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).