logarithmic integral

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

$\mathrm{Li}x:={\displaystyle {\int }_2^x}{\displaystyle \frac{dt}{\ln t}},$ (1)

and the American version is

$\mathrm{li}x:={\displaystyle {\int }_0^x}{\displaystyle \frac{dt}{\ln t}},$ (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

$$\mathrm{li}x=\underset{\epsilon \to 0+}{lim}\left({\int }_0^{1-\epsilon }\frac{dt}{\ln t}+{\int }_{1+\epsilon }^x\frac{dt}{\ln t}\right).$$

The connection between (1) and (2) is

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

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

One has the asymptotic series expansion

$$\mathrm{Li}x=\frac{x}{\ln x}\sum _{n=0}^{\mathrm{\infty }}\frac{n!}{{(\ln x)}^n}.$$

The definition of the logarithmic integral may be extended to the whole complex plane, and one gets the analytic function   $\mathrm{Li}z$  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