Logarithmic Integral

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logarithmic integral

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

Li⁡x:=∫2xd⁢tln⁡t, (1)

and the American version is

li⁡x:=∫0xd⁢tln⁡t, (2)

The integrand 1ln⁡t has a singularity  t=1,  and for  x>1  the latter definition is interpreted as the Cauchy principal value

li⁡x=limε→0+⁡(∫01-εd⁢tln⁡t+∫1+εxd⁢tln⁡t).

The connection between (1) and (2) is

Li⁡x=li⁡x-li⁡2.

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

One has the asymptotic series expansion

Li⁡x=xln⁡x⁢∑n=0∞n!(ln⁡x)n.

The definition of the logarithmic integral may be extended to the whole complex plane, and one gets the analytic function   Li⁡z  having the branch point  z=1  and the derivative  1log⁡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

Tag » Approximation Logarithmic Integral