Logarithmic integral function: Difference between revisions

Revision as of 21:30, 24 September 2014

Revision as of 16:51, 7 November 2014

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

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

:<math> {\rm li} (x) = \lim_{\varepsilon \to 0+} \left( \int_0^{1-\varepsilon} \frac{dt}{\ln t} + \int_{1+\varepsilon}^x \frac{dt}{\ln t} \right). \; </math>