View source for Improper integral

An integral of an unbounded function or of a function over an unbounded set. Suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050370/i0503701.png" /> is a function defined on a finite or infinite half-interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050370/i0503702.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050370/i0503703.png" />, and that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050370/i0503704.png" />, for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050370/i0503705.png" />, is Riemann- (or Lebesgue-) integrable on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050370/i0503706.png" />. Then the limit

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(when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050370/i0503708.png" />, the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050370/i0503709.png" /> is understood as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050370/i05037010.png" />) is called the improper integral

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If the limit (1) exists and is finite, then one says that the improper integral converges and if not, that it diverges. For example, the improper integral

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converges for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050370/i05037013.png" /> and diverges <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050370/i05037014.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050370/i05037015.png" />, then

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converges for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050370/i05037017.png" /> and diverges for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050370/i05037018.png" />.

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050370/i05037019.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050370/i05037020.png" /> is Riemann- (or Lebesgue-) integrable on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050370/i05037021.png" />, then the improper integral (1) is the same as the [[Definite integral|definite integral]].

Similarly, under the corresponding assumptions one defines the improper integral over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050370/i05037022.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050370/i05037023.png" />,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050370/i05037024.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050370/i05037025.png" /> is Riemann- (or Lebesgue-) integrable over every interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050370/i05037026.png" />, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050370/i05037027.png" /> and if the improper integrals

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both converge, then the improper integral

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is defined as their sum,

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and does not depend on the choice of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050370/i05037031.png" />.

If on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050370/i05037032.png" /> there are finitely many points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050370/i05037033.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050370/i05037034.png" />), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050370/i05037035.png" />, such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050370/i05037036.png" /> is Riemann- (or Lebesgue-) integrable on every interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050370/i05037037.png" /> that does not contain any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050370/i05037038.png" /> and if for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050370/i05037039.png" /> the improper integral

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converges, then

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This definition does not depend on the position of the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050370/i05037042.png" />.

The general properties of integrals carry over to improper integrals: linearity, additivity with respect to the intervals over which the integration proceeds, the rule for integrating inequalities, the mean-value theorems, integration by parts, change of variable, and the Newton–Leibniz formula. For example, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050370/i05037043.png" /> coincides almost-everywhere on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050370/i05037044.png" /> with the derivative of a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050370/i05037045.png" /> that is absolutely continuous on every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050370/i05037046.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050370/i05037047.png" />, then

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To decide about the convergence of the indefinite integral of functions of constant sign one uses the comparison test. E.g., for an indefinite integral of the form (1), when

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as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050370/i05037050.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050370/i05037051.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050370/i05037052.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050370/i05037053.png" />, then the convergence of the improper integral

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implies that of

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in this case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050370/i05037056.png" /> is called a comparison function. As a comparison function for integrals (1) in the case of a finite integration limit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050370/i05037057.png" /> one often uses <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050370/i05037058.png" />; for integrals of the form (2) in the case of a finite integration limit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050370/i05037059.png" /> — the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050370/i05037060.png" />; and when there are one or two infinite integration limits — the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050370/i05037061.png" />. For example, when for a non-negative function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050370/i05037062.png" /> defined for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050370/i05037063.png" /> the limit

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exists, the comparison test implies that for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050370/i05037065.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050370/i05037066.png" /> the improper integral

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of the form (1) converges while for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050370/i05037068.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050370/i05037069.png" /> it diverges.

A necessary and sufficient condition for the convergence of an improper integral is given by Cauchy's criterion. I.e., an improper integral of the form (1) converges if and only if for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050370/i05037070.png" /> there is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050370/i05037071.png" /> such that for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050370/i05037072.png" />,

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An improper integral

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is said to be absolutely convergent if the improper integral

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converges. If an improper integral converges absolutely, then it converges and coincides with the [[Lebesgue integral|Lebesgue integral]]. There exist improper integrals that converge, but not absolutely. For example, for a finite interval:

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and for an infinite interval:

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There are several tests to establish the convergence of an improper integral. E.g., if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050370/i05037078.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050370/i05037079.png" /> are defined for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050370/i05037080.png" />, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050370/i05037081.png" /> has on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050370/i05037082.png" /> a bounded primitive, and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050370/i05037083.png" /> is a monotone function tending to zero as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050370/i05037084.png" />, then the improper integral

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converges. Another test: If the improper integral

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converges and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050370/i05037087.png" /> is monotone and bounded for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050370/i05037088.png" />, then the improper integral

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converges.

The convergence of an improper integral can be expressed in terms of the convergence of series. For example, for an improper integral (1) to converge it is necessary and sufficient that for any sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050370/i05037090.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050370/i05037091.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050370/i05037092.png" /> the series

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converges, and if it converges, then the sum of the series is the same as the value of the improper integral (1).

The concept of an improper integral has been generalized to functions of several variables. Suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050370/i05037094.png" /> is defined on an open (bounded or unbounded) set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050370/i05037095.png" /> of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050370/i05037096.png" />-dimensional Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050370/i05037097.png" /> and is Riemann-integrable over any Jordan-measurable set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050370/i05037098.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050370/i05037099.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050370/i050370100.png" /> is said to be integrable in the improper sense over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050370/i050370101.png" /> if for any sequence of Jordan-measurable sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050370/i050370102.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050370/i050370103.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050370/i050370104.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050370/i050370105.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050370/i050370106.png" />, the limit

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exists and is independent of the choice of the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050370/i050370108.png" />. This limit, if it exists and is finite, is called the improper integral

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and, as in the one-dimensional case, one says that this integral converges. It converges if and only if the integral

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is finite. In this case the improper integral

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is the same as the Lebesgue integral. This is connected with the fact that, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050370/i050370112.png" /> and the definition of an improper integral given above, the limit transition is over a very special class of Jordan-measurable sets, namely intervals. The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050370/i050370113.png" /> were taken as arbitrary Jordan-measurable sets. However, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050370/i050370114.png" /> the assertion remains valid when the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050370/i050370115.png" /> are taken to be any Jordan-measurable domains. Thus, in this case the concept of an improper integral does not lead to anything new as compared with the Lebesgue integral.

For improper integrals of functions of several variables a comparison test holds, similar to the one-dimensional case. The integrals

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are taken as comparison integrals, where

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050370/i050370117.png" /></td> </tr></table>

The former converges for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050370/i050370118.png" /> and diverges for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050370/i050370119.png" />, the latter converges for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050370/i050370120.png" /> and diverges for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050370/i050370121.png" />.

Integrals in the sense of the principal value belong to the improper integrals. Suppose that a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050370/i050370122.png" /> is defined on an open set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050370/i050370123.png" /> except possibly at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050370/i050370124.png" /> and suppose that, for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050370/i050370125.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050370/i050370126.png" /> is (Riemann- or Lebesgue-) integrable over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050370/i050370127.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050370/i050370128.png" /> is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050370/i050370129.png" />-neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050370/i050370130.png" />. If the limit

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050370/i050370131.png" /></td> </tr></table>

exists, then it is called the integral in the sense of the principal value (or principal-value integral) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050370/i050370132.png" />, and is denoted by

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050370/i050370133.png" /></td> </tr></table>

If

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050370/i050370134.png" /></td> </tr></table>

exists as an improper integral, then it also exists in the sense of the principal value. The converse is not true, in general. For example, the improper integral

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050370/i050370135.png" /></td> </tr></table>

diverges, whereas

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050370/i050370136.png" /></td> </tr></table>

Similarly one defines integrals in the sense of the principal value at the point at infinity.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  V.A. Il'in,   E.G. Poznyak,   "Fundamentals of mathematical analysis" , '''2''' , MIR  (1982)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  L.D. Kudryavtsev,   "A course in mathematical analysis" , '''2''' , Moscow  (1988)  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  S.M. Nikol'skii,   "A course of mathematical analysis" , '''2''' , MIR  (1977)  (Translated from Russian)</TD></TR></table>



====Comments====
Instead of  "principal value (integral)"  one often says Cauchy principal value (integral). It is also denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050370/i050370137.png" />.

For the method of residues, cf. [[Complex integration, method of|Complex integration, method of]].

A proof of the fact (mentioned above) that a multi-dimensional improper integral of a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050370/i050370138.png" /> over a region <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050370/i050370139.png" /> exists if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050370/i050370140.png" /> exists, can be found in, e.g., [[#References|[a5]]].

An example of a function integrable in the improper sense is the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050370/i050370141.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050370/i050370142.png" />.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G.E. Shilov,   "Mathematical analysis" , '''1–2''' , M.I.T.  (1974)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  G.H. Hardy,   "A course of pure mathematics" , Cambridge Univ. Press  (1975)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  L. Schwartz,   "Méthodes mathématiques pour les sciences physiques" , Hermann  (1965)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  R.C. Buck,   "Advanced calculus" , McGraw-Hill  (1965)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  G. Valiron,   "Théorie des fonctions" , Masson  (1948)  pp. 287ff</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  T.M. Apostol,   "Calculus" , '''1–2''' , Blaisdell  (1969)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  T.M. Apostol,   "Mathematical analysis" , Addison-Wesley  (1963)</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  W. Rudin,   "Real and complex analysis" , McGraw-Hill  (1974)  pp. 24</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  A.C. Zaanen,   "Integration" , North-Holland  (1967)</TD></TR></table>