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单词 LogarithmicIntegral
释义

logarithmic integral


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

Lix:=2xdtlnt,(1)

and the American version is

lix:=0xdtlnt,(2)

The integrand 1lnt has a singularity  t=1, and for  x>1  the latter definition is interpreted asthe Cauchy principal valueDlmfMathworld

lix=limε0+(01-εdtlnt+1+εxdtlnt).

The connection between (1) and (2) is

Lix=lix-li2.

The logarithmic integral appears in some physical problemsand in a formulation of the prime number theorem (Lix  givesa slightly better approximation for the prime counting function than  lix).

One has the asymptotic series expansion

Lix=xlnxn=0n!(lnx)n.

The definition of the logarithmic integral may be extended to the wholecomplex plane, and one gets the analytic functionMathworldPlanetmathLiz  havingthe branch pointMathworldPlanetmathz=1  and the derivative  1logz.

Titlelogarithmic integral
Canonical nameLogarithmicIntegral
Date of creation2013-03-22 17:03:05
Last modified on2013-03-22 17:03:05
Ownerpahio (2872)
Last modified bypahio (2872)
Numerical id14
Authorpahio (2872)
Entry typeDefinition
Classificationmsc 30E20
Classificationmsc 33E20
Classificationmsc 26A36
SynonymLi
Related topicSineIntegral
Related topicPrimeNumberTheorem
Related topicPrimeCountingFunction
Related topicLaTeXSymbolForCauchyPrincipalValue
Related topicConvergenceOfIntegrals
Defineslogarithmic integral
Defineslogarithmus integralis
DefinesEulerian logarithmic integral
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更新时间:2025/5/4 20:38:29