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 EllipticF

 http://functions.wolfram.com/08.05.06.0030.01

 Input Form

 EllipticF[z, m] \[Proportional] (2 (2 Re[Subscript[z, 0]/(2 Pi) - 3/4] + 1) EllipticK[m] + (1/Sqrt[m]) EllipticK[1/m]) (1 - Exp[(-Pi) I (Floor[1/4 + Arg[z - Subscript[z, 0]]/(2 Pi)] + Floor[1/4 - Arg[z - Subscript[z, 0]]/(2 Pi)])]) + Exp[(-Pi) I (Floor[1/4 + Arg[z - Subscript[z, 0]]/(2 Pi)] + Floor[1/4 - Arg[z - Subscript[z, 0]]/(2 Pi)])] EllipticF[Subscript[z, 0], m] + (1/Sqrt[1 - m Sin[Subscript[z, 0]]^2]) (z - Subscript[z, 0]) + ((m Sin[2 Subscript[z, 0]])/ (4 (1 - m Sin[Subscript[z, 0]]^2)^(3/2))) (z - Subscript[z, 0])^2 + \[Ellipsis] /; (z -> Subscript[z, 0]) && Element[Re[Subscript[z, 0]/(2 Pi) - 3/4], Integers]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["EllipticF", "[", RowBox[List["z", ",", "m"]], "]"]], "\[Proportional]", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", RowBox[List["(", RowBox[List[RowBox[List["2", RowBox[List["Re", "[", RowBox[List[FractionBox[SubscriptBox["z", "0"], RowBox[List["2", "\[Pi]"]]], "-", FractionBox["3", "4"]]], "]"]]]], "+", "1"]], ")"]], " ", RowBox[List["EllipticK", "[", "m", "]"]]]], "+", RowBox[List[FractionBox["1", SqrtBox["m"]], " ", RowBox[List["EllipticK", "[", FractionBox["1", "m"], "]"]]]]]], ")"]], RowBox[List["(", RowBox[List["1", "-", RowBox[List["Exp", "[", RowBox[List[RowBox[List["-", "\[Pi]"]], " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "4"], "+", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", SubscriptBox["z", "0"]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]]]], "]"]], "+", RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "4"], "-", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", SubscriptBox["z", "0"]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]]]], "]"]]]], ")"]]]], "]"]]]], ")"]]]], "+", RowBox[List[RowBox[List["Exp", "[", RowBox[List[RowBox[List["-", "\[Pi]"]], " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "4"], "+", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", SubscriptBox["z", "0"]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]]]], "]"]], "+", RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "4"], "-", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", SubscriptBox["z", "0"]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]]]], "]"]]]], ")"]]]], "]"]], RowBox[List["EllipticF", "[", RowBox[List[SubscriptBox["z", "0"], ",", "m"]], "]"]]]], "+", RowBox[List[FractionBox["1", SqrtBox[RowBox[List["1", "-", RowBox[List["m", " ", SuperscriptBox[RowBox[List["Sin", "[", SubscriptBox["z", "0"], "]"]], "2"]]]]]]], " ", RowBox[List["(", RowBox[List["z", "-", SubscriptBox["z", "0"]]], ")"]]]], "+", RowBox[List[FractionBox[RowBox[List["m", " ", RowBox[List["Sin", "[", RowBox[List["2", SubscriptBox["z", "0"]]], "]"]]]], RowBox[List["4", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", RowBox[List["m", " ", SuperscriptBox[RowBox[List["Sin", "[", SubscriptBox["z", "0"], "]"]], "2"]]]]], ")"]], RowBox[List["3", "/", "2"]]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", SubscriptBox["z", "0"]]], ")"]], "2"]]], "+", "\[Ellipsis]"]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List["z", "\[Rule]", SubscriptBox["z", "0"]]], ")"]], "\[And]", RowBox[List[RowBox[List["Re", "[", RowBox[List[FractionBox[SubscriptBox["z", "0"], RowBox[List["2", "\[Pi]"]]], "-", FractionBox["3", "4"]]], "]"]], "\[Element]", "Integers"]]]]]]]]

 MathML Form

 F ( z m ) ( 1 m K ( 1 m ) + 2 ( 2 Re ( z 0 2 π - 3 4 ) + 1 ) K ( m ) ) ( 1 - - π ( arg ( z - z 0 ) 2 π + 1 4 + 1 4 - arg ( z - z 0 ) 2 π ) ) + F ( z 0 m ) - π ( arg ( z - z 0 ) 2 π + 1 4 + 1 4 - arg ( z - z 0 ) 2 π ) + 1 1 - m sin 2 ( z 0 ) ( z - z 0 ) + m sin ( 2 z 0 ) 4 ( 1 - m sin 2 ( z 0 ) ) 3 / 2 ( z - z 0 ) 2 + /; ( z "\[Rule]" z 0 ) Re ( z 0 2 π - 3 4 ) TagBox["\[DoubleStruckCapitalZ]", Function[List[], Integers]] Condition Proportional EllipticF z m 1 m 1 2 -1 EllipticK 1 m -1 2 2 Subscript z 0 2 -1 -1 3 4 1 EllipticK m 1 -1 -1 z -1 Subscript z 0 2 -1 1 4 1 4 -1 z -1 Subscript z 0 2 -1 EllipticF Subscript z 0 m -1 z -1 Subscript z 0 2 -1 1 4 1 4 -1 z -1 Subscript z 0 2 -1 1 1 -1 m Subscript z 0 2 1 2 -1 z -1 Subscript z 0 m 2 Subscript z 0 4 1 -1 m Subscript z 0 2 3 2 -1 z -1 Subscript z 0 2 Rule z Subscript z 0 Subscript z 0 2 -1 -1 3 4 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["EllipticF", "[", RowBox[List["z_", ",", "m_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", RowBox[List["Re", "[", RowBox[List[FractionBox[SubscriptBox["zz", "0"], RowBox[List["2", " ", "\[Pi]"]]], "-", FractionBox["3", "4"]]], "]"]]]], "+", "1"]], ")"]], " ", RowBox[List["EllipticK", "[", "m", "]"]]]], "+", FractionBox[RowBox[List["EllipticK", "[", FractionBox["1", "m"], "]"]], SqrtBox["m"]]]], ")"]], " ", RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "\[Pi]"]], " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "4"], "+", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]]]], "]"]], "+", RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "4"], "-", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]]]], "]"]]]], ")"]]]]]]], ")"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "\[Pi]"]], " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "4"], "+", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]]]], "]"]], "+", RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "4"], "-", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]]]], "]"]]]], ")"]]]]], " ", RowBox[List["EllipticF", "[", RowBox[List[SubscriptBox["zz", "0"], ",", "m"]], "]"]]]], "+", FractionBox[RowBox[List["z", "-", SubscriptBox["zz", "0"]]], SqrtBox[RowBox[List["1", "-", RowBox[List["m", " ", SuperscriptBox[RowBox[List["Sin", "[", SubscriptBox["zz", "0"], "]"]], "2"]]]]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["m", " ", RowBox[List["Sin", "[", RowBox[List["2", " ", SubscriptBox["zz", "0"]]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], ")"]], "2"]]], RowBox[List["4", " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", RowBox[List["m", " ", SuperscriptBox[RowBox[List["Sin", "[", SubscriptBox["zz", "0"], "]"]], "2"]]]]], ")"]], RowBox[List["3", "/", "2"]]]]]], "+", "\[Ellipsis]"]], "/;", RowBox[List[RowBox[List["(", RowBox[List["z", "\[Rule]", SubscriptBox["zz", "0"]]], ")"]], "&&", RowBox[List[RowBox[List["Re", "[", RowBox[List[FractionBox[SubscriptBox["zz", "0"], RowBox[List["2", " ", "\[Pi]"]]], "-", FractionBox["3", "4"]]], "]"]], "\[Element]", "Integers"]]]]]]]]]]

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

 2007-05-02