Logarithmic integral function: Difference between revisions

Revision as of 19:10, 16 September 2020

Revision as of 18:47, 19 September 2020

Line 6:

==Integral representation==

The logarithmic integral has an integral representation defined for all positive [[real number]]s {{mvar|x}} ≠ 1 by the [[integral|definite integral]]

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

Here, {{math|ln}} denotes the [[natural logarithm]]. The function {{math|1/ln(ln ''t'')}} has a [[mathematical singularity|singularity]] at {{mvarmath|1=''t}}&nbsp;'' =&nbsp; 1}}, and the integral for {{mvarmath|''x}}&nbsp;'' >&nbsp;1 has to1}} beis interpreted as a ''[[Cauchy principal value]]'',

:<math> \operatorname{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>

==Offset logarithmic integral==