Limit[SinhIntegral[a + b x], x -> Infinity] == Piecewise[{{(Pi I)/2, Arg[b] == Pi/2}, {-((Pi I)/2), Arg[b] == -(Pi/2)}, {Infinity, Im[a] == 0 && Arg[b] == 0}, {-Infinity, Im[a] == 0 && Arg[b] == Pi}}, ComplexInfinity]

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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> <munder> <mi> lim </mi> <mrow> <mi> x </mi> <semantics> <mo> &#8594; </mo> <annotation encoding='Mathematica'> &quot;\[Rule]&quot; </annotation> </semantics> <mi> &#8734; </mi> </mrow> </munder> <mo> &#8290; </mo> <mtext> &#8201; </mtext> <mrow> <mi> Shi </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mi> x </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> &#63449; </mo> <mrow> <mo> &#62305; </mo> <mtable> <mtr> <mtd> <mfrac> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#8520; </mi> </mrow> <mn> 2 </mn> </mfrac> </mtd> <mtd> <mrow> <mrow> <mi> arg </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> b </mi> <mo> ) </mo> </mrow> <mo> &#63449; </mo> <mfrac> <mi> &#960; </mi> <mn> 2 </mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#8520; </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mtd> <mtd> <mrow> <mrow> <mi> arg </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> b </mi> <mo> ) </mo> </mrow> <mo> &#63449; </mo> <mrow> <mo> - </mo> <mfrac> <mi> &#960; </mi> <mn> 2 </mn> </mfrac> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mi> &#8734; </mi> </mtd> <mtd> <mrow> <mrow> <mrow> <mi> Im </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> <mo> &#63449; </mo> <mn> 0 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mi> arg </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> b </mi> <mo> ) </mo> </mrow> <mo> &#63449; </mo> <mn> 0 </mn> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo> - </mo> <mi> &#8734; </mi> </mrow> </mtd> <mtd> <mrow> <mrow> <mrow> <mi> Im </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> <mo> &#63449; </mo> <mn> 0 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mi> arg </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> b </mi> <mo> ) </mo> </mrow> <mo> &#63449; </mo> <mi> &#960; </mi> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mover> <mi> &#8734; </mi> <mo> ~ </mo> </mover> </mtd> <mtd> <semantics> <mi> True </mi> <annotation encoding='Mathematica'> TagBox[&quot;True&quot;, &quot;PiecewiseDefault&quot;, 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> SinhIntegral </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> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <eq /> <apply> <arg /> <ci> b </ci> </apply> <apply> <times /> <pi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </piece> <piece> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <times /> <pi /> <imaginaryi /> </apply> </apply> <apply> <eq /> <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> </piece> <piece> <infinity /> <apply> <and /> <apply> <eq /> <apply> <imaginary /> <ci> a </ci> </apply> <cn type='integer'> 0 </cn> </apply> <apply> <eq /> <apply> <arg /> <ci> b </ci> </apply> <cn type='integer'> 0 </cn> </apply> </apply> </piece> <piece> <apply> <times /> <cn type='integer'> -1 </cn> <infinity /> </apply> <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>

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2007-05-02