|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
http://functions.wolfram.com/01.19.21.0352.01
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Integrate[z^n E^(b z + e) Sinh[c z + d], z] == (1/2) E^(-d + e) z^(1 + n)
(ExpIntegralE[-n, (c - b) z] - E^(2 d) ExpIntegralE[-n, (-(c + b)) z]) /;
Element[n, Integers]
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["z", "n"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["b", " ", "z"]], "+", "e"]]], " ", RowBox[List["Sinh", "[", RowBox[List[RowBox[List["c", " ", "z"]], "+", "d"]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox["1", "2"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "d"]], "+", "e"]]], " ", SuperscriptBox["z", RowBox[List["1", "+", "n"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["ExpIntegralE", "[", RowBox[List[RowBox[List["-", "n"]], ",", RowBox[List[RowBox[List["(", RowBox[List["c", "-", "b"]], ")"]], " ", "z"]]]], "]"]], "-", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "d"]]], " ", RowBox[List["ExpIntegralE", "[", RowBox[List[RowBox[List["-", "n"]], ",", RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List["c", "+", "b"]], ")"]]]], " ", "z"]]]], "]"]]]]]], ")"]]]]]], "/;", RowBox[List["n", "\[Element]", "Integers"]]]]]]
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <mo> ∫ </mo> <mrow> <msup> <mi> z </mi> <mi> n </mi> </msup> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mrow> <mi> e </mi> <mo> + </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mi> sinh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> d </mi> <mo> + </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mrow> <mi> e </mi> <mo> - </mo> <mi> d </mi> </mrow> </msup> <mo> ⁢ </mo> <msup> <mi> z </mi> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msub> <semantics> <mi> E </mi> <annotation encoding='Mathematica'> TagBox["E", ExpIntegralE] </annotation> </semantics> <mrow> <mo> - </mo> <mi> n </mi> </mrow> </msub> <mo> ( </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> c </mi> <mo> - </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <msup> <mi> ⅇ </mi> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> d </mi> </mrow> </msup> <mo> ⁢ </mo> <mrow> <msub> <semantics> <mi> E </mi> <annotation encoding='Mathematica'> TagBox["E", ExpIntegralE] </annotation> </semantics> <mrow> <mo> - </mo> <mi> n </mi> </mrow> </msub> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mrow> <mo> ( </mo> <mrow> <mi> c </mi> <mo> + </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> n </mi> <mo> ∈ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] </annotation> </semantics> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <int /> <bvar> <ci> z </ci> </bvar> <apply> <times /> <apply> <power /> <ci> z </ci> <ci> n </ci> </apply> <apply> <power /> <exponentiale /> <apply> <plus /> <ci> e </ci> <apply> <times /> <ci> b </ci> <ci> z </ci> </apply> </apply> </apply> <apply> <sinh /> <apply> <plus /> <ci> d </ci> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <power /> <exponentiale /> <apply> <plus /> <ci> e </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> d </ci> </apply> </apply> </apply> <apply> <power /> <ci> z </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <plus /> <apply> <ci> ExpIntegralE </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <apply> <times /> <apply> <plus /> <ci> c </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> d </ci> </apply> </apply> <apply> <ci> ExpIntegralE </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <plus /> <ci> c </ci> <ci> b </ci> </apply> </apply> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <in /> <ci> n </ci> <integers /> </apply> </apply> </annotation-xml> </semantics> </math>
|
|
|
|
|
|
|
|
|
|
| |
|
|
|
|
| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox["z_", "n_"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["b_", " ", "z_"]], "+", "e_"]]], " ", RowBox[List["Sinh", "[", RowBox[List[RowBox[List["c_", " ", "z_"]], "+", "d_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "d"]], "+", "e"]]], " ", SuperscriptBox["z", RowBox[List["1", "+", "n"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["ExpIntegralE", "[", RowBox[List[RowBox[List["-", "n"]], ",", RowBox[List[RowBox[List["(", RowBox[List["c", "-", "b"]], ")"]], " ", "z"]]]], "]"]], "-", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "d"]]], " ", RowBox[List["ExpIntegralE", "[", RowBox[List[RowBox[List["-", "n"]], ",", RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List["c", "+", "b"]], ")"]]]], " ", "z"]]]], "]"]]]]]], ")"]]]], "/;", RowBox[List["n", "\[Element]", "Integers"]]]]]]]] |
|
|
|
|
|
|
|
|
|
|
Date Added to functions.wolfram.com (modification date)
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|