EllipticF[z, m] \[Proportional] (-(1/Sqrt[m])) EllipticK[1/m] + 2 u EllipticK[m] - (Sqrt[2]/Sqrt[-1 + m]) Sqrt[(z - Subscript[z, 0]) Sqrt[-1 + m]] (1 + O[z - Subscript[z, 0]]) /; (z -> Subscript[z, 0]) && Subscript[z, 0] == -ArcCsc[Sqrt[m]] + Pi u && Element[u, Integers]

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["EllipticF", "[", RowBox[List["z", ",", "m"]], "]"]], "\[Proportional]", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", SqrtBox["m"]]]], RowBox[List["EllipticK", "[", FractionBox["1", "m"], "]"]]]], "+", RowBox[List["2", "u", " ", RowBox[List["EllipticK", "[", "m", "]"]]]], "-", RowBox[List[FractionBox[SqrtBox["2"], SqrtBox[RowBox[List[RowBox[List["-", "1"]], "+", "m"]]]], SqrtBox[RowBox[List[RowBox[List["(", RowBox[List["z", "-", SubscriptBox["z", "0"]]], ")"]], SqrtBox[RowBox[List[RowBox[List["-", "1"]], "+", "m"]]]]]], RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", RowBox[List["z", "-", SubscriptBox["z", "0"]]], "]"]]]], ")"]]]]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List["z", "->", SubscriptBox["z", "0"]]], ")"]], "\[And]", RowBox[List[SubscriptBox["z", "0"], "\[Equal]", RowBox[List[RowBox[List["-", RowBox[List["ArcCsc", "[", SqrtBox["m"], "]"]]]], "+", RowBox[List["\[Pi]", " ", "u"]]]]]], "\[And]", RowBox[List["u", "\[Element]", "Integers"]]]]]]]]

<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> &#8733; </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <msqrt> <mi> m </mi> </msqrt> </mfrac> </mrow> <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> 2 </mn> <mo> &#8290; </mo> <mi> u </mi> <mo> &#8290; </mo> <mrow> <mi> K </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mfrac> <mrow> <msqrt> <mn> 2 </mn> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msqrt> <mrow> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> </mrow> </msqrt> <mtext> </mtext> </mrow> <msqrt> <mrow> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mrow> <mi> O </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <semantics> <mo> &#8594; </mo> <annotation encoding='Mathematica'> &quot;\[Rule]&quot; </annotation> </semantics> <msub> <mi> z </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mo> &#8743; </mo> <mrow> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> &#63449; </mo> <mrow> <mrow> <mo> - </mo> <mrow> <msup> <mi> csc </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <msqrt> <mi> m </mi> </msqrt> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> u </mi> </mrow> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <mi> u </mi> <mo> &#8712; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[List[], Integers]] </annotation> </semantics> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <ci> Proportional </ci> <apply> <ci> EllipticF </ci> <ci> z </ci> <ci> m </ci> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <ci> m </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </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> <times /> <cn type='integer'> 2 </cn> <ci> u </ci> <apply> <ci> EllipticK </ci> <ci> m </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <ci> O </ci> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <ci> Rule </ci> <ci> z </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> <apply> <eq /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <arccsc /> <apply> <power /> <ci> m </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <times /> <pi /> <ci> u </ci> </apply> </apply> </apply> <apply> <in /> <ci> u </ci> <integers /> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["EllipticF", "[", RowBox[List["z_", ",", "m_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["-", FractionBox[RowBox[List["EllipticK", "[", FractionBox["1", "m"], "]"]], SqrtBox["m"]]]], "+", RowBox[List["2", " ", "u", " ", RowBox[List["EllipticK", "[", "m", "]"]]]], "-", FractionBox[RowBox[List[SqrtBox["2"], " ", SqrtBox[RowBox[List[RowBox[List["(", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], ")"]], " ", SqrtBox[RowBox[List[RowBox[List["-", "1"]], "+", "m"]]]]]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], "]"]]]], ")"]]]], SqrtBox[RowBox[List[RowBox[List["-", "1"]], "+", "m"]]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List["z", "\[Rule]", SubscriptBox["zz", "0"]]], ")"]], "&&", RowBox[List[SubscriptBox["zz", "0"], "\[Equal]", RowBox[List[RowBox[List["-", RowBox[List["ArcCsc", "[", SqrtBox["m"], "]"]]]], "+", RowBox[List["\[Pi]", " ", "u"]]]]]], "&&", RowBox[List["u", "\[Element]", "Integers"]]]]]]]]]]

2007-05-02