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LucasL




Mathematica Notation

Traditional Notation









Integer Functions > LucasL[nu] > Integration > Indefinite integration > Involving one direct function and elementary functions > Involving power function





http://functions.wolfram.com/04.22.21.0003.01









  


  








Input Form





Integrate[\[Nu]^(\[Alpha] - 1) LucasL[a \[Nu]], \[Nu]] == (-(1/2)) \[Nu]^\[Alpha] (2 ExpIntegralE[1 - \[Alpha], (-a) \[Nu] ArcCsch[2]] + ExpIntegralE[1 - \[Alpha], a \[Nu] ((-I) Pi + ArcCsch[2])] + ExpIntegralE[1 - \[Alpha], a \[Nu] (I Pi + ArcCsch[2])])








Standard Form





Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["\[Nu]", RowBox[List["\[Alpha]", "-", "1"]]], RowBox[List["LucasL", "[", RowBox[List["a", " ", "\[Nu]"]], "]"]], RowBox[List["\[DifferentialD]", "\[Nu]"]]]]]], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", SuperscriptBox["\[Nu]", "\[Alpha]"], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", RowBox[List["ExpIntegralE", "[", RowBox[List[RowBox[List["1", "-", "\[Alpha]"]], ",", RowBox[List[RowBox[List["-", "a"]], " ", "\[Nu]", " ", RowBox[List["ArcCsch", "[", "2", "]"]]]]]], "]"]]]], "+", RowBox[List["ExpIntegralE", "[", RowBox[List[RowBox[List["1", "-", "\[Alpha]"]], ",", RowBox[List["a", " ", "\[Nu]", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "\[Pi]"]], "+", RowBox[List["ArcCsch", "[", "2", "]"]]]], ")"]]]]]], "]"]], "+", RowBox[List["ExpIntegralE", "[", RowBox[List[RowBox[List["1", "-", "\[Alpha]"]], ",", RowBox[List["a", " ", "\[Nu]", " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "\[Pi]"]], "+", RowBox[List["ArcCsch", "[", "2", "]"]]]], ")"]]]]]], "]"]]]], ")"]]]]]]]]








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> &#957; </mi> <mrow> <mi> &#945; </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> &#8290; </mo> <msub> <semantics> <mi> L </mi> <annotation encoding='Mathematica'> TagBox[&quot;L&quot;, LucasL] </annotation> </semantics> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> &#957; </mi> </mrow> </msub> </mrow> <mo> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> &#957; </mi> </mrow> </mrow> </mrow> <mo> &#63449; </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> &#8290; </mo> <msup> <mi> &#957; </mi> <mi> &#945; </mi> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <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> &#957; </mi> <mo> &#8290; </mo> <mrow> <msup> <mi> csch </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mn> 2 </mn> <mo> ) </mo> </mrow> </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> &#957; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mi> &#8520; </mi> </mrow> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mo> + </mo> <mrow> <msup> <mi> csch </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mn> 2 </mn> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </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> &#957; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mo> + </mo> <mrow> <msup> <mi> csch </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mn> 2 </mn> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <ci> &#957; </ci> </bvar> <apply> <times /> <apply> <power /> <ci> &#957; </ci> <apply> <plus /> <ci> &#945; </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> LucasL </ci> <apply> <times /> <ci> a </ci> <ci> &#957; </ci> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <ci> &#957; </ci> <ci> &#945; </ci> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </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 /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> <ci> &#957; </ci> <apply> <arccsch /> <cn type='integer'> 2 </cn> </apply> </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 /> <ci> a </ci> <ci> &#957; </ci> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <pi /> </apply> <apply> <arccsch /> <cn type='integer'> 2 </cn> </apply> </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 /> <ci> a </ci> <ci> &#957; </ci> <apply> <plus /> <apply> <times /> <imaginaryi /> <pi /> </apply> <apply> <arccsch /> <cn type='integer'> 2 </cn> </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[SuperscriptBox["\[Nu]_", RowBox[List["\[Alpha]_", "-", "1"]]], " ", RowBox[List["LucasL", "[", RowBox[List["a_", " ", "\[Nu]_"]], "]"]]]], RowBox[List["\[DifferentialD]", "\[Nu]_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", SuperscriptBox["\[Nu]", "\[Alpha]"], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", RowBox[List["ExpIntegralE", "[", RowBox[List[RowBox[List["1", "-", "\[Alpha]"]], ",", RowBox[List[RowBox[List["-", "a"]], " ", "\[Nu]", " ", RowBox[List["ArcCsch", "[", "2", "]"]]]]]], "]"]]]], "+", RowBox[List["ExpIntegralE", "[", RowBox[List[RowBox[List["1", "-", "\[Alpha]"]], ",", RowBox[List["a", " ", "\[Nu]", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "\[Pi]"]], "+", RowBox[List["ArcCsch", "[", "2", "]"]]]], ")"]]]]]], "]"]], "+", RowBox[List["ExpIntegralE", "[", RowBox[List[RowBox[List["1", "-", "\[Alpha]"]], ",", RowBox[List["a", " ", "\[Nu]", " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "\[Pi]"]], "+", RowBox[List["ArcCsch", "[", "2", "]"]]]], ")"]]]]]], "]"]]]], ")"]]]]]]]]


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





2007-05-02