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Cosh




Mathematica Notation

Traditional Notation









Elementary Functions > Cosh[z] > Integration > Indefinite integration > Involving one direct function and elementary functions > Involving power function > Involving power > Involving zalpha-1and arguments a z+b





http://functions.wolfram.com/01.20.21.0060.01









  


  








Input Form





Integrate[z^(\[Alpha] - 1) Cosh[a z + b], z] == ((1/2) z^\[Alpha] ((-E^(2 b)) ExpIntegralE[1 - \[Alpha], (-a) z] - ExpIntegralE[1 - \[Alpha], a z]))/E^b








Standard Form





Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["z", RowBox[List["\[Alpha]", "-", "1"]]], " ", RowBox[List["Cosh", "[", RowBox[List[RowBox[List["a", " ", "z"]], "+", "b"]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox["1", "2"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", "b"]]], " ", SuperscriptBox["z", "\[Alpha]"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "b"]]]]], " ", RowBox[List["ExpIntegralE", "[", RowBox[List[RowBox[List["1", "-", "\[Alpha]"]], ",", RowBox[List[RowBox[List["-", "a"]], " ", "z"]]]], "]"]]]], "-", RowBox[List["ExpIntegralE", "[", RowBox[List[RowBox[List["1", "-", "\[Alpha]"]], ",", RowBox[List["a", " ", "z"]]]], "]"]]]], ")"]]]]]]]]








MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mo> &#8747; </mo> <mrow> <mrow> <msup> <mi> z </mi> <mrow> <mi> &#945; </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mi> cosh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> b </mi> <mo> + </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mo> - </mo> <mi> b </mi> </mrow> </msup> <mo> &#8290; </mo> <msup> <mi> z </mi> <mi> &#945; </mi> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <msup> <mi> &#8519; </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> b </mi> </mrow> </msup> </mrow> <mo> &#8290; </mo> <mrow> <msub> <semantics> <mi> E </mi> <annotation encoding='Mathematica'> TagBox[&quot;E&quot;, ExpIntegralE] </annotation> </semantics> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> &#945; </mi> </mrow> </msub> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> a </mi> </mrow> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <msub> <semantics> <mi> E </mi> <annotation encoding='Mathematica'> TagBox[&quot;E&quot;, ExpIntegralE] </annotation> </semantics> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> &#945; </mi> </mrow> </msub> <mo> ( </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <ci> z </ci> </bvar> <apply> <times /> <apply> <power /> <ci> z </ci> <apply> <plus /> <ci> &#945; </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <cosh /> <apply> <plus /> <ci> b </ci> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> <apply> <power /> <ci> z </ci> <ci> &#945; </ci> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> b </ci> </apply> </apply> </apply> <apply> <ci> ExpIntegralE </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#945; </ci> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> ExpIntegralE </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#945; </ci> </apply> </apply> <apply> <times /> <ci> a </ci> <ci> z </ci> </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[SuperscriptBox["z_", RowBox[List["\[Alpha]_", "-", "1"]]], " ", RowBox[List["Cosh", "[", RowBox[List[RowBox[List["a_", " ", "z_"]], "+", "b_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox["1", "2"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", "b"]]], " ", SuperscriptBox["z", "\[Alpha]"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "b"]]]]], " ", RowBox[List["ExpIntegralE", "[", RowBox[List[RowBox[List["1", "-", "\[Alpha]"]], ",", RowBox[List[RowBox[List["-", "a"]], " ", "z"]]]], "]"]]]], "-", RowBox[List["ExpIntegralE", "[", RowBox[List[RowBox[List["1", "-", "\[Alpha]"]], ",", RowBox[List["a", " ", "z"]]]], "]"]]]], ")"]]]]]]]]


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





2002-12-18