Exponential integral Ei: Operations (formula 06.35.25.0001)

ExpIntegralEi

$Mathematica Notation$

http://functions.wolfram.com/06.35.25.0001.01

Input Form

Limit[ExpIntegralEi[a + b x], x -> Infinity] == Piecewise[{{Pi I, (0 < Arg[a] < Pi && Pi/2 <= Arg[b] <= Pi) || (Im[a] <= 0 && Inequality[Pi/2, LessEqual, Arg[b], Less, Pi])}, {(-Pi) I, (0 <= Arg[a] <= Pi && Arg[b] <= -(Pi/2)) || (-Pi < Arg[a] < 0 && (Arg[b] <= -(Pi/2) || Arg[b] == Pi))}, {0, Im[a] == 0 && Arg[b] == Pi}}, ComplexInfinity]

Standard Form

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MathML Form

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <munder> <mi> lim </mi> <mrow> <mi> x </mi> <semantics> <mo> → </mo> <annotation encoding='Mathematica'> "\[Rule]" </annotation> </semantics> <mi> ∞ </mi> </mrow> </munder> <mo> ⁢ </mo> <mtext>   </mtext> <mrow> <mi> Ei </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mi> x </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo>  </mo> <mrow> <mo>  </mo> <mtable> <mtr> <mtd> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> ⅈ </mi> </mrow> </mtd> <mtd> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 0 </mn> <mo> < </mo> <mrow> <mi> arg </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> <mo> < </mo> <mi> π </mi> </mrow> <mo> ∧ </mo> <mrow> <mfrac> <mi> π </mi> <mn> 2 </mn> </mfrac> <mo> ≤ </mo> <mrow> <mi> arg </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> b </mi> <mo> ) </mo> </mrow> <mo> ≤ </mo> <mi> π </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ∨ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mi> Im </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> <mo> ≤ </mo> <mn> 0 </mn> </mrow> <mo> ∧ </mo> <mrow> <mfrac> <mi> π </mi> <mn> 2 </mn> </mfrac> <mo> ≤ </mo> <mrow> <mi> arg </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> b </mi> <mo> ) </mo> </mrow> <mo> < </mo> <mi> π </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mrow> <mo> - </mo> <mi> π </mi> </mrow> <mo> ⁢ </mo> <mi> ⅈ </mi> </mrow> </mtd> <mtd> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 0 </mn> <mo> ≤ </mo> <mrow> <mi> arg </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> <mo> ≤ </mo> <mi> π </mi> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mi> arg </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> b </mi> <mo> ) </mo> </mrow> <mo> ≤ </mo> <mrow> <mo> - </mo> <mfrac> <mi> π </mi> <mn> 2 </mn> </mfrac> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ∨ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mi> π </mi> </mrow> <mo> < </mo> <mrow> <mi> arg </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> <mo> < </mo> <mn> 0 </mn> </mrow> <mo> ∧ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mi> arg </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> b </mi> <mo> ) </mo> </mrow> <mo> ≤ </mo> <mrow> <mo> - </mo> <mfrac> <mi> π </mi> <mn> 2 </mn> </mfrac> </mrow> </mrow> <mo> ∨ </mo> <mrow> <mrow> <mi> arg </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> b </mi> <mo> ) </mo> </mrow> <mo>  </mo> <mi> π </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mn> 0 </mn> </mtd> <mtd> <mrow> <mrow> <mrow> <mi> Im </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> <mo>  </mo> <mn> 0 </mn> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mi> arg </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> b </mi> <mo> ) </mo> </mrow> <mo>  </mo> <mi> π </mi> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mover> <mi> ∞ </mi> <mo> ~ </mo> </mover> </mtd> <mtd> <semantics> <mi> True </mi> <annotation encoding='Mathematica'> TagBox["True", "PiecewiseDefault", Rule[AutoDelete, False], Rule[DeletionWarning, True]] </annotation> </semantics> </mtd> </mtr> </mtable> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <limit /> <bvar> <ci> x </ci> </bvar> <condition> <apply> <tendsto /> <ci> x </ci> <infinity /> </apply> </condition> <apply> <ci> ExpIntegralEi </ci> <apply> <plus /> <ci> a </ci> <apply> <times /> <ci> b </ci> <ci> x </ci> </apply> </apply> </apply> </apply> <piecewise> <piece> <apply> <times /> <pi /> <imaginaryi /> </apply> <apply> <or /> <apply> <and /> <apply> <lt /> <cn type='integer'> 0 </cn> <apply> <arg /> <ci> a </ci> </apply> <pi /> </apply> <apply> <leq /> <apply> <times /> <pi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <arg /> <ci> b </ci> </apply> <pi /> </apply> </apply> <apply> <and /> <apply> <leq /> <apply> <imaginary /> <ci> a </ci> </apply> <cn type='integer'> 0 </cn> </apply> <apply> <ci> Inequality </ci> <apply> <times /> <pi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <leq /> <apply> <arg /> <ci> b </ci> </apply> <lt /> <pi /> </apply> </apply> </apply> </piece> <piece> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <pi /> </apply> <imaginaryi /> </apply> <apply> <or /> <apply> <and /> <apply> <leq /> <cn type='integer'> 0 </cn> <apply> <arg /> <ci> a </ci> </apply> <pi /> </apply> <apply> <leq /> <apply> <arg /> <ci> b </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <pi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <lt /> <apply> <times /> <cn type='integer'> -1 </cn> <pi /> </apply> <apply> <arg /> <ci> a </ci> </apply> <cn type='integer'> 0 </cn> </apply> <apply> <or /> <apply> <leq /> <apply> <arg /> <ci> b </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <pi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <eq /> <apply> <arg /> <ci> b </ci> </apply> <pi /> </apply> </apply> </apply> </apply> </piece> <piece> <cn type='integer'> 0 </cn> <apply> <and /> <apply> <eq /> <apply> <imaginary /> <ci> a </ci> </apply> <cn type='integer'> 0 </cn> </apply> <apply> <eq /> <apply> <arg /> <ci> b </ci> </apply> <pi /> </apply> </apply> </piece> <otherwise> <apply> <ci> OverTilde </ci> <infinity /> </apply> </otherwise> </piecewise> </apply> </annotation-xml> </semantics> </math>

Rule Form

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Date Added to functions.wolfram.com (modification date)

2007-05-02