integral transform

integral transform
In mathematics, a function that results when a given function is multiplied by a so-called kernel function, and the product is integrated (see integration) between suitable limits.

Its value lies in its ability to simplify intractable differential equations (subject to particular boundary conditions) by transforming the derivatives and boundary conditions into terms of an algebraic equation that may easily be solved. The solution yielded must be converted to the final solution using an inverse transformation. Several integral transforms are named for the mathematicians who introduced them (Fourier transform, Laplace transform).

* * *

      mathematical operator that produces a new function f(y) by integrating the product of an existing function F(x) and a so-called kernel function K(x, y) between suitable limits. The process, which is called transformation, is symbolized by the equation f(y) = ∫K(x, y)F(x)dx. Several transforms are commonly named for the mathematicians who introduced them: in the Laplace transform, the kernel is exy and the limits of integration are zero and plus infinity; in the Fourier transform, the kernel is (2π)−1/2eixy and the limits are minus and plus infinity.

      Integral transforms are valuable for the simplification that they bring about, most often in dealing with differential equations (differential equation) subject to particular boundary conditions. Proper choice of the class of transformation usually makes it possible to convert not only the derivatives (derivative) in an intractable differential equation but also the boundary values into terms of an algebraic equation that can be easily solved. The solution obtained is, of course, the transform of the solution of the original differential equation, and it is necessary to invert this transform to complete the operation. For the common transformations, tables are available that list many functions and their transforms.

* * *


Universalium. 2010.

Игры ⚽ Нужно решить контрольную?

Look at other dictionaries:

  • Integral transform — In mathematics, an integral transform is any transform T of the following form:: (Tf)(u) = int {t 1}^{t 2} K(t, u), f(t), dt.The input of this transform is a function f , and the output is another function Tf . An integral transform is a… …   Wikipedia

  • Integral geometry — In mathematics, the term integral geometry is used in two ways, which, although related, imply different views of the content of the subject. Cases The more traditional usage is that of Santalo and Blaschke. It follows from the classic theorem of …   Wikipedia

  • integral — integrality, n. integrally, adv. /in ti greuhl, in teg reuhl/, adj. 1. of, pertaining to, or belonging as a part of the whole; constituent or component: integral parts. 2. necessary to the completeness of the whole: This point is integral to his… …   Universalium

  • transform — transformable, adj. transformative, adj. v. /trans fawrm /; n. /trans fawrm/, v.t. 1. to change in form, appearance, or structure; metamorphose. 2. to change in condition, nature, or character; convert. 3. to change into another substance;… …   Universalium

  • Integral yoga — Yogaschool|name=Integral yoga color=green bgcolor=white religious origins=Hinduism, Vedanta regional origins=Sri Aurobindo Ashram, India founding guru=Sri Aurobindo, The Mother popularity= millions, both in India and abroad practice… …   Wikipedia

  • Integral — This article is about the concept of integrals in calculus. For the set of numbers, see integer. For other uses, see Integral (disambiguation). A definite integral of a function can be represented as the signed area of the region bounded by its… …   Wikipedia

  • Laplace transform — In mathematics, the Laplace transform is one of the best known and most widely used integral transforms. It is commonly used to produce an easily soluble algebraic equation from an ordinary differential equation. It has many important… …   Wikipedia

  • Fourier transform — Fourier transforms Continuous Fourier transform Fourier series Discrete Fourier transform Discrete time Fourier transform Related transforms The Fourier transform is a mathematical operation that decomposes a function into its constituent… …   Wikipedia

  • Mellin transform — In mathematics, the Mellin transform is an integral transform that may be regarded as the multiplicative version of the two sided Laplace transform. This integral transform is closely connected to the theory of Dirichlet series, and is often used …   Wikipedia

  • Weierstrass transform — In mathematics, the Weierstrass transform [Ahmed I. Zayed, Handbook of Function and Generalized Function Transformations , Chapter 18. CRC Press, 1996.] of a function f : R rarr; R is the function F defined by:F(x)=frac{1}{sqrt{4piint {… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”