Concepts inA note about Laplace transform tables for computer use
Laplace transform
The Laplace transform is a widely used integral transform with many applications in physics and engineering. Denoted, it is a linear operator of a function f(t) with a real argument t (t ¿ 0) that transforms it to a function F(s) with a complex argument s. This transformation is essentially bijective for the majority of practical uses; the respective pairs of f(t) and F(s) are matched in tables.
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Fourier transform
The Fourier transform expresses a mathematical function of time as a function of frequency, known as its frequency spectrum. Named for Joseph Fourier, it is a mathematical transform with many applications in physics and engineering. The Fourier integral theorem details this relationship.
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Inverse Laplace transform
[edit] Mellin's inverse formula An integral formula for the inverse Laplace transform, called the Bromwich integral, the Fourier¿Mellin integral, and Mellin's inverse formula, is given by the line integral: where the integration is done along the vertical line in the complex plane such that is greater than the real part of all singularities of F(s). This ensures that the contour path is in the region of convergence.
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Constant coefficients
In mathematics, constant coefficients is a term applied to differential operators, and also some difference operators, to signify that they contain no functions of the independent variables, other than constant functions. In other words, it singles out special operators, within the larger class of operators having variable coefficients. Such constant coefficient operators have been found to be the easiest to handle, in several respects.
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Ordinary differential equation
In mathematics, an ordinary differential equation (abbreviated ODE) is an equation containing a function of one independent variable and its derivatives. There are many general forms an ODE can take, and these are classified in practice (see below). The derivatives are ordinary because partial derivatives only apply to functions of many independent variables.
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Reduce (computer algebra system)
Reduce is a general-purpose computer algebra system geared towards applications in physics. The development of the Reduce computer algebra system was started in the 1960s by Anthony C. Hearn. Since then, many scientists from all over the world have contributed to its development under his direction. Reduce is written entirely in its own LISP dialect called Standard LISP, expressed in an Algol-like syntax called RLISP. The latter is used as a basis for Reduce's user-level language.
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Algebra
Algebra (from Arabic al-jebr meaning "reunion of broken parts") is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures. Together with geometry, analysis, topology, combinatorics, and number theory, algebra is one of the main branches of pure mathematics.
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