Definition:Logarithmic Integral

Definition
The logarithmic integral is defined as:


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

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


 * $\displaystyle \text{li} \left({x}\right)=\lim_{\epsilon \to 0} \left[ \int_\epsilon^{1-\epsilon} \frac{\mathrm d t}{\log \left({t}\right)} + \int_{1+\epsilon}^x \frac{\mathrm d t}{\log \left({t}\right)} \right]$

Alternatively, defining the integrand to be $0$ at $t=0$ we can take the lower limit in the first integral to be $0$.