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variants of this functions
EllipticPi






Mathematica Notation

Traditional Notation









Elliptic Integrals > EllipticPi[n,z,m] > Identities > Functional identities





http://functions.wolfram.com/08.06.17.0002.01









  


  










Input Form





Cot[\[Theta]] (EllipticPi[m Sin[\[Theta]]^2, \[Phi], m] - EllipticF[\[Phi], m]) Sqrt[1 - m Sin[\[Theta]]^2] + EllipticE[\[Phi], m] EllipticF[\[Theta], m] == Cot[\[Phi]] (EllipticPi[m Sin[\[Phi]]^2, \[Theta], m] - EllipticF[\[Theta], m]) Sqrt[1 - m Sin[\[Phi]]^2] + EllipticE[\[Theta], m] EllipticF[\[Phi], m]










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List["Cot", "[", "\[Theta]", "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["EllipticPi", "[", RowBox[List[RowBox[List["m", " ", SuperscriptBox[RowBox[List["Sin", "[", "\[Theta]", "]"]], "2"]]], ",", "\[Phi]", ",", "m"]], "]"]], "-", RowBox[List["EllipticF", "[", RowBox[List["\[Phi]", ",", "m"]], "]"]]]], ")"]], " ", SqrtBox[RowBox[List["1", "-", RowBox[List["m", " ", SuperscriptBox[RowBox[List["Sin", "[", "\[Theta]", "]"]], "2"]]]]]]]], "+", RowBox[List[RowBox[List["EllipticE", "[", RowBox[List["\[Phi]", ",", "m"]], "]"]], " ", RowBox[List["EllipticF", "[", RowBox[List["\[Theta]", ",", "m"]], "]"]]]]]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["Cot", "[", "\[Phi]", "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["EllipticPi", "[", RowBox[List[RowBox[List["m", " ", SuperscriptBox[RowBox[List["Sin", "[", "\[Phi]", "]"]], "2"]]], ",", "\[Theta]", ",", "m"]], "]"]], "-", RowBox[List["EllipticF", "[", RowBox[List["\[Theta]", ",", "m"]], "]"]]]], ")"]], " ", SqrtBox[RowBox[List["1", "-", RowBox[List["m", " ", SuperscriptBox[RowBox[List["Sin", "[", "\[Phi]", "]"]], "2"]]]]]]]], "+", RowBox[List[RowBox[List["EllipticE", "[", RowBox[List["\[Theta]", ",", "m"]], "]"]], " ", RowBox[List["EllipticF", "[", RowBox[List["\[Phi]", ",", "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> <mrow> <mrow> <mi> E </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> &#981; </mi> <mo> &#10072; </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> F </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> &#952; </mi> <mo> &#10072; </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <mi> cot </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> &#952; </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> &#928; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mi> m </mi> <mo> &#8290; </mo> <mrow> <msup> <mi> sin </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mi> &#952; </mi> <mo> ) </mo> </mrow> </mrow> <mo> ; </mo> <mrow> <mi> &#981; </mi> <mo> &#10072; </mo> <mi> m </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> F </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> &#981; </mi> <mo> &#10072; </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mi> m </mi> <mo> &#8290; </mo> <mrow> <msup> <mi> sin </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mi> &#952; </mi> <mo> ) </mo> </mrow> </mrow> </mrow> </msqrt> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mrow> <mi> cot </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> &#981; </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mi> m </mi> <mo> &#8290; </mo> <mrow> <msup> <mi> sin </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mi> &#981; </mi> <mo> ) </mo> </mrow> </mrow> </mrow> </msqrt> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> &#928; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mi> m </mi> <mo> &#8290; </mo> <mrow> <msup> <mi> sin </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mi> &#981; </mi> <mo> ) </mo> </mrow> </mrow> <mo> ; </mo> <mrow> <mi> &#952; </mi> <mo> &#10072; </mo> <mi> m </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> F </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> &#952; </mi> <mo> &#10072; </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <mi> E </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> &#952; </mi> <mo> &#10072; </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> F </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> &#981; </mi> <mo> &#10072; </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <plus /> <apply> <times /> <apply> <ci> EllipticE </ci> <ci> &#981; </ci> <ci> m </ci> </apply> <apply> <ci> EllipticF </ci> <ci> &#952; </ci> <ci> m </ci> </apply> </apply> <apply> <times /> <apply> <cot /> <ci> &#952; </ci> </apply> <apply> <plus /> <apply> <ci> EllipticPi </ci> <apply> <times /> <ci> m </ci> <apply> <power /> <apply> <sin /> <ci> &#952; </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <ci> &#981; </ci> <ci> m </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> EllipticF </ci> <ci> &#981; </ci> <ci> m </ci> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> m </ci> <apply> <power /> <apply> <sin /> <ci> &#952; </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <cot /> <ci> &#981; </ci> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> m </ci> <apply> <power /> <apply> <sin /> <ci> &#981; </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <ci> EllipticPi </ci> <apply> <times /> <ci> m </ci> <apply> <power /> <apply> <sin /> <ci> &#981; </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <ci> &#952; </ci> <ci> m </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> EllipticF </ci> <ci> &#952; </ci> <ci> m </ci> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <ci> EllipticE </ci> <ci> &#952; </ci> <ci> m </ci> </apply> <apply> <ci> EllipticF </ci> <ci> &#981; </ci> <ci> m </ci> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[RowBox[List[RowBox[List["Cot", "[", "\[Theta]_", "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["EllipticPi", "[", RowBox[List[RowBox[List["m_", " ", SuperscriptBox[RowBox[List["Sin", "[", "\[Theta]_", "]"]], "2"]]], ",", "\[Phi]_", ",", "m_"]], "]"]], "-", RowBox[List["EllipticF", "[", RowBox[List["\[Phi]_", ",", "m_"]], "]"]]]], ")"]], " ", SqrtBox[RowBox[List["1", "-", RowBox[List["m_", " ", SuperscriptBox[RowBox[List["Sin", "[", "\[Theta]_", "]"]], "2"]]]]]]]], "+", RowBox[List[RowBox[List["EllipticE", "[", RowBox[List["\[Phi]_", ",", "m_"]], "]"]], " ", RowBox[List["EllipticF", "[", RowBox[List["\[Theta]_", ",", "m_"]], "]"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["Cot", "[", "\[Phi]", "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["EllipticPi", "[", RowBox[List[RowBox[List["m", " ", SuperscriptBox[RowBox[List["Sin", "[", "\[Phi]", "]"]], "2"]]], ",", "\[Theta]", ",", "m"]], "]"]], "-", RowBox[List["EllipticF", "[", RowBox[List["\[Theta]", ",", "m"]], "]"]]]], ")"]], " ", SqrtBox[RowBox[List["1", "-", RowBox[List["m", " ", SuperscriptBox[RowBox[List["Sin", "[", "\[Phi]", "]"]], "2"]]]]]]]], "+", RowBox[List[RowBox[List["EllipticE", "[", RowBox[List["\[Theta]", ",", "m"]], "]"]], " ", RowBox[List["EllipticF", "[", RowBox[List["\[Phi]", ",", "m"]], "]"]]]]]]]]]]










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





2001-10-29





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