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

, 

where "PV-int" means Cauchy Principal Value integral.

This definition is extended to complex arguments  via the formula .  Note that the resulting branch cuts are the intervals  and . However, since  is defined as a Cauchy principal value integral, the values on the branch cuts are "isolated". That is, the complex function  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.