PlanetMath: logarithmic integral

(more info)

Math for the people, by the people.

Main Menu

sections Encyclopædia
Papers
Books
Expositions

meta Requests (198)
Orphanage
Unclass'd
Unproven (498)
Corrections (25)
Classification

talkback Polls
Forums
Feedback
Bug Reports

downloads Snapshots
PM Book

information News
Docs
Wiki
ChangeLog
TODO List
Legalese
About

Entry average rating: No information on entry rating

logarithmic integral

(Definition)

The European or Eulerian version of logarithmic integral (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

, and for

the latter definition is interpreted as the Cauchy principal value

$\displaystyle \operatorname{li}{x} = \lim_{\varepsilon\to 0+}\left(\int_0^{1-\varepsilon}\!\frac{dt}{\ln{t}} +\int_{1+\varepsilon}^x\frac{dt}{\ln{t}}\right).$

The connection between (1) and (2) is

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

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

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

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 and the derivative $\displaystyle\frac{1}{\log{z}}$ .

Anyone with an account can edit this entry. Please help improve it!

"logarithmic integral" is owned by pahio. [ full author list (2) ]

(view preamble)

Other names:

Also defines:

logarithmic integral, logarithmus integralis, Eulerian logarithmic integral

This object's parent.

Log in to rate this entry.
(view current ratings)

Cross-references: derivative, branch point, analytic function, complex plane, series, prime counting function, approximation, prime number theorem, connection, Cauchy principal value
There are 6 references to this entry.

This is version 11 of logarithmic integral, born on 2007-05-06, modified 2008-06-04.
Object id is 9341, canonical name is LogarithmicIntegral2.
Accessed 1944 times total.

Classification:

AMS MSC:	26A36 (Real functions :: Functions of one variable :: Antidifferentiation)
	30E20 (Functions of a complex variable :: Miscellaneous topics of analysis in the complex domain :: Integration, integrals of Cauchy type, integral representations of analytic functions)
	33E20 (Special functions :: Other special functions :: Other functions defined by series and integrals)

Pending Errata and Addenda

None.

Discussion

forum policy

Interact