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An alternate notation for the Laplace transform is {} instead of F, [3] often written as () = {()} in an abuse of notation. The meaning of the integral depends on types of functions of interest. A necessary condition for existence of the integral is that f must be locally integrable on [0, ∞) .
The unilateral Laplace transform takes as input a function whose time domain is the non-negative reals, which is why all of the time domain functions in the table below are multiples of the Heaviside step function, u(t). The entries of the table that involve a time delay τ are required to be causal (meaning that τ > 0).
The Laplace operator is named after the French mathematician Pierre-Simon de Laplace (1749–1827), who first applied the operator to the study of celestial mechanics: the Laplacian of the gravitational potential due to a given mass density distribution is a constant multiple of that density distribution.
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 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.
In keeping with the triangular notation for the Laplacian, sometimes is used. Another way to write the d'Alembertian in flat standard coordinates is ∂ 2 {\displaystyle \partial ^{2}} . This notation is used extensively in quantum field theory , where partial derivatives are usually indexed, so the lack of an index with the squared partial ...
On functions, the Laplace–de Rham operator is actually the negative of the Laplace–Beltrami operator, as the conventional normalization of the codifferential assures that the Laplace–de Rham operator is (formally) positive definite, whereas the Laplace–Beltrami operator is typically negative. The sign is merely a convention, and both ...
The equation is normally expressed as a polynomial in the parameter s of the Laplace transform, hence the name s-plane. Points in the s-plane take the form s = σ + jω , where ' j ' is used instead of the usual ' i ' to represent the imaginary component (the variable ' i ' is often used to denote electrical current in engineering contexts).