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Improper integral
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{{Short description|Concept in mathematical analysis}} [[File:Improperintegral2.png|thumb|200px|An improper Riemann integral of the first kind, where the region in the plane implied by the integral is infinite in extent horizontally. The area of such a region, which the integral represents, may be finite (as here) or infinite.]] [[File:Improperintegral1.png|thumb|200px|An improper Riemann integral of the second kind, where the implied region is infinite vertically. The region may have either finite (as here) or infinite area.]] {{Calculus|Integral}} 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.<ref name="Buck">{{cite book|title=Advanced Calculus |first=R. Creighton |last=Buck |edition=2nd |year=1965 |publisher=McGraw-Hill |pages=133–134}}</ref> In the context of [[Riemann integral]]s (or, equivalently, [[Darboux integral]]s), this typically involves unboundedness, either of the set over which the integral is taken or of the integrand (the function being integrated), or both. It may also involve bounded but not closed sets or bounded but not [[continuous function]]s. While an improper integral is typically written symbolically just like a standard definite integral, it actually represents a [[limit (mathematics)|limit]] of a definite integral or a sum of such limits; thus improper integrals are said to converge or diverge.<ref name="Schaums">{{cite book|title=Schaum's Outline of Theory and Problems of Advanced Calculus |first=Murray R. |last=Spiegel |publisher=McGraw-Hill |year=1963 |page=260 |isbn=0-07-060229-8}}</ref><ref name="Buck" /> If a regular definite integral (which may [[retronym]]ically be called a '''proper integral'''<!-- redirects to here -->) is worked out as if it is improper, the same answer will result. In the simplest case of a real-valued function of a single variable integrated in the sense of Riemann (or Darboux) over a single interval, improper integrals may be in any of the following forms: # <math>\int_a^\infty f(x)\, dx</math> # <math>\int_{-\infty}^b f(x)\, dx</math> # <math>\int_{-\infty}^\infty f(x)\, dx</math> # <math>\int_a^b f(x)\, dx</math>, where <math>f(x)</math> is undefined or discontinuous somewhere on <math>[a,b]</math> The first three forms are improper because the integrals are taken over an unbounded interval. (They may be improper for other reasons, as well, as explained below.) Such an integral is sometimes described as being of the "first" type or kind if the integrand otherwise satisfies the assumptions of integration.<ref name="Schaums" /> Integrals in the fourth form that are improper because <math>f(x)</math> has a [[vertical asymptote]] somewhere on the interval <math>[a,b]</math> may be described as being of the "second" type or kind.<ref name="Schaums" /> Integrals that combine aspects of both types are sometimes described as being of the "third" type or kind.<ref name="Schaums" /> In each case above, the improper integral must be rewritten using one or more limits, depending on what is causing the integral to be improper. For example, in case 1, if <math>f(x)</math> is continuous on the entire interval <math>[a,\infty)</math>, then : <math>\int_a^\infty f(x)\, dx = \lim_{b \to \infty} \int_a^b f(x)\, dx.</math> The limit on the right is taken to be the definition of the integral notation on the left. If <math>f(x)</math> is only continuous on <math>(a,\infty)</math> and not at <math>a</math> itself, then typically this is rewritten as : <math>\int_a^\infty f(x)\, dx = \lim_{t \to a^+} \int_t^c f(x)\, dx + \lim_{b \to \infty} \int_c^b f(x)\, dx,</math> for any choice of <math>c>a</math>. Here both limits must converge to a finite value for the improper integral to be said to converge. This requirement avoids the ambiguous case of adding positive and negative infinities (i.e., the "<math>\infty-\infty</math>" [[indeterminate form]]). Alternatively, an [[iterated limit]] could be used or a single limit based on the [[Cauchy principal value]]. If <math>f(x)</math> is continuous on <math>[a,d)</math> and <math>(d,\infty)</math>, with a [[Classification of discontinuities|discontinuity of any kind]] at <math>d</math>, then : <math>\int_a^\infty f(x)\, dx = \lim_{t \to d^-} \int_a^t f(x)\, dx + \lim_{u \to d^+} \int_u^c f(x)\, dx + \lim_{b \to \infty} \int_c^b f(x)\, dx,</math> for any choice of <math>c>d</math>. The previous remarks about indeterminate forms, iterated limits, and the Cauchy principal value also apply here. The function <math>f(x)</math> can have more discontinuities, in which case even more limits would be required (or a more complicated principal value expression). Cases 2–4 are handled similarly. See the examples below. Improper integrals can also be evaluated in the context of complex numbers, in higher dimensions, and in other theoretical frameworks such as [[Lebesgue integration]] or [[Henstock–Kurzweil integral|Henstock–Kurzweil integration]]. Integrals that are considered improper in one framework may not be in others.
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