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The special function defined, for positive real  $  x $, 
$  x \neq 1 $, 
by

$$ 
 \mathop{\rm li} ( x)  = \ 
\int\limits _ { 0 } ^ { x } 

\frac{dt}{ \mathop{\rm ln}  t }
 ;
$$

for  $  x > 1 $
the integrand has at  $  t = 1 $
an infinite discontinuity and the integral logarithm is taken to be the principal value

$$ 
 \mathop{\rm li} ( x)  = \ 
\lim\limits _ {\epsilon \downarrow 0 } \ 
\left \{
\int\limits _ { 0 } ^ { {1 }  - \epsilon }

\frac{dt}{ \mathop{\rm ln}  t }
 +
\int\limits _ {1 + \epsilon } ^ { x } 

\frac{dt}{ \mathop{\rm ln}  t }

\right \} .
$$

The graph of the integral logarithm is given in the article [[Integral exponential function|Integral exponential function]]. For  $  x $
small:

$$ 
 \mathop{\rm li} ( x)  \approx  
\frac{x}{ \mathop{\rm ln} ( 1 / x ) }
 .
$$

The integral logarithm has for positive real  $  x $
the series representation

$$ 
 \mathop{\rm li} ( x)  =  c
+  \mathop{\rm ln}  |  \mathop{\rm ln}  x | +
\sum _ { k= 1} ^  \infty  

\frac{(  \mathop{\rm ln}  x )  ^ {k} }{k ! k }
 ,\ \ 
k > 0 ,\ \ 
x \neq 1 ,
$$

where  $  c = 0.5772 \dots $
is the [[Euler constant|Euler constant]]. As a function of the complex variable  $  z $,

$$ 
 \mathop{\rm li} ( z)  =  c +
 \mathop{\rm ln} ( -  \mathop{\rm ln}  z ) +
\sum _ { k=1 } ^  \infty  

\frac{(  \mathop{\rm ln}  z )  ^ {k} }{k ! k }

$$

is a single-valued analytic function in the complex  $  z $-
plane with slits along the real axis from  $  - \infty $
to 0 and from 1 to  $  + \infty $(
the imaginary part of the logarithms is taken within the limits  $  - \pi $
and  $  \pi $). 
The behaviour of  $   \mathop{\rm li}  x $
along  $  ( 1 , + \infty ) $
is described by

$$ 
\lim\limits _ {\eta \downarrow 0 }   \mathop{\rm li} ( x \pm  i \eta )
 =   \mathop{\rm li}  x \mp \pi i ,\ \ 
x > 1 .
$$

The integral logarithm is related to the [[Integral exponential function|integral exponential function]]  $   \mathop{\rm Ei} ( x) $
by

$$ 
 \mathop{\rm li} ( x)  = \ 
 \mathop{\rm Ei} (  \mathop{\rm ln}  x ) ,\ \ 
x < 1 ; \ \ 
 \mathop{\rm Ei} ( x)  = \ 
 \mathop{\rm li} ( e  ^ {x} ) ,\ \ 
x < 0 .
$$

For real  $  x > 0 $
one sometimes uses the notation

$$ 
 \mathop{\rm Li} ( x)  = \ 
\left \{ 
\begin{array}{ll}
 \mathop{\rm li} ( x)  =   \mathop{\rm Ei} (  \mathop{\rm ln}  x )   &\textrm{ for }  0 < x < 1 ,  \\
 \mathop{\rm li} ( x) + \pi i  =   \mathop{\rm Ei}  ^ {*} (  \mathop{\rm ln}  x )   &\textrm{ for }  x > 1 .  \\
\end{array}

 \right .$$

For references, see [[Integral cosine|Integral cosine]].

====Comments====
The function  $   \mathop{\rm li} $
is better known as the logarithmic integral. It can, of course, be defined by the integral (as above) for  $  z \in \mathbf C \setminus  \{ {x \in \mathbf R } : {x \leq  0 \textrm{ or }   x \geq  1 } \} $.

The series representation for positive  $  x $, 
$  x \neq 1 $, 
is then also said to define the modified logarithmic integral, and is the boundary value of  $   \mathop{\rm li} ( x + i \eta ) \pm  \pi i $, 
$  x > 1 $, 
$  \eta \rightarrow 0 $. 
For real  $  x > 1 $
the value  $   \mathop{\rm li} ( x) $
is a good approximation of  $  \pi ( x) $, 
the number of primes smaller than  $  x $
(see [[De la Vallée-Poussin theorem|de la Vallée-Poussin theorem]]; [[Distribution of prime numbers|Distribution of prime numbers]]; [[Prime number|Prime number]]).