# Definition:Logarithmic Integral

## Definition

The logarithmic integral is defined as:

$\displaystyle \map {\operatorname {li} } x = \PV_0^x \frac {\d t} {\map \ln t}$

where:

$\ln$ denotes the natural logarithm function
$\operatorname {PV}$ denotes the Cauchy principal value of the proceeding integral.

That is, as $\dfrac 1 {\ln t}$ has discontinuities at $t = 0$ and $t = 1$:

$\map {\operatorname {li} } x = \begin{cases}\displaystyle \lim_{\varepsilon \to 0^+} \paren {\int_{\varepsilon}^x \frac {\rd t} {\ln t} } & 0 < x < 1 \\ \displaystyle \lim_{\varepsilon \mathop \to 0^+} \paren {\int_\varepsilon^{1 - \varepsilon} \frac {\rd t} {\ln t} + \int_{1 + \varepsilon}^x \frac {\rd t} {\ln t} } & x > 1\end{cases}$

### Eulerian Logarithmic Interval

The Eulerian logarithmic integral is defined as:

$\displaystyle \map \Li x = \int_2^x \frac {\d t} {\map \ln t}$

## Also defined as

By defining the integrand to be $0$ at $t = 0$, the lower limit can be taken in the first integral to be $0$.

The logarithmic integral and the Eulerian logarithmic integral are not consistently denoted in the literature (some sources use $\map {\operatorname {li} } x$ to indicate the Eulerian version, for example).

It is therefore important to take care which is being referred to at any point.

## Also known as

The logarithmic integral is also seen referred to as the integral logarithm.