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EllipticF






Mathematica Notation

Traditional Notation









Elliptic Integrals > EllipticF[z,m] > Identities > Functional identities





http://functions.wolfram.com/08.05.17.0002.01









  


  










Input Form





EllipticF[z, m] == (2/(Sqrt[m] + 1)) EllipticF[(1/2) ArcSin[(Sqrt[m] Tan[z] + Sqrt[Tan[z]^2 + (1 - m) Tan[z]^4])/(1 + Tan[z]^2)], (4 Sqrt[m])/(1 + Sqrt[m])^2] /; Inequality[0, LessEqual, m, Less, 1] && Inequality[0, LessEqual, z, Less, 1]










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["EllipticF", "[", RowBox[List["z", ",", "m"]], "]"]], "\[Equal]", RowBox[List[FractionBox["2", RowBox[List[SqrtBox["m"], "+", "1"]]], " ", RowBox[List["EllipticF", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["ArcSin", "[", " ", FractionBox[RowBox[List[RowBox[List[SqrtBox["m"], " ", RowBox[List["Tan", "[", "z", "]"]]]], "+", SqrtBox[RowBox[List[SuperscriptBox[RowBox[List["Tan", "[", "z", "]"]], "2"], "+", RowBox[List[RowBox[List["(", RowBox[List["1", "-", "m"]], ")"]], " ", SuperscriptBox[RowBox[List["Tan", "[", "z", "]"]], "4"]]]]]]]], RowBox[List["1", "+", SuperscriptBox[RowBox[List["Tan", "[", "z", "]"]], "2"]]]], "]"]]]], ",", FractionBox[RowBox[List["4", " ", SqrtBox["m"]]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", SqrtBox["m"]]], ")"]], "2"]]]], "]"]]]]]], "/;", RowBox[List[RowBox[List["0", "\[LessEqual]", "m", "<", "1"]], "\[And]", RowBox[List["0", "\[LessEqual]", "z", "<", "1"]]]]]]]]










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> <mi> F </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> &#10072; </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <mrow> <mn> 2 </mn> <mtext> </mtext> </mrow> <mrow> <msqrt> <mi> m </mi> </msqrt> <mo> + </mo> <mn> 1 </mn> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mi> F </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <msup> <mi> sin </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <mrow> <mrow> <msqrt> <mi> m </mi> </msqrt> <mo> &#8290; </mo> <mrow> <mi> tan </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <msqrt> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msup> <mi> tan </mi> <mn> 4 </mn> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <msup> <mi> tan </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </msqrt> </mrow> <mrow> <mrow> <msup> <mi> tan </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> &#10072; </mo> <mfrac> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <msqrt> <mi> m </mi> </msqrt> </mrow> <msup> <mrow> <mo> ( </mo> <mrow> <msqrt> <mi> m </mi> </msqrt> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mn> 0 </mn> <mo> &#8804; </mo> <mi> m </mi> <mo> &lt; </mo> <mn> 1 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mn> 0 </mn> <mo> &#8804; </mo> <mi> z </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> z </ci> <ci> m </ci> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> m </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> EllipticF </ci> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <arcsin /> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> m </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <tan /> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> <apply> <power /> <apply> <tan /> <ci> z </ci> </apply> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <power /> <apply> <tan /> <ci> z </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <power /> <apply> <tan /> <ci> z </ci> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <ci> m </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> m </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <ci> Inequality </ci> <cn type='integer'> 0 </cn> <leq /> <ci> m </ci> <lt /> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Inequality </ci> <cn type='integer'> 0 </cn> <leq /> <ci> z </ci> <lt /> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["EllipticF", "[", RowBox[List["z_", ",", "m_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List["2", " ", RowBox[List["EllipticF", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["ArcSin", "[", FractionBox[RowBox[List[RowBox[List[SqrtBox["m"], " ", RowBox[List["Tan", "[", "z", "]"]]]], "+", SqrtBox[RowBox[List[SuperscriptBox[RowBox[List["Tan", "[", "z", "]"]], "2"], "+", RowBox[List[RowBox[List["(", RowBox[List["1", "-", "m"]], ")"]], " ", SuperscriptBox[RowBox[List["Tan", "[", "z", "]"]], "4"]]]]]]]], RowBox[List["1", "+", SuperscriptBox[RowBox[List["Tan", "[", "z", "]"]], "2"]]]], "]"]]]], ",", FractionBox[RowBox[List["4", " ", SqrtBox["m"]]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", SqrtBox["m"]]], ")"]], "2"]]]], "]"]]]], RowBox[List[SqrtBox["m"], "+", "1"]]], "/;", RowBox[List[RowBox[List["0", "\[LessEqual]", "m", "<", "1"]], "&&", RowBox[List["0", "\[LessEqual]", "z", "<", "1"]]]]]]]]]]










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





2001-10-29