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SinIntegral






Mathematica Notation

Traditional Notation









Gamma, Beta, Erf > SinIntegral[z] > Integration > Indefinite integration > Involving one direct function and elementary functions > Involving hyperbolic functions and a power function > Involving sinh and power





http://functions.wolfram.com/06.37.21.0037.01









  


  










Input Form





Integrate[z Sinh[b z] SinIntegral[a z], z] == (1/(4 b^2 (a^2 + b^2))) (4 a b Cos[a z] Cosh[b z] - 4 b^2 Sin[a z] Sinh[b z] + (a^2 + b^2) (I (ExpIntegralEi[((-I) a + b) z] - ExpIntegralEi[(I a + b) z] - ExpIntegralEi[(-I) a z - b z] + ExpIntegralEi[I a z - b z]) + 2 (-Gamma[2, (-b) z] + Gamma[2, b z]) SinIntegral[a z]))










Standard Form





Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List["z", " ", RowBox[List["Sinh", "[", RowBox[List["b", " ", "z"]], "]"]], RowBox[List["SinIntegral", "[", RowBox[List["a", " ", "z"]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox["1", RowBox[List["4", " ", SuperscriptBox["b", "2"], " ", RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "+", SuperscriptBox["b", "2"]]], ")"]]]]], RowBox[List["(", RowBox[List[RowBox[List["4", " ", "a", " ", "b", " ", RowBox[List["Cos", "[", RowBox[List["a", " ", "z"]], "]"]], " ", RowBox[List["Cosh", "[", RowBox[List["b", " ", "z"]], "]"]]]], "-", RowBox[List["4", " ", SuperscriptBox["b", "2"], " ", RowBox[List["Sin", "[", RowBox[List["a", " ", "z"]], "]"]], " ", RowBox[List["Sinh", "[", RowBox[List["b", " ", "z"]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "+", SuperscriptBox["b", "2"]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["ExpIntegralEi", "[", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a"]], "+", "b"]], ")"]], " ", "z"]], "]"]], "-", RowBox[List["ExpIntegralEi", "[", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a"]], "+", "b"]], ")"]], " ", "z"]], "]"]], "-", RowBox[List["ExpIntegralEi", "[", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a", " ", "z"]], "-", RowBox[List["b", " ", "z"]]]], "]"]], "+", RowBox[List["ExpIntegralEi", "[", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a", " ", "z"]], "-", RowBox[List["b", " ", "z"]]]], "]"]]]], ")"]]]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["Gamma", "[", RowBox[List["2", ",", RowBox[List[RowBox[List["-", "b"]], " ", "z"]]]], "]"]]]], "+", RowBox[List["Gamma", "[", RowBox[List["2", ",", RowBox[List["b", " ", "z"]]]], "]"]]]], ")"]], " ", RowBox[List["SinIntegral", "[", RowBox[List["a", " ", "z"]], "]"]]]]]], ")"]]]]]], ")"]]]]]]]]










MathML Form







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</mo> <mo> ( </mo> <mrow> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> a </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> - </mo> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> , </mo> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> , </mo> <mrow> <mrow> <mo> - </mo> <mi> b </mi> </mrow> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> Si </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <ci> z </ci> </bvar> <apply> <times /> <ci> z </ci> <apply> <sinh /> <apply> <times /> <ci> b </ci> <ci> z </ci> </apply> </apply> <apply> <ci> SinIntegral </ci> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -4 </cn> <apply> <sin /> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> <apply> <sinh /> <apply> <times /> <ci> b </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 4 </cn> <ci> a </ci> <apply> <cos /> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> <apply> <cosh /> <apply> <times /> <ci> b </ci> <ci> z </ci> </apply> </apply> <ci> b </ci> </apply> <apply> <times /> <apply> <plus /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <imaginaryi /> <apply> <plus /> <apply> <ci> ExpIntegralEi </ci> <apply> <times /> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> a </ci> </apply> </apply> </apply> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> ExpIntegralEi </ci> <apply> <times /> <apply> <plus /> <ci> b </ci> <apply> <times /> <ci> a </ci> <imaginaryi /> </apply> </apply> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> ExpIntegralEi </ci> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> <ci> z </ci> </apply> <apply> <times /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <ci> ExpIntegralEi </ci> <apply> <plus /> <apply> <times /> <imaginaryi /> <ci> a </ci> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> b </ci> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <ci> Gamma </ci> <cn type='integer'> 2 </cn> <apply> <times /> <ci> b </ci> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Gamma </ci> <cn type='integer'> 2 </cn> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <ci> SinIntegral </ci> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List["z_", " ", RowBox[List["Sinh", "[", RowBox[List["b_", " ", "z_"]], "]"]], " ", RowBox[List["SinIntegral", "[", RowBox[List["a_", " ", "z_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List["4", " ", "a", " ", "b", " ", RowBox[List["Cos", "[", RowBox[List["a", " ", "z"]], "]"]], " ", RowBox[List["Cosh", "[", RowBox[List["b", " ", "z"]], "]"]]]], "-", RowBox[List["4", " ", SuperscriptBox["b", "2"], " ", RowBox[List["Sin", "[", RowBox[List["a", " ", "z"]], "]"]], " ", RowBox[List["Sinh", "[", RowBox[List["b", " ", "z"]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "+", SuperscriptBox["b", "2"]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["ExpIntegralEi", "[", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a"]], "+", "b"]], ")"]], " ", "z"]], "]"]], "-", RowBox[List["ExpIntegralEi", "[", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a"]], "+", "b"]], ")"]], " ", "z"]], "]"]], "-", RowBox[List["ExpIntegralEi", "[", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a", " ", "z"]], "-", RowBox[List["b", " ", "z"]]]], "]"]], "+", RowBox[List["ExpIntegralEi", "[", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a", " ", "z"]], "-", RowBox[List["b", " ", "z"]]]], "]"]]]], ")"]]]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["Gamma", "[", RowBox[List["2", ",", RowBox[List[RowBox[List["-", "b"]], " ", "z"]]]], "]"]]]], "+", RowBox[List["Gamma", "[", RowBox[List["2", ",", RowBox[List["b", " ", "z"]]]], "]"]]]], ")"]], " ", RowBox[List["SinIntegral", "[", RowBox[List["a", " ", "z"]], "]"]]]]]], ")"]]]]]], RowBox[List["4", " ", SuperscriptBox["b", "2"], " ", RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "+", SuperscriptBox["b", "2"]]], ")"]]]]]]]]]










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





2001-10-29