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http://functions.wolfram.com/08.03.21.0004.01
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Integrate[m EllipticPi[n, m^2], m] == EllipticE[m^2] - EllipticK[m^2] +
(m^2 - n) EllipticPi[n, m^2]
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Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List["m", " ", RowBox[List["EllipticPi", "[", RowBox[List["n", ",", SuperscriptBox["m", "2"]]], "]"]], RowBox[List["\[DifferentialD]", "m"]]]]]], "\[Equal]", RowBox[List[RowBox[List["EllipticE", "[", SuperscriptBox["m", "2"], "]"]], "-", RowBox[List["EllipticK", "[", SuperscriptBox["m", "2"], "]"]], "+", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["m", "2"], "-", "n"]], ")"]], " ", RowBox[List["EllipticPi", "[", RowBox[List["n", ",", SuperscriptBox["m", "2"]]], "]"]]]]]]]]]]
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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mo> ∫ </mo> <mrow> <mrow> <mi> m </mi> <mo> ⁢ </mo> <mrow> <mi> Π </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> n </mi> <mo> ❘ </mo> <msup> <mi> m </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> m </mi> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mi> E </mi> <mo> ⁡ </mo> <mo> ( </mo> <msup> <mi> m </mi> <mn> 2 </mn> </msup> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> K </mi> <mo> ⁡ </mo> <mo> ( </mo> <msup> <mi> m </mi> <mn> 2 </mn> </msup> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <msup> <mi> m </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> Π </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> n </mi> <mo> ❘ </mo> <msup> <mi> m </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <ci> m </ci> </bvar> <apply> <times /> <ci> m </ci> <apply> <ci> EllipticPi </ci> <ci> n </ci> <apply> <power /> <ci> m </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <apply> <plus /> <apply> <ci> EllipticE </ci> <apply> <power /> <ci> m </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> EllipticK </ci> <apply> <power /> <ci> m </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <power /> <ci> m </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> </apply> <apply> <ci> EllipticPi </ci> <ci> n </ci> <apply> <power /> <ci> m </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List["m_", " ", RowBox[List["EllipticPi", "[", RowBox[List["n_", ",", SuperscriptBox["m_", "2"]]], "]"]]]], RowBox[List["\[DifferentialD]", "m_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["EllipticE", "[", SuperscriptBox["m", "2"], "]"]], "-", RowBox[List["EllipticK", "[", SuperscriptBox["m", "2"], "]"]], "+", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["m", "2"], "-", "n"]], ")"]], " ", RowBox[List["EllipticPi", "[", RowBox[List["n", ",", SuperscriptBox["m", "2"]]], "]"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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