Definition:Logarithmic Integral

Definition
The logarithmic integral is defined as:


 * $\displaystyle \operatorname {li} \left({x}\right) = \int_0^x \frac {\mathrm d t} {\ln \left({t}\right)}$

where $\ln$ denotes the natural logarithm function.

Since $\dfrac 1 {\ln \left({t}\right)}$ is undefined at $t = 0$ and $t = 1$, this is interpreted to mean:


 * $\displaystyle \operatorname {li} \left({x}\right) = \lim_{\epsilon \mathop \to 0} \left({\int_\epsilon^{1 - \epsilon} \frac {\mathrm d t} {\ln \left({t}\right)} + \int_{1 + \epsilon}^x \frac {\mathrm d t} {\ln \left({t}\right)} }\right)$

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 $0$.

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