Logarithmic integral function: Difference between revisions

Revision as of 18:46, 15 November 2005

Revision as of 18:29, 2 December 2005

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In [[mathematics]], the '''logarithmic integral function''' or '''integral logarithm''' li(''x'') is a [[function (mathematics)|non-elementary function]] defined for all positive [[real number]]s ''<math>x''&ne; 1\ne1</math> by the [[integral|definite integral]]:

:<math> {\rm li} (x) = \int_{0}^{x} \frac{dt}{\ln (t)} \; . </math>

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The logarithmic integral finds application in many areas, in particular it is used is in estimates of [[prime number]] densities, such as the [[prime number theorem]]:

:&<math>\pi;(''x'') ~ \sim\hbox{li}(''x'') ~ \sim\hbox{Li}(''x'') </math>

where &pi;(''x'') denotes the number of primes smaller than or equal to ''x''.

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The function li(''x'') is related to the ''[[exponential integral]]'' Ei(''x'') via the equation

:<math>\hbox{li}(''x'') = \hbox{Ei }(\ln (''x'')) &nbsp;&nbsp; \quad\hbox{for all positive real ''}x'' &ne; 1\ne1.</math>

This leads to series expansions of li(''x''), for instance: