Poincaré-Bertrand formula

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)