Logarithmic integral function: Difference between revisions

Revision as of 19:07, 11 May 2004

Revision as of 10:42, 14 May 2004

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

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