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EllipticK




Mathematica Notation

Traditional Notation









Elliptic Integrals > EllipticK[z] > Integration > Indefinite integration > Involving one direct function and elementary functions > Involving power function > Involving power > Power arguments





http://functions.wolfram.com/08.02.21.0011.01









  


  








Input Form





Integrate[z^3 EllipticK[a z^2], z] == (1/(9 a^2)) ((4 + a z^2) EllipticE[a z^2] + (-4 + a z^2 + 3 a^2 z^4) EllipticK[a z^2])








Standard Form





Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["z", "3"], " ", RowBox[List["EllipticK", "[", RowBox[List["a", " ", SuperscriptBox["z", "2"]]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox["1", RowBox[List["9", " ", SuperscriptBox["a", "2"]]]], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["4", "+", RowBox[List["a", " ", SuperscriptBox["z", "2"]]]]], ")"]], " ", RowBox[List["EllipticE", "[", RowBox[List["a", " ", SuperscriptBox["z", "2"]]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "4"]], "+", RowBox[List["a", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["3", " ", SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "4"]]]]], ")"]], " ", RowBox[List["EllipticK", "[", RowBox[List["a", " ", SuperscriptBox["z", "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> z </mi> <mn> 3 </mn> </msup> <mo> &#8290; </mo> <mrow> <mi> K </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mfrac> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> a </mi> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mn> 4 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> E </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mn> 4 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> K </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mrow> <mn> 9 </mn> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> </mrow> </mfrac> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <ci> z </ci> </bvar> <apply> <times /> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> <apply> <ci> EllipticK </ci> <apply> <times /> <ci> a </ci> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <plus /> <apply> <times /> <ci> a </ci> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> 4 </cn> </apply> <apply> <ci> EllipticE </ci> <apply> <times /> <ci> a </ci> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <ci> a </ci> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -4 </cn> </apply> <apply> <ci> EllipticK </ci> <apply> <times /> <ci> a </ci> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 9 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </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_", "3"], " ", RowBox[List["EllipticK", "[", RowBox[List["a_", " ", SuperscriptBox["z_", "2"]]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["4", "+", RowBox[List["a", " ", SuperscriptBox["z", "2"]]]]], ")"]], " ", RowBox[List["EllipticE", "[", RowBox[List["a", " ", SuperscriptBox["z", "2"]]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "4"]], "+", RowBox[List["a", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["3", " ", SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "4"]]]]], ")"]], " ", RowBox[List["EllipticK", "[", RowBox[List["a", " ", SuperscriptBox["z", "2"]]], "]"]]]]]], RowBox[List["9", " ", SuperscriptBox["a", "2"]]]]]]]]


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





2001-10-29