EllipticF[\[Phi], m] == (1/EllipticE[m]) ((Sqrt[1 - n] Cot[\[Phi]] + EllipticE[\[Phi], m]) EllipticK[m] - Sqrt[1 - n] Cot[\[Phi]] EllipticPi[n, m]) /; \[Phi] == ArcSin[Sqrt[n/m]] && 0 < n < 1 && 0 < m < 1

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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> <mrow> <mi> F </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> &#981; </mi> <mo> &#10072; </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mfrac> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> n </mi> </mrow> </msqrt> <mo> &#8290; </mo> <mrow> <mi> cot </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> &#981; </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> E </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> <mrow> <mi> K </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> n </mi> </mrow> </msqrt> <mo> &#8290; </mo> <mrow> <mi> cot </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> &#981; </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> &#928; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> n </mi> <mo> &#10072; </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mrow> <mi> E </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> </mfrac> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> &#981; </mi> <mo> &#10869; </mo> <mrow> <msup> <mi> sin </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <msqrt> <mfrac> <mi> n </mi> <mi> m </mi> </mfrac> </msqrt> <mo> ) </mo> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <mn> 0 </mn> <mo> &lt; </mo> <mi> n </mi> <mo> &lt; </mo> <mn> 1 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mn> 0 </mn> <mo> &lt; </mo> <mi> m </mi> <mo> &lt; </mo> <mn> 1 </mn> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> EllipticF </ci> <ci> &#981; </ci> <ci> m </ci> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <cot /> <ci> &#981; </ci> </apply> </apply> <apply> <ci> EllipticE </ci> <ci> &#981; </ci> <ci> m </ci> </apply> </apply> <apply> <ci> EllipticK </ci> <ci> m </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <cot /> <ci> &#981; </ci> </apply> <apply> <ci> EllipticPi </ci> <ci> n </ci> <ci> m </ci> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <ci> EllipticE </ci> <ci> m </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <and /> <apply> <eq /> <ci> &#981; </ci> <apply> <arcsin /> <apply> <power /> <apply> <times /> <ci> n </ci> <apply> <power /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <lt /> <cn type='integer'> 0 </cn> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> <apply> <lt /> <cn type='integer'> 0 </cn> <ci> m </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["EllipticF", "[", RowBox[List["\[Phi]_", ",", "m_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[SqrtBox[RowBox[List["1", "-", "n"]]], " ", RowBox[List["Cot", "[", "\[Phi]", "]"]]]], "+", RowBox[List["EllipticE", "[", RowBox[List["\[Phi]", ",", "m"]], "]"]]]], ")"]], " ", RowBox[List["EllipticK", "[", "m", "]"]]]], "-", RowBox[List[SqrtBox[RowBox[List["1", "-", "n"]]], " ", RowBox[List["Cot", "[", "\[Phi]", "]"]], " ", RowBox[List["EllipticPi", "[", RowBox[List["n", ",", "m"]], "]"]]]]]], RowBox[List["EllipticE", "[", "m", "]"]]], "/;", RowBox[List[RowBox[List["\[Phi]", "\[Equal]", RowBox[List["ArcSin", "[", SqrtBox[FractionBox["n", "m"]], "]"]]]], "&&", RowBox[List["0", "<", "n", "<", "1"]], "&&", RowBox[List["0", "<", "m", "<", "1"]]]]]]]]]]

2001-10-29