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http://functions.wolfram.com/08.04.21.0007.01
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Integrate[m EllipticE[z, m^2], m] ==
(1/3) ((1 + m^2) EllipticE[z, m^2] + (-1 + m^2) EllipticF[z, m^2] +
Cot[z] Sqrt[1 - m^2 Sin[z]^2])
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Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List["m", " ", RowBox[List["EllipticE", "[", RowBox[List["z", ",", SuperscriptBox["m", "2"]]], "]"]], RowBox[List["\[DifferentialD]", "m"]]]]]], "\[Equal]", RowBox[List[FractionBox["1", "3"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["1", "+", SuperscriptBox["m", "2"]]], ")"]], " ", RowBox[List["EllipticE", "[", RowBox[List["z", ",", SuperscriptBox["m", "2"]]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox["m", "2"]]], ")"]], " ", RowBox[List["EllipticF", "[", RowBox[List["z", ",", SuperscriptBox["m", "2"]]], "]"]]]], "+", RowBox[List[RowBox[List["Cot", "[", "z", "]"]], " ", SqrtBox[RowBox[List["1", "-", RowBox[List[SuperscriptBox["m", "2"], " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "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> z </mi> <mo> ⁢ </mo> <mrow> <mi> E </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> ❘ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 3 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <mrow> <msup> <mi> sin </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> </mrow> </mrow> </msqrt> <mo> ⁢ </mo> <mrow> <mi> cot </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> E </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> ❘ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> F </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> ❘ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <ci> z </ci> </bvar> <apply> <times /> <ci> z </ci> <apply> <ci> EllipticE </ci> <ci> a </ci> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 3 </cn> <apply> <plus /> <apply> <times /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <sin /> <ci> a </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <cot /> <ci> a </ci> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <ci> EllipticE </ci> <ci> a </ci> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <ci> EllipticF </ci> <ci> a </ci> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </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["EllipticE", "[", RowBox[List["z_", ",", SuperscriptBox["m_", "2"]]], "]"]]]], RowBox[List["\[DifferentialD]", "m_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox["1", "3"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["1", "+", SuperscriptBox["m", "2"]]], ")"]], " ", RowBox[List["EllipticE", "[", RowBox[List["z", ",", SuperscriptBox["m", "2"]]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox["m", "2"]]], ")"]], " ", RowBox[List["EllipticF", "[", RowBox[List["z", ",", SuperscriptBox["m", "2"]]], "]"]]]], "+", RowBox[List[RowBox[List["Cot", "[", "z", "]"]], " ", SqrtBox[RowBox[List["1", "-", RowBox[List[SuperscriptBox["m", "2"], " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]]]]]]]]], ")"]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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