The logarithmic integral, Li(x), is defined as:

$\mathrm{Li}\left(x\right)=\mathrm{PV}{\int }_0^x\frac{1}{\ln \left(t\right)}\ⅆt$ , $x\ge 0$ 

$=\mathrm{Ei}\left(\ln \left(x\right)\right)$ 

where the integral is a Cauchy Principal Value integral.

This definition is extended to complex arguments $z$ via the formula $\mathrm{Li}\left(x\right)=\mathrm{Ei}\left(\ln \left(x\right)\right)$.  Note that the resulting branch cuts are the intervals $\left(-\mathrm{\infty }\,0\right)$ and $\left(0\,1\right)$. However, since $\mathrm{Li}\left(x\right)$ is defined as a Cauchy principal value integral, the values on the branch cuts are "isolated". That is, the complex function $\mathrm{Li}\left(z\right)$ is not continuous onto the branch cuts from either above or below.

Li(x) provides an approximation to the number of primes less than or equal to x.

$\mathrm{Li}\left(x\right)$

$\mathrm{Li}\left(x\right)$

$\mathrm{Li}\left(10.\right)$

$6.165599505$

$\mathrm{Li}\left(1000.\right)$

$177.6096580$

$\mathrm{nops}\left(\mathrm{select}\left(\mathrm{isprime}\,\left[\mathrm{`\$`}\left(1..1000\right)\right]\right)\right)$

$168$

$\mathrm{convert}\left(\mathrm{Li}\left(x\right)\,\mathrm{Ei}\right)$

$\mathrm{Ei}\left(\ln \left(x\right)\right)$

$\mathrm{Li}\left(1.+\mathrm{I}\right)$

$0.6139116692+2.059584214\mathrm{I}$

$\mathrm{Li}\left(0.5\right)$

$\mathrm{-0.3786710431}$

$\mathrm{Li}\left(0.5+0.\mathrm{I}\right)$

$\mathrm{-0.3786710431}+3.141592654\mathrm{I}$

$\mathrm{Li}\left(0.5-0.\mathrm{I}\right)$

$\mathrm{-0.3786710431}-3.141592654\mathrm{I}$

$\mathrm{Li}\left(-3.123\right)$

$\mathrm{-0.06158134361}+4.063328884\mathrm{I}$