e^{-st}f(t)\,dt.} The integral is most commonly understood as an improper integral, which converges if and only if both integrals ∫ 0 ∞ e − s t f ( t ) d t , ∫ − ∞ 0 e − s t f ( t ) d t {\displaystyle \int _{0}^{\infty }e^{-st}f(t)\,dt,\quad \int _{-\infty }^{0}e^{-st}f(t)\,dt} exist. There seems to be no generally accepted notation for the two-sided transform; the B {\displaystyle {\mathcal {B}}} used here recalls "bilateral". The two-sided transform used by some authors is T { f } ( s ) = s B { f } ( s ) = s F ( s ) = s ∫ − ∞ ∞ e − s t f ( t ) d t . {\displaystyle {\mathcal {T}}\{f\}(s)=s{\mathcal {B}}\{f\}(s)=sF(s)=s\int _{-\infty }^{\infty }e^{-st}f(t)\,dt.} In pure mathematics the argument t can be any variable, and Laplace transforms are used to study how differential operators transform the function. In science and engineering applications, the argument t often represents time (in seconds), and the function f(t) often represents a signal or waveform that varies with time. In these cases, the signals are transformed by filters, that work like a mathematical operator, but with a restriction. They have to be causal, which means that the output in a given time t cannot depend on an output which is a higher value of t. In population ecology, the argument t often represents spatial displacement in a dispersal kernel. When working with functions of time, f(t) is called the time domain representation of the signal, while F(s) is called the s-domain (or Laplace domain) representation. The inverse transformation then represents a synthesis of the signal as the sum of its frequency components taken over all frequencies, whereas the forward transformation represents the analysis of the signal into its frequency components.
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