# logarithmic integral

The European or Eulerian version of (in Latin logarithmus integralis) is defined as

 $\displaystyle\operatorname{Li}{x}:=\int_{2}^{x}\frac{dt}{\ln{t}},$ (1)

and the American version is

 $\displaystyle\operatorname{li}{x}:=\int_{0}^{x}\frac{dt}{\ln{t}},$ (2)

The integrand $\displaystyle\frac{1}{\ln{t}}$ has a singularity  $t=1$,  and for  $x>1$  the latter definition is interpreted as the Cauchy principal value

 $\operatorname{li}{x}=\lim_{\varepsilon\to 0+}\left(\int_{0}^{1-\varepsilon}\!% \frac{dt}{\ln{t}}+\int_{1+\varepsilon}^{x}\frac{dt}{\ln{t}}\right).$

The connection between (1) and (2) is

 $\operatorname{Li}{x}=\operatorname{li}{x}-\operatorname{li}{2}.$

The logarithmic integral appears in some physical problems and in a formulation of the prime number theorem ($\operatorname{Li}{x}$  gives a slightly better approximation for the prime counting function than  $\operatorname{li}{x}$).

One has the asymptotic series expansion

 $\operatorname{Li}{x}=\frac{x}{\ln{x}}\sum_{n=0}^{\infty}\frac{n!}{(\ln{x})^{n}}.$

The definition of the logarithmic integral may be extended to the whole complex plane, and one gets the analytic function$\operatorname{Li}{z}$  having the branch point$z=1$  and the derivative$\displaystyle\frac{1}{\log{z}}$.

 Title logarithmic integral Canonical name LogarithmicIntegral Date of creation 2013-03-22 17:03:05 Last modified on 2013-03-22 17:03:05 Owner pahio (2872) Last modified by pahio (2872) Numerical id 14 Author pahio (2872) Entry type Definition Classification msc 30E20 Classification msc 33E20 Classification msc 26A36 Synonym Li Related topic SineIntegral Related topic PrimeNumberTheorem Related topic PrimeCountingFunction Related topic LaTeXSymbolForCauchyPrincipalValue Related topic ConvergenceOfIntegrals Defines logarithmic integral Defines logarithmus integralis Defines Eulerian logarithmic integral