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variants of this functions
EllipticPi




Mathematica Notation

Traditional Notation









Elliptic Integrals > EllipticPi[n,z,m] > Integration > Indefinite integration > Involving only one direct function with respect to m





http://functions.wolfram.com/08.06.21.0005.01









  


  








Input Form





Integrate[EllipticPi[n, z, m], m] == Sqrt[4 - 2 m + 2 m Cos[2 z]] Cot[z] + 2 EllipticE[z, m] - 2 EllipticF[z, m] + 2 (m - n) EllipticPi[n, z, m]








Standard Form





Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[RowBox[List["EllipticPi", "[", RowBox[List["n", ",", "z", ",", "m"]], "]"]], RowBox[List["\[DifferentialD]", "m"]]]]]], "\[Equal]", RowBox[List[RowBox[List[SqrtBox[RowBox[List["4", "-", RowBox[List["2", " ", "m"]], "+", RowBox[List["2", " ", "m", " ", RowBox[List["Cos", "[", RowBox[List["2", " ", "z"]], "]"]]]]]]], " ", RowBox[List["Cot", "[", "z", "]"]]]], "+", RowBox[List["2", " ", RowBox[List["EllipticE", "[", RowBox[List["z", ",", "m"]], "]"]]]], "-", RowBox[List["2", " ", RowBox[List["EllipticF", "[", RowBox[List["z", ",", "m"]], "]"]]]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List["m", "-", "n"]], ")"]], " ", RowBox[List["EllipticPi", "[", RowBox[List["n", ",", "z", ",", "m"]], "]"]]]]]]]]]]








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> <mi> &#928; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> n </mi> <mo> ; </mo> <mrow> <mi> z </mi> <mo> &#10072; </mo> <mi> m </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> m </mi> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <msqrt> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> m </mi> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> m </mi> </mrow> <mo> + </mo> <mn> 4 </mn> </mrow> </msqrt> <mo> &#8290; </mo> <mrow> <mi> cot </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mi> E </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> &#10072; </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mi> F </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> &#10072; </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> &#928; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> n </mi> <mo> ; </mo> <mrow> <mi> z </mi> <mo> &#10072; </mo> <mi> m </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <ci> m </ci> </bvar> <apply> <ci> EllipticPi </ci> <ci> n </ci> <ci> z </ci> <ci> m </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <cos /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> </apply> <ci> m </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> m </ci> </apply> </apply> <cn type='integer'> 4 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <cot /> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> EllipticE </ci> <ci> z </ci> <ci> m </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> EllipticF </ci> <ci> z </ci> <ci> m </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> m </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> </apply> <apply> <ci> EllipticPi </ci> <ci> n </ci> <ci> z </ci> <ci> m </ci> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List["EllipticPi", "[", RowBox[List["n_", ",", "z_", ",", "m_"]], "]"]], RowBox[List["\[DifferentialD]", "m_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SqrtBox[RowBox[List["4", "-", RowBox[List["2", " ", "m"]], "+", RowBox[List["2", " ", "m", " ", RowBox[List["Cos", "[", RowBox[List["2", " ", "z"]], "]"]]]]]]], " ", RowBox[List["Cot", "[", "z", "]"]]]], "+", RowBox[List["2", " ", RowBox[List["EllipticE", "[", RowBox[List["z", ",", "m"]], "]"]]]], "-", RowBox[List["2", " ", RowBox[List["EllipticF", "[", RowBox[List["z", ",", "m"]], "]"]]]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List["m", "-", "n"]], ")"]], " ", RowBox[List["EllipticPi", "[", RowBox[List["n", ",", "z", ",", "m"]], "]"]]]]]]]]]]


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





2001-10-29