Definition:Logarithmic Integral/Also defined as

Logarithmic Integral: Also defined as

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

Hence:

$\map \li x = \begin {cases} \ds \int_0^x \frac {\rd t} {\ln t} & : 0 \le x < 1 \\ \ds \lim_{\varepsilon \mathop \to 0^+} \paren {\int_0^{1 - \varepsilon} \frac {\rd t} {\ln t} + \int_{1 + \varepsilon}^x \frac {\rd t} {\ln t} } & : x > 1 \end {cases}$

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