Logarithmic integral function: Difference between revisions

Revision as of 14:58, 18 July 2010

Revision as of 19:32, 24 July 2010

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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'' = 1, and the integral for ''x'' > 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>