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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
+
{{MSC|28A06}}
  
<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/i0503707.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
[[Category:Real functions]]
  
(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
+
{{TEX|done}}
  
<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/i05037011.png" /></td> </tr></table>
+
===Definition===
 +
The term usually denotes a limiting process which yields a definition of integral of an unbounded function or of a function over an unbounded set, even when the function is not [[Absolutely integrable function|summable]].  
  
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
+
Assume that $f$ is a function defined on an half-open interval $[a, b[\subset \mathbb R$, where $b$ is allowed
 +
to take the value $+\infty$. If $f$ is Riemann- (or Lebesgue-) integrable on every interval $[a, \beta]\subset [a,b]$
 +
and the limit  
 +
\[
 +
\lim_{\beta\uparrow b} \int_a^b f(x)\, dx
 +
\]
 +
exists, then such limit is called the improper integral of $f$ over $[a,b[$. If the limit exists and is finite, then one says that the improper integral converges and if not, that it diverges. A similar definition is possible for
 +
the cases $]a,b]$ and $]a,b[$. In the latter the improper integral is the sum of the limits
 +
\[
 +
\lim_{\beta\uparrow b} \int_c^\beta f(x)\, dx
 +
\]
 +
and
 +
\[
 +
\lim_{\alpha\downarrow a} \int_\alpha^c f(x)\, dx\,
 +
\]
 +
which are assumed to exist for some point $c\in ]a,b[$ and not to give the indeterminate form $+\infty-\infty$.
 +
Under these assumptions the result is independent of the point $c$.
  
<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/i05037012.png" /></td> </tr></table>
+
Some generalizations are used even for functions defined on domains of type $]a_0, a_1[\cup ]a_1, a_2[\cup
 +
\ldots \cup ]a_{k-1}, a_k[$. In this case it is required that the improper integral exists on every
 +
separate interval and that in the resulting $k$ values $+\infty$ and $-\infty$ do not both appear.
  
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
+
===Comparison with Riemann- and Lebesgue- integrals===
 +
If the function $f$ is Riemann-integrable over $[a,b]$, then the improper integral coincides with the
 +
Riemann integral. The same holds with the Lebesgue integral if $f$ is Lebesgue-integrable over $[a,b[$.
 +
A partial converse to the last statement holds: if $f$ is Lebesgue-measurable for every $[a,\beta]\subset [a,b[$
 +
and the improper integral of $|f|$ exists and is finite, then $f$ is [[Absolutely integrable function|summable]] and the improper integral coincides with the Lebesgue integral. However the improper integral might exist even when $f$
 +
is not summable, as it is the case of
 +
\[
 +
\int_0^\infty \frac{\sin x}{x}\, dx\, .
 +
\]
 +
===Properties===
 +
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 $f$ coincides almost-everywhere on $[a,b[$ with the derivative of a function $F$ that is [[Absolutely continuous function|absolutely continuous]] on every $[a,\beta]\subset [a,b[$ then
 +
\[
 +
\int_a^b f(x)\, dx = F(b)-F(a)\, .
 +
\]
  
<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/i05037016.png" /></td> </tr></table>
+
===Criteria===
 +
To decide about the convergence of the indefinite integral of functions of constant sign one uses the comparison test.
 +
That is, if $0\leq f\leq g$ and the improper integral of $g$ converges, then so does the improper integral of $f$.
  
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" />.
+
A useful general criterion is that of Cauchy: the improper integral of $f$ on $[a,b[$ converges if and only if for every $\varepsilon>0$ $\exists \eta\in [a,b[$ such that
 
+
\[
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]].
+
\left|\int_\alpha^\beta f(x)\, dx\right| < \varepsilon \qquad \forall \beta>\alpha> \eta\, .
 
+
\]
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
 
 
 
<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/i05037028.png" /></td> </tr></table>
 
 
 
both converge, then 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/i05037029.png" /></td> </tr></table>
 
 
 
is defined as their sum,
 
 
 
<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/i05037030.png" /></td> </tr></table>
 
 
 
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
 
 
 
<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/i05037040.png" /></td> </tr></table>
 
 
 
converges, then
 
 
 
<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/i05037041.png" /></td> </tr></table>
 
 
 
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
 
 
 
<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/i05037048.png" /></td> </tr></table>
 
 
 
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
 
 
 
<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/i05037049.png" /></td> </tr></table>
 
 
 
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
 
 
 
<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/i05037054.png" /></td> </tr></table>
 
 
 
implies that of
 
 
 
<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/i05037055.png" /></td> </tr></table>
 
 
 
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
 
 
 
<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/i05037064.png" /></td> </tr></table>
 
 
 
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
 
 
 
<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/i05037067.png" /></td> </tr></table>
 
 
 
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" />,
 
 
 
<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/i05037073.png" /></td> </tr></table>
 
 
 
An 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/i05037074.png" /></td> </tr></table>
 
 
 
is said to be absolutely convergent if 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/i05037075.png" /></td> </tr></table>
 
 
 
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:
 
 
 
<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/i05037076.png" /></td> </tr></table>
 
 
 
and for an infinite interval:
 
 
 
<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/i05037077.png" /></td> </tr></table>
 
 
 
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
 
 
 
<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/i05037085.png" /></td> </tr></table>
 
 
 
converges. Another test: If 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/i05037086.png" /></td> </tr></table>
 
 
 
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
 
 
 
<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/i05037089.png" /></td> </tr></table>
 
  
 +
The convergence of the improper integral can be turned into deciding the convergence of certain series: the
 +
improper integral of $f$ over $[a,b[$ converges if and only if for every sequence $b_n \uparrow b$ the
 +
corresponding series
 +
\[
 +
\sum_{i=1}^\infty \int_{b_{i-1}}^{b_i} f(x)\, dx
 +
\]
 
converges.
 
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
+
===Higher dimensions and Cauchy principal value===
 
+
The concept of improper integral can be generalized to integrals of several variables. However such generalization hinges on deciding for a given domain $\Omega$ in which way it should be approximated by a sequence of ''cannonical domains'' and this is not so clear in more than one variable. Moreover, the fact that the higher-dimensional version
<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/i05037093.png" /></td> </tr></table>
+
of the Riemann integral is quite involved has made some definitions of improper integral seldomly used.  
 
 
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
 
 
 
<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/i050370107.png" /></td> </tr></table>
 
 
 
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
 
 
 
<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/i050370109.png" /></td> </tr></table>
 
 
 
and, as in the one-dimensional case, one says that this integral converges. It converges if and only if the 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/i050370110.png" /></td> </tr></table>
 
 
 
is finite. In this case 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/i050370111.png" /></td> </tr></table>
 
 
 
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
 
 
 
<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/i050370116.png" /></td> </tr></table>
 
 
 
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]]].
+
A popular version of integrating functions with a point singularity, which is of uttermost importance in potential
 +
theory, harmonic analysis and partial differential equations, leads to the Cauchy [[Principal value|principal value]].
 +
Assume $f: \Omega \to \mathbb R$ is a function which is Lebesgue integrable on $\Omega\setminus B_r (x_0)$ for any
 +
$r>0$. The Cauchy principal value of the integral of $f$ over $\Omega$, which is denoted by
 +
\[
 +
{\rm PV}\, \int_\Omega f
 +
\]
 +
is given by the limit
 +
\[
 +
\lim_{r\downarrow 0} \int_{\Omega \setminus B_r (x_0)} f
 +
\]
 +
(when it exists).
  
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" />.
+
However the Cauchy principal value is rarely called improper integral, especially in one space dimension. In fact,
 +
if we consider the function $\frac{1}{x}$ on $]-1,1[$, given its symmetry it is obvious that
 +
\[
 +
{\rm PV}\, \int_{-1}^1 \frac{1}{x}\, dx = 0\, .
 +
\]
 +
On the other hand most authors say that the improper integral of $\frac{1}{x}$ does not exist, since for the improper
 +
integral of $f$ to be well defined for a function which is singular at $0$ it is usually required that both limits
 +
\[
 +
\lim_{\alpha\downarrow 0} \int_\alpha^1 f(x)\, dx
 +
\]
 +
and
 +
\[
 +
\lim_{a\uparrow 0} \int_{-1}^a f(x)\, dx
 +
\]
 +
exist and their sum gives not the indeterminacy $+\infty-\infty$.
  
====References====
+
===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>
+
{|
 +
|-
 +
|valign="top"|{{Ref|Ap1}}|| T. M. Apostol, "Calculus" , '''1–2''' , Blaisdell (1969)
 +
|-
 +
|valign="top"|{{Ref|Ap1}}|| T. M. Apostol, "Mathematical analysis" , Addison-Wesley (1963)
 +
|-
 +
|valign="top"|{{Ref|Ru1}}|| R.C. Buck,  "Advanced calculus" , McGraw-Hill  (1965)
 +
|-
 +
|valign="top"|{{Ref|Ru1}}|| G.H. Hardy,  "A course of pure mathematics" , Cambridge Univ. Press  (1975)  
 +
|-
 +
|valign="top"|{{Ref|IP}}|| V.A. Il'in,    E.G. Poznyak,  "Fundamentals of mathematical analysis" , '''1–2''' ,  MIR  (1982) (Translated from Russian) {{MR|}} {{ZBL|}}
 +
|-
 +
|valign="top"|{{Ref|Ku}}||  L.D. Kudryavtsev,  "Mathematical analysis" , '''1''' , Moscow (1973)  (In Russian{{MR|1617334}} {{MR|1070567}} {{MR|1070566}}  {{MR|1070565}} {{MR|0866891}} {{MR|0767983}} {{MR|0767982}}  {{MR|0628614}} {{MR|0619214}} {{ZBL|1080.00002}} {{ZBL|1080.00001}}  {{ZBL|1060.26002}} {{ZBL|0869.00003}} {{ZBL|0696.26002}} {{ZBL|0703.26001}} {{ZBL|0609.00001}} {{ZBL|0632.26001}} {{ZBL|0485.26002}} {{ZBL|0485.26001}}
 +
|-
 +
|valign="top"|{{Ref|Nik}}||  S.M. Nikol'skii,  "A course of mathematical analysis" , '''1''' , MIR  (1977) (Translated from Russian) {{MR|}} {{ZBL|0397.00003}}  {{ZBL|0384.00004}}
 +
|-
 +
|valign="top"|{{Ref|Ru}}|| W. Rudin,  "Real and complex analysis" , McGraw-Hill (1966) pp. 98  {{MR|0210528}} {{ZBL|0142.01701}}
 +
|-
 +
|valign="top"|{{Ref|Sch}}||  L. Schwartz, L. Schwartz,  "Méthodes mathématiques pour les sciences physiques" , Hermann  (1965)
 +
|-
 +
|valign="top"|{{Ref|Sh}}|| G.E. Shilov,  "Mathematical analysis" , '''1–2''' , M.I.T. (1974)  (Translated from Russian)
 +
|-
 +
|valign="top"|{{Ref|Roy}}|| G. Valiron,  "Théorie des fonctions" , Masson  (1948)
 +
|-
 +
|valign="top"|{{Ref|Zaa}}|| A.C. Zaanen,  "Integration" , North-Holland  (1967) {{MR|0222234}} {{ZBL|0175.05002}}
 +
|-
 +
|}

Latest revision as of 12:30, 7 July 2013

2020 Mathematics Subject Classification: Primary: 28A06 [MSN][ZBL]

Definition

The term usually denotes a limiting process which yields a definition of integral of an unbounded function or of a function over an unbounded set, even when the function is not summable.

Assume that $f$ is a function defined on an half-open interval $[a, b[\subset \mathbb R$, where $b$ is allowed to take the value $+\infty$. If $f$ is Riemann- (or Lebesgue-) integrable on every interval $[a, \beta]\subset [a,b]$ and the limit \[ \lim_{\beta\uparrow b} \int_a^b f(x)\, dx \] exists, then such limit is called the improper integral of $f$ over $[a,b[$. If the limit exists and is finite, then one says that the improper integral converges and if not, that it diverges. A similar definition is possible for the cases $]a,b]$ and $]a,b[$. In the latter the improper integral is the sum of the limits \[ \lim_{\beta\uparrow b} \int_c^\beta f(x)\, dx \] and \[ \lim_{\alpha\downarrow a} \int_\alpha^c f(x)\, dx\, \] which are assumed to exist for some point $c\in ]a,b[$ and not to give the indeterminate form $+\infty-\infty$. Under these assumptions the result is independent of the point $c$.

Some generalizations are used even for functions defined on domains of type $]a_0, a_1[\cup ]a_1, a_2[\cup \ldots \cup ]a_{k-1}, a_k[$. In this case it is required that the improper integral exists on every separate interval and that in the resulting $k$ values $+\infty$ and $-\infty$ do not both appear.

Comparison with Riemann- and Lebesgue- integrals

If the function $f$ is Riemann-integrable over $[a,b]$, then the improper integral coincides with the Riemann integral. The same holds with the Lebesgue integral if $f$ is Lebesgue-integrable over $[a,b[$. A partial converse to the last statement holds: if $f$ is Lebesgue-measurable for every $[a,\beta]\subset [a,b[$ and the improper integral of $|f|$ exists and is finite, then $f$ is summable and the improper integral coincides with the Lebesgue integral. However the improper integral might exist even when $f$ is not summable, as it is the case of \[ \int_0^\infty \frac{\sin x}{x}\, dx\, . \]

Properties

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 $f$ coincides almost-everywhere on $[a,b[$ with the derivative of a function $F$ that is absolutely continuous on every $[a,\beta]\subset [a,b[$ then \[ \int_a^b f(x)\, dx = F(b)-F(a)\, . \]

Criteria

To decide about the convergence of the indefinite integral of functions of constant sign one uses the comparison test. That is, if $0\leq f\leq g$ and the improper integral of $g$ converges, then so does the improper integral of $f$.

A useful general criterion is that of Cauchy: the improper integral of $f$ on $[a,b[$ converges if and only if for every $\varepsilon>0$ $\exists \eta\in [a,b[$ such that \[ \left|\int_\alpha^\beta f(x)\, dx\right| < \varepsilon \qquad \forall \beta>\alpha> \eta\, . \]

The convergence of the improper integral can be turned into deciding the convergence of certain series: the improper integral of $f$ over $[a,b[$ converges if and only if for every sequence $b_n \uparrow b$ the corresponding series \[ \sum_{i=1}^\infty \int_{b_{i-1}}^{b_i} f(x)\, dx \] converges.

Higher dimensions and Cauchy principal value

The concept of improper integral can be generalized to integrals of several variables. However such generalization hinges on deciding for a given domain $\Omega$ in which way it should be approximated by a sequence of cannonical domains and this is not so clear in more than one variable. Moreover, the fact that the higher-dimensional version of the Riemann integral is quite involved has made some definitions of improper integral seldomly used.

A popular version of integrating functions with a point singularity, which is of uttermost importance in potential theory, harmonic analysis and partial differential equations, leads to the Cauchy principal value. Assume $f: \Omega \to \mathbb R$ is a function which is Lebesgue integrable on $\Omega\setminus B_r (x_0)$ for any $r>0$. The Cauchy principal value of the integral of $f$ over $\Omega$, which is denoted by \[ {\rm PV}\, \int_\Omega f \] is given by the limit \[ \lim_{r\downarrow 0} \int_{\Omega \setminus B_r (x_0)} f \] (when it exists).

However the Cauchy principal value is rarely called improper integral, especially in one space dimension. In fact, if we consider the function $\frac{1}{x}$ on $]-1,1[$, given its symmetry it is obvious that \[ {\rm PV}\, \int_{-1}^1 \frac{1}{x}\, dx = 0\, . \] On the other hand most authors say that the improper integral of $\frac{1}{x}$ does not exist, since for the improper integral of $f$ to be well defined for a function which is singular at $0$ it is usually required that both limits \[ \lim_{\alpha\downarrow 0} \int_\alpha^1 f(x)\, dx \] and \[ \lim_{a\uparrow 0} \int_{-1}^a f(x)\, dx \] exist and their sum gives not the indeterminacy $+\infty-\infty$.

References

[Ap1] T. M. Apostol, "Calculus" , 1–2 , Blaisdell (1969)
[Ap1] T. M. Apostol, "Mathematical analysis" , Addison-Wesley (1963)
[Ru1] R.C. Buck, "Advanced calculus" , McGraw-Hill (1965)
[Ru1] G.H. Hardy, "A course of pure mathematics" , Cambridge Univ. Press (1975)
[IP] V.A. Il'in, E.G. Poznyak, "Fundamentals of mathematical analysis" , 1–2 , MIR (1982) (Translated from Russian)
[Ku] L.D. Kudryavtsev, "Mathematical analysis" , 1 , Moscow (1973) (In Russian) MR1617334 MR1070567 MR1070566 MR1070565 MR0866891 MR0767983 MR0767982 MR0628614 MR0619214 Zbl 1080.00002 Zbl 1080.00001 Zbl 1060.26002 Zbl 0869.00003 Zbl 0696.26002 Zbl 0703.26001 Zbl 0609.00001 Zbl 0632.26001 Zbl 0485.26002 Zbl 0485.26001
[Nik] S.M. Nikol'skii, "A course of mathematical analysis" , 1 , MIR (1977) (Translated from Russian) Zbl 0397.00003 Zbl 0384.00004
[Ru] W. Rudin, "Real and complex analysis" , McGraw-Hill (1966) pp. 98 MR0210528 Zbl 0142.01701
[Sch] L. Schwartz, L. Schwartz, "Méthodes mathématiques pour les sciences physiques" , Hermann (1965)
[Sh] G.E. Shilov, "Mathematical analysis" , 1–2 , M.I.T. (1974) (Translated from Russian)
[Roy] G. Valiron, "Théorie des fonctions" , Masson (1948)
[Zaa] A.C. Zaanen, "Integration" , North-Holland (1967) MR0222234 Zbl 0175.05002
How to Cite This Entry:
Improper integral. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Improper_integral&oldid=29898
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article