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EllipticF






Mathematica Notation

Traditional Notation









Elliptic Integrals > EllipticF[z,m] > General characteristics > Branch cuts > With respect to z > Formulas for vertical intervals > For m>0





http://functions.wolfram.com/08.05.04.0019.01









  


  










Input Form





Limit[EllipticF[Pi/2 + 2 k Pi + I x + \[Epsilon], m], \[Epsilon] -> Plus[0]] == -EllipticF[Pi/2 + I x, m] - (2/Sqrt[m]) EllipticK[1/m] + 4 (k + 1) EllipticK[m] /; (Element[m, Reals] && Element[x, Reals] && (0 < m < 1 && x > -Im[ArcCsc[Sqrt[m]]])) || ((m > 1 && x < 0) && Element[k, Integers])










Standard Form





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MathML Form







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</mo> <mi> x </mi> </mrow> <mo> + </mo> <mfrac> <mi> &#960; </mi> <mn> 2 </mn> </mfrac> </mrow> <mo> &#10072; </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mfrac> <mn> 2 </mn> <msqrt> <mi> m </mi> </msqrt> </mfrac> <mo> &#8290; </mo> <mrow> <mi> K </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mn> 1 </mn> <mi> m </mi> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> K </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <mi> m </mi> <mo> &#8712; </mo> <semantics> <mi> &#8477; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalR]&quot;, Function[List[], Reals]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mi> x </mi> <mo> &#8712; </mo> <semantics> <mi> &#8477; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalR]&quot;, Function[List[], Reals]] </annotation> </semantics> </mrow> <mo> &#8743; 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</ci> </bvar> <condition> <apply> <tendsto /> <ci> &#1013; </ci> <apply> <plus /> <cn type='integer'> 0 </cn> </apply> </apply> </condition> <apply> <ci> EllipticF </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> <ci> k </ci> </apply> <apply> <times /> <imaginaryi /> <ci> x </ci> </apply> <apply> <times /> <pi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <ci> &#1013; </ci> </apply> <ci> m </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> F </ci> </apply> <apply> <ci> VerticalSeparator </ci> <apply> <plus /> <apply> <times /> <imaginaryi /> <ci> x </ci> </apply> <apply> <times /> <pi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <ci> m </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <power /> <ci> m </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> EllipticK </ci> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <plus /> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> EllipticK </ci> <ci> m </ci> </apply> </apply> </apply> </apply> <apply> <or /> <apply> <and /> <apply> <in /> <ci> m </ci> <reals /> </apply> <apply> <in /> <ci> x </ci> <reals /> </apply> <apply> <and /> <apply> <lt /> <cn type='integer'> 0 </cn> <ci> m </ci> <cn type='integer'> 1 </cn> </apply> <apply> <gt /> <ci> x </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <imaginary /> <apply> <arccsc /> <apply> <power /> <ci> m </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <and /> <apply> <gt /> <ci> m </ci> <cn type='integer'> 1 </cn> </apply> <apply> <lt /> <ci> x </ci> <cn type='integer'> 0 </cn> </apply> </apply> <apply> <in /> <ci> k </ci> <integers /> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Limit", "[", RowBox[List[RowBox[List["EllipticF", "[", RowBox[List[RowBox[List[FractionBox["\[Pi]", "2"], "+", RowBox[List["2", " ", "k", " ", "\[Pi]"]], "+", RowBox[List["\[ImaginaryI]", " ", "x_"]], "+", "\[Epsilon]_"]], ",", "m_"]], "]"]], ",", RowBox[List["\[Epsilon]_", "\[Rule]", RowBox[List["+", "0"]]]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["-", RowBox[List["EllipticF", "[", RowBox[List[RowBox[List[FractionBox["\[Pi]", "2"], "+", RowBox[List["\[ImaginaryI]", " ", "x"]]]], ",", "m"]], "]"]]]], "-", FractionBox[RowBox[List["2", " ", RowBox[List["EllipticK", "[", FractionBox["1", "m"], "]"]]]], SqrtBox["m"]], "+", RowBox[List["4", " ", RowBox[List["(", RowBox[List["k", "+", "1"]], ")"]], " ", RowBox[List["EllipticK", "[", "m", "]"]]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["m", "\[Element]", "Reals"]], "&&", RowBox[List["x", "\[Element]", "Reals"]], "&&", RowBox[List["(", RowBox[List[RowBox[List["0", "<", "m", "<", "1"]], "&&", RowBox[List["x", ">", RowBox[List["-", RowBox[List["Im", "[", RowBox[List["ArcCsc", "[", SqrtBox["m"], "]"]], "]"]]]]]]]], ")"]]]], ")"]], "||", RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["m", ">", "1"]], "&&", RowBox[List["x", "<", "0"]]]], ")"]], "&&", RowBox[List["k", "\[Element]", "Integers"]]]], ")"]]]]]]]]]]










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





2007-05-02





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