Integral logarithm
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. 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. 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 $ \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.
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; Distribution of prime numbers; Prime number).
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