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An improper integral such that the integral of the absolute value of the integrand converges. If an improper integral is absolutely convergent, it is also convergent. To take a concrete example, let an improper integral be given by
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{{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/a/a010/a010380/a0103801.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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[[Category:Real functions]]
  
where the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010380/a0103802.png" /> is Riemann- (or Lebesgue-) integrable on all intervals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010380/a0103803.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010380/a0103804.png" />.
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{{TEX|done}}
  
A necessary and sufficient condition for the absolute convergence of the integral (*) (Cauchy's criterion for the absolute convergence of an improper integral) is that for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010380/a0103805.png" /> there must exist an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010380/a0103806.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010380/a0103807.png" />, such that for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010380/a0103808.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010380/a0103809.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010380/a01038010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010380/a01038011.png" />, the inequality
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===Definition===
 +
Let $f:[a,b[\to \mathbb R$ be a function which is Riemann-integrable on every subinterval $[a,\beta]\subset [a,b[$ (where we
 +
also allow the case $b=\infty$). If
 +
\begin{equation}\label{e:abs_conv}
 +
\lim_{\beta\to b} \int_a^\beta |f(x)|\, dx < \infty\, ,
 +
\end{equation}
 +
then the [[Improper integral|improper integral]]
 +
\[
 +
\lim_{\beta \to b} \int_a^\beta f(x)\, dx
 +
\]
 +
exists and is finite and it is said to be ''absolutely convergent''. Similar variants define absolutely convergent improper integrals
 +
on $]a,b]$ and $]a,b[$.  
  
<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/a/a010/a010380/a01038012.png" /></td> </tr></table>
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Since Riemann-integrable functions are Lebesgue integrable, it actually follows from the definition that $f$ is
 +
[[Absolutely integrable function|absolutely integrable]] and the improper integral of $f$ is just its Lebesgue integral:  
 +
the notion of absolutely convergent improper integral is therefore seldomly used nowadays.
  
is true.
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The existence of the improper integral does not guarantee its absolute convergence, as for instance is the case for
 +
the function $\frac{sin x}{x}$ on $]0, \infty$[.
 +
===Criteria===
 +
Since the absolute convergence of the improper integral of $f$ is guaranteed to the converge of the improper integral of $|f|$, the
 +
classical criteria might be easily derived from those for the [[Improper integral|improper integral. In particular we have the
  
If the improper integral is absolutely convergent, then it is equal to the Lebesgue integral of the integrand. There exist improper integrals which are convergent but not absolutely convergent, for example
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'''Comparison principle'''
 +
If $g$ is Riemann-integrable, $|g|\leq f$ and the improper integral of $f$ converges, the improper integral of $g$ converges absolutely.
  
<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/a/a010/a010380/a01038013.png" /></td> </tr></table>
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'''Cauchy criterion'''
 +
The improper integral of $f$ converges if and only if for every $\varepsilon >0$ there exists $\eta\in [a, b[$ such that
 +
\[
 +
\int_\eta^\beta |g(x)|\, dx < \varepsilon \qquad \forall \beta\in ]\eta, b[\, .
 +
\]
 +
===Higher dimensions===
 +
The  concept of [[Improper integral|improper integral]] is difficult to generalize to integrals of several variables, given a certain arbitrariness on how
 +
to approximate a general domain. Instead the absence of a canonical approximating sequence of sets is irrelevant for absolutely convergent improper integrals, since one can use the theory of Lebesgue. Indeed, consider any Lebesgue measurable set $\Omega$ and a function $f: \Omega\to \R$ which is Lebesgue integrable on a sequence of measurable sets $\Omega_k\uparrow \Omega$. If
 +
\[
 +
\lim_{k\to\infty} \int_{\Omega_k} |f| < \infty
 +
\]
 +
then $f$ is [[Absolutely integrable function|summable]] and
 +
\[
 +
\lim_{k\to\infty} \int_{\Omega_k} f
 +
\]
 +
is the Lebesgue integral of $f$ over $\Omega$.
  
In order to find out whether or not a given integral is absolutely convergent it is expedient to use the tests for convergence of improper integrals of non-negative functions; for example, the absolute convergence of
+
===References===
 
+
{|
<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/a/a010/a010380/a01038014.png" /></td> </tr></table>
+
|-
 
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|valign="top"|{{Ref|Ap1}}|| T. M. Apostol, "Calculus" , '''1–2''' , Blaisdell  (1969)
is established with the aid of the [[Comparison criterion of convergence|comparison criterion of convergence]].
+
|-
 
+
|valign="top"|{{Ref|Ap1}}|| T. M. Apostol, "Mathematical analysis" , Addison-Wesley  (1963)
For most of the available definitions of multiple improper integrals there is a different relation between convergence and absolute convergence. Let a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010380/a01038015.png" /> be defined on an open set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010380/a01038016.png" /> in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010380/a01038017.png" />-dimensional Euclidean space. If, for any sequence of cube-filled domains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010380/a01038018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010380/a01038019.png" /> which monotonically exhaust <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010380/a01038020.png" /> (i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010380/a01038021.png" />, and
+
|-
 
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|valign="top"|{{Ref|Ru1}}|| R.C. Buck,   "Advanced calculus" , McGraw-Hill  (1965)
<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/a/a010/a010380/a01038022.png" /></td> </tr></table>
+
|-
 
+
|valign="top"|{{Ref|Ru1}}|| G.H. Hardy,  "A course of pure mathematics" , Cambridge Univ. Press  (1975)
the limit of the Riemann integrals
+
|-
 
+
|valign="top"|{{Ref|IP}}|| V.A. Il'in,   E.G. Poznyak,  "Fundamentals of mathematical analysis" , '''1–2''' , MIR  (1982)  (Translated from Russian) {{MR|}} {{ZBL|}}
<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/a/a010/a010380/a01038023.png" /></td> </tr></table>
+
|-
 
+
|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}}
as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010380/a01038024.png" />, exists and if this limit is independent of the choice of the above sequence of domains, then it is usually called the improper integral
+
|-
 
+
|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}}
<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/a/a010/a010380/a01038025.png" /></td> </tr></table>
+
|-
 
+
|valign="top"|{{Ref|Ru}}||  W. Rudin,    "Real and complex analysis" , McGraw-Hill  (1966)  pp. 98    {{MR|0210528}} {{ZBL|0142.01701}}
The integral thus defined is convergent if and only if it is absolutely convergent. There are also other definitions of multiple improper integrals. For example, for a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010380/a01038026.png" /> which is defined on the whole space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010380/a01038027.png" /> and which is Riemann-integrable on any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010380/a01038028.png" />-dimensional sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010380/a01038029.png" /> of radius <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010380/a01038030.png" /> with centre at the origin, one can define the improper integral over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010380/a01038031.png" /> by the equation:
+
|-
 
+
|valign="top"|{{Ref|Sch}}||  L. Schwartz, L. Schwartz,  "Méthodes mathématiques pour les sciences physiques" , Hermann  (1965)
<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/a/a010/a010380/a01038032.png" /></td> </tr></table>
+
|-
 
+
|valign="top"|{{Ref|Sh}}|| G.E. Shilov,  "Mathematical analysis" , '''1–2''' , M.I.T.  (1974)  (Translated from Russian)
The absolute convergence of the integral then implies its convergence, but the converse proposition is not true.
+
|-
 
+
|valign="top"|{{Ref|Roy}}|| G. Valiron,  "Théorie des fonctions" , Masson  (1948)
====References====
+
|-
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> V.A. Il'in,   E.G. Poznyak,  "Fundamentals of mathematical analysis" , '''1–2''' , MIR  (1982)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  L.D. Kudryavtsev,  "Mathematical analysis" , '''1''' , Moscow (1973) (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  S.M. Nikol'skii,  "A course of mathematical analysis" , '''1''' , MIR (1977)  (Translated from Russian)</TD></TR></table>
+
|valign="top"|{{Ref|Zaa}}|| A.C. Zaanen,  "Integration" , North-Holland (1967) {{MR|0222234}} {{ZBL|0175.05002}}
 
+
|-
 
+
|}
 
 
====Comments====
 
 
 
 
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  T.M. Apostol,  "Mathematical analysis" , Addison-Wesley (1969)</TD></TR></table>
 

Revision as of 21:41, 9 July 2013

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

Definition

Let $f:[a,b[\to \mathbb R$ be a function which is Riemann-integrable on every subinterval $[a,\beta]\subset [a,b[$ (where we also allow the case $b=\infty$). If \begin{equation}\label{e:abs_conv} \lim_{\beta\to b} \int_a^\beta |f(x)|\, dx < \infty\, , \end{equation} then the improper integral \[ \lim_{\beta \to b} \int_a^\beta f(x)\, dx \] exists and is finite and it is said to be absolutely convergent. Similar variants define absolutely convergent improper integrals on $]a,b]$ and $]a,b[$.

Since Riemann-integrable functions are Lebesgue integrable, it actually follows from the definition that $f$ is absolutely integrable and the improper integral of $f$ is just its Lebesgue integral: the notion of absolutely convergent improper integral is therefore seldomly used nowadays.

The existence of the improper integral does not guarantee its absolute convergence, as for instance is the case for the function $\frac{sin x}{x}$ on $]0, \infty$[.

Criteria

Since the absolute convergence of the improper integral of $f$ is guaranteed to the converge of the improper integral of $|f|$, the classical criteria might be easily derived from those for the [[Improper integral|improper integral. In particular we have the

Comparison principle If $g$ is Riemann-integrable, $|g|\leq f$ and the improper integral of $f$ converges, the improper integral of $g$ converges absolutely.

Cauchy criterion The improper integral of $f$ converges if and only if for every $\varepsilon >0$ there exists $\eta\in [a, b[$ such that \[ \int_\eta^\beta |g(x)|\, dx < \varepsilon \qquad \forall \beta\in ]\eta, b[\, . \]

Higher dimensions

The concept of improper integral is difficult to generalize to integrals of several variables, given a certain arbitrariness on how to approximate a general domain. Instead the absence of a canonical approximating sequence of sets is irrelevant for absolutely convergent improper integrals, since one can use the theory of Lebesgue. Indeed, consider any Lebesgue measurable set $\Omega$ and a function $f: \Omega\to \R$ which is Lebesgue integrable on a sequence of measurable sets $\Omega_k\uparrow \Omega$. If \[ \lim_{k\to\infty} \int_{\Omega_k} |f| < \infty \] then $f$ is summable and \[ \lim_{k\to\infty} \int_{\Omega_k} f \] is the Lebesgue integral of $f$ over $\Omega$.

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:
Absolutely convergent improper integral. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Absolutely_convergent_improper_integral&oldid=12585
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article