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The [[Integral transform|integral transform]]

$$ \tag{* }
f ( x)  =  
\frac{1} \pi 

\int\limits _ {- \infty } ^ {+\infty }

\frac{1}{1 + ( x - t )  ^ {2} }
  d \alpha ( t)
$$

where  $  \alpha ( t) $
is a [[Function of bounded variation|function of bounded variation]] in every finite interval, and also the transform

$$ 
f ( x)  =  
\frac{1} \pi 

\int\limits _ {- \infty } ^  \infty  

\frac{\phi ( t) }{1 + ( x - t )  ^ {2} }
 \ 
d t
$$

which results from (*) if  $  \alpha ( t) $
is an absolutely-continuous function (cf. [[Absolute continuity|Absolute continuity]]). Let

$$ 
\widehat{g}  ( x)  =  - 
\frac{1} \pi 

\int\limits _ { 0 } ^  \infty  

\frac{g ( x + u ) - 2 g ( x) + g( x - u ) }{u  ^ {2} }
  d u
$$

and let

$$ 
T _ {t} g ( x)  = \ 
\sum_{k=0}^  \infty   ( - 1 )  ^ {k}

\frac{t  ^ {2k} }{( 2 k ) ! }

g ^ {( 2 k ) } ( x) +
$$

$$ 
+ 
\sum_{k=0}^  \infty   ( - 1 )  ^ {k} 
\frac{t  ^ {2k+1} }{( 2 k + 1 ) ! }
 \widehat{g}  {}  ^ {(2k)} ( x) .
$$

The following inversion formulas hold for the Poisson transform:

$$ 

\frac{\alpha ( x + 0 ) + \alpha ( x - 0 ) }{2}
 -

\frac{\alpha ( + 0 ) + \alpha ( - 0 ) }{2\ }
 =
$$

$$ 
= \ 
\lim\limits _ {t \uparrow 1 }  \int\limits _ { 0 } ^ { x }  T _ {t} f ( u)  d u
$$

for all  $  x $, 
and

$$ 
\phi ( x)  = \ 
\lim\limits _ {t \uparrow 1 } \ 
T _ {t} f ( x)
$$

almost everywhere.

Let  $  C $
be a convex open acute cone in  $  \mathbf R  ^ {n} $
with vertex at zero and let  $  C  ^ {*} $
be the dual cone, that is,

$$ 
C  ^ {*}  =  \{  \xi  : {\xi _ {1} x _ {1} + \dots + \xi _ {n} x _ {n} \geq  0  \textrm{ for  all  }  x \in C } \}
.
$$

The function

$$ 
{\mathcal K} _ {C} ( z)  = \ 
\int\limits _ { C }
e ^ {i ( z _ {1} \xi _ {1} + \dots + z _ {n} \xi _ {n} ) }  d \xi
$$

is called the Cauchy kernel of the tube domain  $  T  ^ {C} = \{ {z = x + i y } : {x \in \mathbf R  ^ {n} ,  y \in C } \} $. 
The Poisson transform of a (generalized) function  $  f $
is the convolution (cf. [[Convolution of functions|Convolution of functions]])

$$ 
f \star {\mathcal P} _ {C} ( x , y ) ,\ \ 
( x , y ) \in T  ^ {C} ,
$$

where

$$ 
{\mathcal P} _ {C} ( x , y )  = \ 

\frac{| {\mathcal K} _ {C} ( x + i y ) |  ^ {2} }{( 2 \pi )  ^ {n} {\mathcal K} _ {C} ( i y ) }

$$

is the Poisson kernel of the tube domain  $  T  ^ {C} $(
see [[#References|[2]]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Pollard,   "The Poisson transform"  ''Trans. Amer. Math. Soc.'' , '''78''' :  2  (1955)  pp. 541–550</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V.S. Vladimirov,   "Generalized functions in mathematical physics" , MIR  (1977)  (Translated from Russian)</TD></TR></table>