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http://functions.wolfram.com/06.37.25.0001.01
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Limit[SinIntegral[a + b x], x -> Infinity] ==
Piecewise[{{Pi/2, Arg[b] == 0}, {-(Pi/2), Arg[b] == Pi},
{I Infinity, Abs[Arg[a]] == Pi/2 && Arg[b] == Pi/2},
{(-I) Infinity, Abs[Arg[a]] == Pi/2 && Arg[b] == -(Pi/2)}},
ComplexInfinity]
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Cell[BoxData[RowBox[List[RowBox[List["Limit", "[", RowBox[List[RowBox[List["SinIntegral", "[", RowBox[List["a", " ", "+", RowBox[List["b", " ", "x"]]]], "]"]], ",", RowBox[List["x", "\[Rule]", "\[Infinity]"]]]], "]"]], "\[Equal]", RowBox[List["Piecewise", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["\[Pi]", "2"], ",", RowBox[List[RowBox[List["Arg", "[", "b", "]"]], "\[Equal]", "0"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["-", FractionBox["\[Pi]", "2"]]], ",", RowBox[List[RowBox[List["Arg", "[", "b", "]"]], "\[Equal]", "\[Pi]"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "Infinity"]], ",", RowBox[List[RowBox[List[RowBox[List["Abs", "[", RowBox[List["Arg", "[", "a", "]"]], "]"]], "\[Equal]", FractionBox["\[Pi]", "2"]]], "&&", RowBox[List[RowBox[List["Arg", "[", "b", "]"]], "\[Equal]", FractionBox["\[Pi]", "2"]]]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "Infinity"]], ",", RowBox[List[RowBox[List[RowBox[List["Abs", "[", RowBox[List["Arg", "[", "a", "]"]], "]"]], "\[Equal]", FractionBox["\[Pi]", "2"]]], "&&", RowBox[List[RowBox[List["Arg", "[", "b", "]"]], "\[Equal]", RowBox[List["-", FractionBox["\[Pi]", "2"]]]]]]]]], "}"]]]], "}"]], ",", "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> → </mo> <annotation encoding='Mathematica'> "\[Rule]" </annotation> </semantics> <mi> ∞ </mi> </mrow> </munder> <mo> ⁢ </mo> <mtext>   </mtext> <mrow> <mi> Si </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> <mfrac> <mi> π </mi> <mn> 2 </mn> </mfrac> </mtd> <mtd> <mrow> <mrow> <mi> arg </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> b </mi> <mo> ) </mo> </mrow> <mo>  </mo> <mn> 0 </mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo> - </mo> <mfrac> <mi> π </mi> <mn> 2 </mn> </mfrac> </mrow> </mtd> <mtd> <mrow> <mrow> <mi> arg </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> b </mi> <mo> ) </mo> </mrow> <mo>  </mo> <mi> π </mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> ∞ </mi> </mrow> </mtd> <mtd> <mrow> <mrow> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mrow> <mi> arg </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> <mo>  </mo> <mfrac> <mi> π </mi> <mn> 2 </mn> </mfrac> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mi> arg </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> b </mi> <mo> ) </mo> </mrow> <mo>  </mo> <mfrac> <mi> π </mi> <mn> 2 </mn> </mfrac> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mrow> <mo> - </mo> <mi> ⅈ </mi> </mrow> <mo> ⁢ </mo> <mi> ∞ </mi> </mrow> </mtd> <mtd> <mrow> <mrow> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mrow> <mi> arg </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> <mo>  </mo> <mfrac> <mi> π </mi> <mn> 2 </mn> </mfrac> </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> </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> SinIntegral </ci> <apply> <plus /> <ci> a </ci> <apply> <times /> <ci> b </ci> <ci> x </ci> </apply> </apply> </apply> </apply> <piecewise> <piece> <apply> <times /> <pi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <eq /> <apply> <arg /> <ci> b </ci> </apply> <cn type='integer'> 0 </cn> </apply> </piece> <piece> <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> <eq /> <apply> <arg /> <ci> b </ci> </apply> <pi /> </apply> </piece> <piece> <apply> <times /> <imaginaryi /> <infinity /> </apply> <apply> <and /> <apply> <eq /> <apply> <abs /> <apply> <arg /> <ci> a </ci> </apply> </apply> <apply> <times /> <pi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </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> </apply> </piece> <piece> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <infinity /> </apply> <apply> <and /> <apply> <eq /> <apply> <abs /> <apply> <arg /> <ci> a </ci> </apply> </apply> <apply> <times /> <pi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </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> </apply> </piece> <otherwise> <apply> <ci> OverTilde </ci> <infinity /> </apply> </otherwise> </piecewise> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Limit", "[", RowBox[List[RowBox[List["SinIntegral", "[", RowBox[List["a_", "+", RowBox[List["b_", " ", "x_"]]]], "]"]], ",", RowBox[List["x_", "\[Rule]", "\[Infinity]"]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List["\[Piecewise]", GridBox[List[List[FractionBox["\[Pi]", "2"], RowBox[List[RowBox[List["Arg", "[", "b", "]"]], "\[Equal]", "0"]]], List[RowBox[List["-", FractionBox["\[Pi]", "2"]]], RowBox[List[RowBox[List["Arg", "[", "b", "]"]], "\[Equal]", "\[Pi]"]]], List[RowBox[List["\[ImaginaryI]", " ", "\[Infinity]"]], RowBox[List[RowBox[List[RowBox[List["Abs", "[", RowBox[List["Arg", "[", "a", "]"]], "]"]], "\[Equal]", FractionBox["\[Pi]", "2"]]], "&&", RowBox[List[RowBox[List["Arg", "[", "b", "]"]], "\[Equal]", FractionBox["\[Pi]", "2"]]]]]], List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "\[Infinity]"]], RowBox[List[RowBox[List[RowBox[List["Abs", "[", RowBox[List["Arg", "[", "a", "]"]], "]"]], "\[Equal]", FractionBox["\[Pi]", "2"]]], "&&", RowBox[List[RowBox[List["Arg", "[", "b", "]"]], "\[Equal]", RowBox[List["-", FractionBox["\[Pi]", "2"]]]]]]]], List["ComplexInfinity", TagBox["True", "PiecewiseDefault", Rule[AutoDelete, False], Rule[DeletionWarning, True]]]], Rule[ColumnAlignments, List[Left]], Rule[ColumnSpacings, 1.2`], Rule[ColumnWidths, Automatic]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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){kind=small}
