ExpIntegralEi[x] == -Integrate[1/(E^t t), {t, -x, Infinity}, PrincipalValue -> True] /; Element[x, Reals]

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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <mi> Ei </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mo> - </mo> <mi> &#119979; </mi> </mrow> <mo> &#8290; </mo> <mrow> <msubsup> <mo> &#8747; </mo> <mrow> <mo> - </mo> <mi> x </mi> </mrow> <mi> &#8734; </mi> </msubsup> <mrow> <mfrac> <msup> <mi> &#8519; </mi> <mrow> <mo> - </mo> <mi> t </mi> </mrow> </msup> <mi> t </mi> </mfrac> <mo> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> t </mi> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> x </mi> <mo> &#8712; </mo> <semantics> <mi> &#8477; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalR]&quot;, Function[Reals]] </annotation> </semantics> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> ExpIntegralEi </ci> <ci> x </ci> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#119979; </ci> </apply> <apply> <int /> <bvar> <ci> t </ci> </bvar> <lowlimit> <apply> <times /> <cn type='integer'> -1 </cn> <ci> x </ci> </apply> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> t </ci> </apply> </apply> <apply> <power /> <ci> t </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <in /> <ci> x </ci> <reals /> </apply> </apply> </annotation-xml> </semantics> </math>

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2001-10-29