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In mathematical analysis, an improper integral is an extension of the notion of a definite integral to cases that violate the usual assumptions for that kind of integral. [1] In the context of Riemann integrals (or, equivalently, Darboux integrals ), this typically involves unboundedness, either of the set over which the integral is taken or of ...
In mathematics, the comparison test, sometimes called the direct comparison test to distinguish it from similar related tests (especially the limit comparison test), provides a way of deducing whether an infinite series or an improper integral converges or diverges by comparing the series or integral to one whose convergence properties are known.
An inexact differential is a differential for which the integral over some two paths with the same end points is different. Specifically, there exist integrable paths ,: [,] such that () = (), () = and In this case, we denote the integrals as | and | respectively to make explicit the path dependence of the change of the quantity we are considering as .
The result of the procedure for principal value is the same as the ordinary integral; since it no longer matches the definition, it is technically not a "principal value". The Cauchy principal value can also be defined in terms of contour integrals of a complex-valued function f ( z ) : z = x + i y , {\displaystyle f(z):z=x+i\,y\;,} with x , y ...
Singular or weakly singular: An integral equation is called singular or weakly singular if the integral is an improper integral. [7] This could be either because at least one of the limits of integration is infinite or the kernel becomes unbounded, meaning infinite, on at least one point in the interval or domain over which is being integrated.
6 Integral equations. 7 Integral transforms. 8 Integral geometry. 9 Other. 10 See also. Toggle the table of contents. ... Improper integrals. Cauchy principal value;
In mathematics, a nonelementary antiderivative of a given elementary function is an antiderivative (or indefinite integral) that is, itself, not an elementary function. [1] A theorem by Liouville in 1835 provided the first proof that nonelementary antiderivatives exist. [2]
In calculus and mathematical analysis the limits of integration (or bounds of integration) of the integral () of a Riemann integrable function f {\displaystyle f} defined on a closed and bounded interval are the real numbers a {\displaystyle a} and b {\displaystyle b} , in which a {\displaystyle a} is called the lower limit and b {\displaystyle ...