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 The integral is most commonly understood as an improper integral, which converges if and only if both integrals exist. There seems to be no generally accepted notation for the two-sided transform; the used here recalls "bilateral". The two-sided transform used by some authors is 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. Relationship to the Fourier transform The Fourier transform can be defined in terms of the two-sided Laplace transform: Note that definitions of the Fourier transform differ, and in particular is often used instead. In terms of the Fourier transform, we may also obtain the two-sided Laplace transform, as The Fourier transform is normally defined so that it exists for real values; the above definition defines the image in a strip which may not include the real axis where the Fourier transform is supposed to converge. This is then why Laplace transforms retain their value in control theory and signal processing: the convergence of a Fourier transform integral within its domain only means that a linear,