\begin{definition}[Definition:Logarithmic Integral]
The '''logarithmic integral''' is defined as:
:$\ds \map {\operatorname {li} } x = \PV_0^x \frac {\d t} {\map \ln t}$
where:
:$\ln$ denotes the natural logarithm function
:$\operatorname {PV}$ denotes the Cauchy principal value.
That is, as $\dfrac 1 {\ln t}$ has discontinuities at $t = 0$ and $t = 1$:
:$\map {\operatorname {li} } x = \begin {cases} \ds \lim_{\varepsilon \to 0^+} \paren {\int_{\varepsilon}^x \frac {\rd t} {\ln t} } & 0 < x < 1 \\ \ds \lim_{\varepsilon \mathop \to 0^+} \paren {\int_\varepsilon^{1 - \varepsilon} \frac {\rd t} {\ln t} + \int_{1 + \varepsilon}^x \frac {\rd t} {\ln t} } & x > 1 \end {cases}$
\end{definition}