Poincaré-Bertrand formula

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

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

(1)

The formula is true under more general assumptions on the curve and the function (see [4]). If , where , , , equation (1) is true for almost-all (see [5], [6]). If the curve is closed and the function depends on one variable only, then equation (1) takes the form

(2)

and holds for all or almost-all , depending (respectively) on whether satisfies a Hölder condition or , . 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)