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The improper integral (in the sense of the Cauchy principal value)

$$ 
\widetilde{f}  ( x)  = \ 
{
\frac{1}{2 \pi }
 }
\int\limits _ { 0 } ^ { {2 }  \pi }
f ( t)   \mathop{\rm cotan}  
\frac{x - t }{2 }
  dt,
$$

where the periodic function  $  f $
is called the density of the Hilbert singular integral, while  $   \mathop{\rm cotan} \{ {( x - t)/2 } \} $
is called its kernel. If  $  f $
is summable,  $  \widetilde{f}  $
exists almost-everywhere; if  $  f $
satisfies the Lipschitz condition of order  $  \alpha $, 
$  0 < \alpha < 1 $, 
$  \widetilde{f}  $
exists for any  $  x $
and satisfies this condition as well. If  $  f $
has summable  $  p $-
th power,  $  p > 1 $, 
$  \widetilde{f}  $
has the same property, and

$$ 
\left \{ \int\limits _ { 0 } ^ { {2 }  \pi }
| \widetilde{f}  ( x) |  ^ {p}  dx \right \}  ^ {1/p}  \leq   M _ {p} \left \{ \int\limits _ { 0 } ^ { {2 }  \pi }
| f ( x) |  ^ {p}  dx \right \}  ^ {1/p} ,
$$

where  $  M _ {p} $
is a constant independent of  $  f $. 
In addition, the inversion formula of Hilbert's singular integral,

$$ 
f ( x)  = \ 
{
\frac{1}{2 \pi }
 }
\int\limits _ { 0 } ^ { {2 }  \pi }
\widetilde{f}  ( t)   \mathop{\rm cotan}  
\frac{t - x }{2 }
  dt +
{
\frac{1}{2 \pi }
 }
\int\limits _ { 0 } ^ { {2 }  \pi } f ( t)  dt ,
$$

is valid. The function  $  \widetilde{f}  $
is said to be conjugate with  $  f $.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  D. Hilbert,   "Grundzüge einer allgemeinen Theorie der linearen Integralgleichungen" , Chelsea, reprint  (1953)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  M. Riesz,   "Sur les fonctions conjugées"  ''Math. Z.'' , '''27'''  (1927)  pp. 218–244</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  N.K. [N.K. Bari] Bary,   "A treatise on trigonometric series" , Pergamon  (1964)  (Translated from Russian)</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></table>

====Comments====
See also [[Hilbert kernel|Hilbert kernel]]; [[Hilbert transform|Hilbert transform]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A. Zygmund,   "Trigonometric series" , '''1–2''' , Cambridge Univ. Press  (1988)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  B.L. Moiseiwitsch,   "Integral equations" , Longman  (1977)</TD></TR></table>