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logarithmic integral
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(Definition)
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The European or Eulerian version of logarithmic integral (in Latin logarithmus integralis) is defined as
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(1) |
and the American version is
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(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 $$\li{x} = \lim_{\varepsilon\to 0+}\left(\int_0^{1-\varepsilon}\!\frac{dt}{\ln{t}} +\int_{1+\varepsilon}^x\frac{dt}{\ln{t}}\right).$$ The connection between (1) and (2) is $$\Li{x} = \li{x}-\li{2}.$$ 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 $$\Li{x} = \frac{x}{\ln{x}}\sum_{n=0}^\infty\frac{n!}{(\ln{x})^n}.$$
The definition of the logarithmic integral may be extended to the whole complex plane, and one gets the analytic function $\Li{z}$ having the branch point $z = 1$ and the derivative $\displaystyle\frac{1}{\log{z}}$ .
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"logarithmic integral" is owned by pahio. [ full author list (2) ]
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Cross-references: derivative, branch point, analytic function, complex plane, series, prime counting function, approximation, prime number theorem, connection, Cauchy principal value, integrand
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 3784 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) |
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Pending Errata and Addenda
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