Definition, Betydelse & Anagram | Engelska ordet INTEGRALS

Definition av INTEGRALS

1. böjningsform av integral

Exempel på hur man kan använda INTEGRALS i en mening

• Antiderivatives are related to definite integrals through the second fundamental theorem of calculus: the definite integral of a function over a closed interval where the function is Riemann integrable is equal to the difference between the values of an antiderivative evaluated at the endpoints of the interval.
• It can be used to approximate integrals by finite sums, or conversely to evaluate finite sums and infinite series using integrals and the machinery of calculus.
• However, with the appropriate reduction formula, every elliptic integral can be brought into a form that involves integrals over rational functions and the three Legendre canonical forms, also known as the elliptic integrals of the first, second and third kind.
• Typical examples of entire functions are polynomials and the exponential function, and any finite sums, products and compositions of these, such as the trigonometric functions sine and cosine and their hyperbolic counterparts sinh and cosh, as well as derivatives and integrals of entire functions such as the error function.
• Separable ordinary differential equation, a class of equations that can be separated into a pair of integrals.
• Period (algebraic geometry), numbers that can be expressed as integrals of algebraic differential forms over algebraically defined domains, forming a ring.
• Inner products of vectors can describe finite sums (via finite-dimensional vector spaces), infinite series (via vectors in sequence spaces), and integrals (via vectors in Hilbert spaces).
• Quadrature (differential equations), expressing a differential equation solution in terms of integrals.
• Haar and Weil (respectively in 1933 and 1940) showed that the integrals and Fourier series are special cases of a very wide class of topological groups.
• For example, they occur implicitly in the epsilon-delta definition of continuity; the intermediate value theorem asserts that the image of an interval by a continuous function is an interval; integrals of real functions are defined over an interval; etc.
• In mathematics, the Cauchy integral theorem (also known as the Cauchy–Goursat theorem) in complex analysis, named after Augustin-Louis Cauchy (and Édouard Goursat), is an important statement about line integrals for holomorphic functions in the complex plane.
• Those integrals are in turn named elliptic because they first were encountered for the calculation of the arc length of an ellipse.
• This may be used for calculation in cases where the integral can be calculated directly, but it is usually the case that residues are used to simplify calculation of integrals, and not the other way around.
• In complex analysis, the residue theorem, sometimes called Cauchy's residue theorem, is a powerful tool to evaluate line integrals of analytic functions over closed curves; it can often be used to compute real integrals and infinite series as well.
• He also introduced the Kontsevich integral, a topological invariant of knots (and links) defined by complicated integrals analogous to Feynman integrals, and generalizing the classical Gauss linking number.
• Integration by parts is a heuristic rather than a purely mechanical process for solving integrals; given a single function to integrate, the typical strategy is to carefully separate this single function into a product of two functions u(x)v(x) such that the residual integral from the integration by parts formula is easier to evaluate than the single function.
• However, different methods of computation of indefinite integrals can result in multiple resulting antiderivatives, each implicitly containing different constants of integration, and no particular option may be considered simplest.
• Chebyshev's inequality can also be obtained directly from a simple comparison of areas, starting from the representation of an expected value as the difference of two improper Riemann integrals (last formula in the definition of expected value for arbitrary real-valued random variables).
• The term numerical quadrature (often abbreviated to quadrature) is more or less a synonym for "numerical integration", especially as applied to one-dimensional integrals.
• In calculus, integration by substitution, also known as u-substitution, reverse chain rule or change of variables, is a method for evaluating integrals and antiderivatives.

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