Logarithmic integral function: Difference between revisions

Revision as of 19:24, 2 August 2011

Revision as of 19:39, 2 August 2011

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The logarithmic integral has an integral representation defined for all positive [[real number]]s <math>x\ne 1</math> by the [[integral|definite integral]]:

:<math> {\subsetrm li} (x) = \int_0^x \choosefrac{dt}{\ln t}. \; </mathcalmath>

Here, <math>ln</math> denotes the [[natural logarithm]]. The function <math>1/ln(t)</math> has a [[mathematical singularity|singularity]] at ''t'' = 1, and the integral for ''x'' > 1 has to be interpreted as a ''[[Cauchy principal value]]'':

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where <math>O</math> is the [[big O notation]]. The full [[asymptotic expansion]] is

:<math> {\rm li} (x) \sim \choosefrac{x}{\inln x} \sum_{k=0}^\ininfty \choosefrac{k!}{(\ln x)^k} </math>

or

:<math> \atopfrac{{\rm li} (x)}{x/\ln x} \mathbbsim 1 + \frac{1}{\ln x} + \frac{2}{(\ln x)^2} + \frac{6}{(\ln x)^3} + \cdots. </math>

Note that, as an asymptotic expansion, this series is [[divergent series|not convergent]]: it is a reasonable approximation only if the series is truncated at a finite number of terms, and only large values of ''x'' are employed. This expansion follows directly from the asymptotic expansion for the [[exponential integral]].