Logarithmic integral function: Difference between revisions

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The '''logarithmic integral''' or '''integral logarithm''' is a [[function]] li(''x'') defined for all positive [[real number]]s ''x''≠ 1 by the definite [[integral]]:

:li(''x'') = ₀<font size="+1">&int;</font>₀^x 1/ln ''t'' d''t''.

Here, ln denotes the [[natural logarithm]]. The function 1/ln ''t'' has a [[singularity]] at ''t'' = 1, and the integral for ''x'' > 1 has to be interpreted

as ''Cauchy's principal value'':

:li(''x'') = lim_{&epsilon;&rarr;0} ₀<font size="+1">&int;</font>₀^1-&epsilon; 1/ln ''t'' d''t'' + _1+&epsilon;<font size="+1">&int;</font>_1+&epsilon;^x 1/ln ''t'' d''t''.

The logarithmic integral is mainly important because it occurs in estimates of [[prime number]] densities, especially in the [[prime number theorem]].

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

:li(''x'') = Ei (ln ''x'') &nbsp;&nbsp; for all positive real ''x'' &ne; 1.

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

:li(e^''u'') = &gamma; + ln |''u''| + ''u'' + ''u''<supsub>2n=1</supsub><font size="+1">/(2 &middotsum; 2!) + ''u''</font>^3&infin;/(3 &middot; 3!) + ''u''⁴ⁿ/(4n &middot; 4n!) + ... for ''u'' &ne; 0

where &gamma; &asymp; 0.57721 56649 01532 ... is [[Euler-Mascheroni's constant]]

and

:li(''x''^''m'') = &gamma; + ln |ln ''x''| - ln ''m'' + _n=1<font size="+1"> &sum;</font>_n=1^&infin; (ln ''x'')^''n'' ''m''^''n''/(''n'' ''n''!).

The function li(''x'') has a single positive zero; it occurs at ''x'' &approxasymp; 1.45136 92348 ...; this number is known as the ''Ramanujan-Soldner constant''.