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A formula for rearranging the order of integration in iterated improper integrals of Cauchy principal value type (cf. [[Improper integral|Improper integral]]).
 
A formula for rearranging the order of integration in iterated improper integrals of Cauchy principal value type (cf. [[Improper integral|Improper integral]]).
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072980/p0729801.png" /> be a simple closed or open smooth curve in the complex plane, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072980/p0729802.png" /> be a function defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072980/p0729803.png" /> (generally speaking complex-valued) and satisfying a uniform [[Hölder condition|Hölder condition]] with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072980/p0729804.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072980/p0729805.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072980/p0729806.png" /> be a fixed point on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072980/p0729807.png" /> which is not an end-point if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072980/p0729808.png" /> is open. Then one has the Poincaré–Bertrand formula
+
Let $  \Gamma $
 +
be a simple closed or open smooth curve in the complex plane, let $  \phi ( t , t _ {1} ) $
 +
be a function defined on $  \Gamma $(
 +
generally speaking complex-valued) and satisfying a uniform [[Hölder condition|Hölder condition]] with respect to $  t $
 +
and $  t _ {1} $,  
 +
and let $  t _ {0} $
 +
be a fixed point on $  \Gamma $
 +
which is not an end-point if $  \Gamma $
 +
is open. Then one has the Poincaré–Bertrand formula
 +
 
 +
$$ \tag{1 }
 +
\int\limits _  \Gamma
 +
 
 +
\frac{dt}{t - t _ {0} }
 +
 
 +
\int\limits _  \Gamma
 +
 
 +
\frac{\phi ( t , t _ {1} ) }{t _ {1} - t }
 +
  d t _ {1\ } =
 +
$$
 +
 
 +
$$
 +
= \
 +
- \pi  ^ {2} \phi ( t _ {0} , t _ {0} ) + \int\limits _  \Gamma  d t _ {1} \int\limits _  \Gamma 
 +
\frac{\phi ( t , t _ {1} ) }{
 +
( t - t _ {0} ) ( t _ {1} - t ) }
 +
  d t .
 +
$$
 +
 
 +
The formula is true under more general assumptions on the curve  $  \Gamma $
 +
and the function  $  \phi $(
 +
see [[#References|[4]]]). If  $  \phi ( t , t _ {1} ) = \alpha ( t) \beta ( t _ {1} ) $,
 +
where  $  \alpha \in L _ {p} $,
 +
$  \beta \in L _ {q }  $,
 +
$  q = p / ( p - 1 ) $,
 +
equation (1) is true for almost-all  $  t _ {0} \in \Gamma $(
 +
see [[#References|[5]]], [[#References|[6]]]). If the curve  $  \Gamma $
 +
is closed and the function  $  \phi $
 +
depends on one variable only, then equation (1) takes the form
 +
 
 +
$$ \tag{2 }
 +
 
 +
\frac{1}{( \pi i )  ^ {2} }
  
<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/p/p072/p072980/p0729809.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
\int\limits _  \Gamma
  
<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/p/p072/p072980/p07298010.png" /></td> </tr></table>
+
\frac{dt}{t - t _ {0} }
  
The formula is true under more general assumptions on the curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072980/p07298011.png" /> and the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072980/p07298012.png" /> (see [[#References|[4]]]). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072980/p07298013.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072980/p07298014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072980/p07298015.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072980/p07298016.png" />, equation (1) is true for almost-all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072980/p07298017.png" /> (see [[#References|[5]]], [[#References|[6]]]). If the curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072980/p07298018.png" /> is closed and the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072980/p07298019.png" /> depends on one variable only, then equation (1) takes the form
+
\int\limits _  \Gamma
  
<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/p/p072/p072980/p07298020.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
\frac{\phi ( t _ {1} ) }{t _ {1} - t }
 +
  d t _ {1}  = \phi ( t _ {0} ) ,
 +
$$
  
and holds for all or almost-all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072980/p07298021.png" />, depending (respectively) on whether <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072980/p07298022.png" /> satisfies a Hölder condition or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072980/p07298023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072980/p07298024.png" />. Equation (2) is also called the Poincaré–Bertrand formula.
+
and holds for all or almost-all $  t _ {0} \in \Gamma $,  
 +
depending (respectively) on whether $  \phi $
 +
satisfies a Hölder condition or $  \phi \in L _ {p} $,  
 +
p > 1 $.  
 +
Equation (2) is also called the Poincaré–Bertrand formula.
  
 
Analogues of formula (1) have been constructed for multiple integrals (see [[#References|[8]]]–[[#References|[11]]]).
 
Analogues of formula (1) have been constructed for multiple integrals (see [[#References|[8]]]–[[#References|[11]]]).

Latest revision as of 08:06, 6 June 2020


A formula for rearranging the order of integration in iterated improper integrals of Cauchy principal value type (cf. Improper integral).

Let $ \Gamma $ be a simple closed or open smooth curve in the complex plane, let $ \phi ( t , t _ {1} ) $ be a function defined on $ \Gamma $( generally speaking complex-valued) and satisfying a uniform Hölder condition with respect to $ t $ and $ t _ {1} $, and let $ t _ {0} $ be a fixed point on $ \Gamma $ which is not an end-point if $ \Gamma $ is open. Then one has the Poincaré–Bertrand formula

$$ \tag{1 } \int\limits _ \Gamma \frac{dt}{t - t _ {0} } \int\limits _ \Gamma \frac{\phi ( t , t _ {1} ) }{t _ {1} - t } d t _ {1\ } = $$

$$ = \ - \pi ^ {2} \phi ( t _ {0} , t _ {0} ) + \int\limits _ \Gamma d t _ {1} \int\limits _ \Gamma \frac{\phi ( t , t _ {1} ) }{ ( t - t _ {0} ) ( t _ {1} - t ) } d t . $$

The formula is true under more general assumptions on the curve $ \Gamma $ and the function $ \phi $( see [4]). If $ \phi ( t , t _ {1} ) = \alpha ( t) \beta ( t _ {1} ) $, where $ \alpha \in L _ {p} $, $ \beta \in L _ {q } $, $ q = p / ( p - 1 ) $, equation (1) is true for almost-all $ t _ {0} \in \Gamma $( see [5], [6]). If the curve $ \Gamma $ is closed and the function $ \phi $ depends on one variable only, then equation (1) takes the form

$$ \tag{2 } \frac{1}{( \pi i ) ^ {2} } \int\limits _ \Gamma \frac{dt}{t - t _ {0} } \int\limits _ \Gamma \frac{\phi ( t _ {1} ) }{t _ {1} - t } d t _ {1} = \phi ( t _ {0} ) , $$

and holds for all or almost-all $ t _ {0} \in \Gamma $, depending (respectively) on whether $ \phi $ satisfies a Hölder condition or $ \phi \in L _ {p} $, $ p > 1 $. Equation (2) is also called the Poincaré–Bertrand formula.

Analogues of formula (1) have been constructed for multiple integrals (see [8][11]).

Formula (1) was obtained, under certain conditions, by G.H. Hardy (see [7]) before H. Poincaré (see [1]) and G. Bertrand (see [2], [3]).

References

[1] H. Poincaré, "Les méthodes nouvelles de la mécanique céleste" , 3 , Gauthier-Villars (1899)
[2] G. Bertrand, "Equations de Fredholm à intégrales principales au sens de Cauchy" C.R. Acad. Sci. Paris , 172 (1921) pp. 1458–1461
[3] G. Bertrand, "La théorie des marées et les équations intégrales" Ann. Sci. Ecole Norm. Sup. , 40 (1923) pp. 151–258
[4] N.I. Muskhelishvili, "Singular integral equations" , Wolters-Noordhoff (1972) (Translated from Russian)
[5] B.V. Khvedelidze, "Some properties of singular integrals in the sense of the Cauchy–Lebesgue principal value" Soobsh. Akad. Nauk. GruzSSR , 8 : 5 (1947) pp. 283–290 (In Russian)
[6] B.V. Khvedelidze, "The method of Cauchy-type integrals in the discontinuous boundary-value problems of the theory of holomorphic functions of a complex variable" J. Soviet Math. , 7 : 3 (1977) pp. 309–415 Itogi Nauk. i Tekhn. Sovrem. Probl. Mat. , 7 (1975) pp. 5–162
[7] G.H. Hardy, "The theory of Cauchy's principal values" Proc. London Math. Soc. , 7 : 2 (1909) pp. 181–208
[8] F. Tricomi, "Equazioni integrali contenenti il valor principale doppio" Math. Z. , 27 (1928) pp. 87–133
[9] G. Giraud, "Sur une classe générale d'équations à intégrales principales" C.R. Acad. Sci. Paris , 202 : 26 (1936) pp. 2124–2127
[10] G. Giraud, "Equations à intégrales principales; étude suivie d'une application" Ann. Sci. Ecole Norm. Sup. , 51 : 3–4 (1934) pp. 251–372
[11] S.G. Mikhlin, "Singular integral equations" Uspekhi Mat. Nauk , 3 : 3 (1948) pp. 29–112 (In Russian)
[12] S.G. Mikhlin, "Multidimensional singular integrals and integral equations" , Pergamon (1965) (Translated from Russian)
How to Cite This Entry:
Poincaré-Bertrand formula. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Poincar%C3%A9-Bertrand_formula&oldid=14405
This article was adapted from an original article by B.V. Khvedelidze (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article