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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]]).

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} }

\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 [[#References|[8]]]–[[#References|[11]]]).

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

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Poincaré,   "Les méthodes nouvelles de la mécanique céleste" , '''3''' , Gauthier-Villars  (1899)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  G. Bertrand,   "Equations de Fredholm à intégrales principales au sens de Cauchy"  ''C.R. Acad. Sci. Paris'' , '''172'''  (1921)  pp. 1458–1461</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  G. Bertrand,   "La théorie des marées et les équations intégrales"  ''Ann. Sci. Ecole Norm. Sup.'' , '''40'''  (1923)  pp. 151–258</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  N.I. Muskhelishvili,   "Singular integral equations" , Wolters-Noordhoff  (1972)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  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)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  G.H. Hardy,   "The theory of Cauchy's principal values"  ''Proc. London Math. Soc.'' , '''7''' :  2  (1909)  pp. 181–208</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  F. Tricomi,   "Equazioni integrali contenenti il valor principale doppio"  ''Math. Z.'' , '''27'''  (1928)  pp. 87–133</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  G. Giraud,   "Sur une classe générale d'équations à intégrales principales"  ''C.R. Acad. Sci. Paris'' , '''202''' :  26  (1936)  pp. 2124–2127</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  G. Giraud,   "Equations à intégrales principales; étude suivie d'une application"  ''Ann. Sci. Ecole Norm. Sup.'' , '''51''' :  3–4  (1934)  pp. 251–372</TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top">  S.G. Mikhlin,   "Singular integral equations"  ''Uspekhi Mat. Nauk'' , '''3''' :  3  (1948)  pp. 29–112  (In Russian)</TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top">  S.G. Mikhlin,   "Multidimensional singular integrals and integral equations" , Pergamon  (1965)  (Translated from Russian)</TD></TR></table>