html, body, form { margin: 0; padding: 0; width: 100%; } #calculate { position: relative; width: 177px; height: 110px; background: transparent url(/images/alphabox/embed_functions_inside.gif) no-repeat scroll 0 0; } #i { position: relative; left: 18px; top: 44px; width: 133px; border: 0 none; outline: 0; font-size: 11px; } #eq { width: 9px; height: 10px; background: transparent; position: absolute; top: 47px; right: 18px; cursor: pointer; }

 EllipticE

 http://functions.wolfram.com/08.04.06.0082.01

 Input Form

 EllipticE[ArcSin[z], m] \[Proportional] (m z Sqrt[-z^2])/Sqrt[(-m) z^2] + ((Sqrt[-z^2] Sqrt[-m])/(2 z)) ((((I Sqrt[-z^2])/Sqrt[z^2]) (Sqrt[1/m] - 1/Sqrt[m]) - Sqrt[1/m] - 1/Sqrt[m]) EllipticE[m] + 2 EllipticE[1/m] + 2 ((1 - m)/m) EllipticK[1/m] + 2 I (1 - Sqrt[m/(m - 1)] Sqrt[(m - 1)/m]) (-EllipticE[1 - 1/m] + (1/m) EllipticK[1 - 1/m])) + (((1 - m) Sqrt[-z^2])/(2 z Sqrt[(-m) z^2])) (1 - (-7 + 2 m - 3 m^2)/(12 (-1 + m) m z^2) + (21 - 11 m - 9 m^2 + 15 m^3)/(120 (-1 + m) m^2 z^4) + \[Ellipsis]) /; (Abs[z] -> Infinity)

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["EllipticE", "[", RowBox[List[RowBox[List["ArcSin", "[", "z", "]"]], ",", "m"]], "]"]], "\[Proportional]", RowBox[List[FractionBox[RowBox[List["m", " ", "z", " ", SqrtBox[RowBox[List["-", SuperscriptBox["z", "2"]]]]]], SqrtBox[RowBox[List[RowBox[List["-", "m"]], " ", SuperscriptBox["z", "2"]]]]], "+", RowBox[List[FractionBox[RowBox[List[SqrtBox[RowBox[List["-", SuperscriptBox["z", "2"]]]], SqrtBox[RowBox[List["-", "m"]]]]], RowBox[List["2", "z"]]], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", SqrtBox[RowBox[List["-", SuperscriptBox["z", "2"]]]]]], SqrtBox[SuperscriptBox["z", "2"]]], RowBox[List["(", RowBox[List[SqrtBox[FractionBox["1", "m"]], "-", FractionBox["1", SqrtBox["m"]]]], ")"]]]], "-", SqrtBox[FractionBox["1", "m"]], "-", FractionBox["1", SqrtBox["m"]]]], ")"]], RowBox[List["EllipticE", "[", "m", "]"]]]], "+", RowBox[List["2", RowBox[List["EllipticE", "[", FractionBox["1", "m"], "]"]]]], "+", RowBox[List["2", FractionBox[RowBox[List["1", "-", "m"]], "m"], " ", RowBox[List["EllipticK", "[", FractionBox["1", "m"], "]"]]]], "+", RowBox[List["2", "\[ImaginaryI]", RowBox[List["(", RowBox[List["1", "-", RowBox[List[SqrtBox[FractionBox["m", RowBox[List["m", "-", "1"]]]], SqrtBox[FractionBox[RowBox[List["m", "-", "1"]], "m"]]]]]], ")"]], RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["EllipticE", "[", RowBox[List["1", "-", FractionBox["1", "m"]]], "]"]]]], "+", RowBox[List[FractionBox["1", "m"], " ", RowBox[List["EllipticK", "[", RowBox[List["1", "-", FractionBox["1", "m"]]], "]"]]]]]], ")"]]]]]], ")"]]]], "+", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "-", "m"]], ")"]], SqrtBox[RowBox[List["-", SuperscriptBox["z", "2"]]]]]], RowBox[List["2", "z", " ", SqrtBox[RowBox[List[RowBox[List["-", "m"]], " ", SuperscriptBox["z", "2"]]]]]]], RowBox[List["(", RowBox[List["1", "-", FractionBox[RowBox[List[RowBox[List["-", "7"]], "+", RowBox[List["2", " ", "m"]], "-", RowBox[List["3", " ", SuperscriptBox["m", "2"]]]]], RowBox[List["12", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "m"]], ")"]], " ", "m", " ", SuperscriptBox["z", "2"]]]], "+", FractionBox[RowBox[List["21", "-", RowBox[List["11", " ", "m"]], "-", RowBox[List["9", " ", SuperscriptBox["m", "2"]]], "+", RowBox[List["15", " ", SuperscriptBox["m", "3"]]]]], RowBox[List["120", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "m"]], ")"]], " ", SuperscriptBox["m", "2"], " ", SuperscriptBox["z", "4"]]]], "+", "\[Ellipsis]"]], ")"]]]]]]]], "/;", RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "\[Rule]", "\[Infinity]"]], ")"]]]]]]

 MathML Form

 E ( sin - 1 ( z ) m ) m z - z 2 - m z 2 + - z 2 - m 2 z ( 2 E ( 1 m ) + ( - z 2 z 2 ( 1 m - 1 m ) - 1 m - 1 m ) E ( m ) + 2 ( 1 - m m - 1 m - 1 m ) ( 1 m K ( 1 - 1 m ) - E ( 1 - 1 m ) ) + 2 ( 1 - m ) m K ( 1 m ) ) + ( 1 - m ) - z 2 2 z - m z 2 ( 1 - - 3 m 2 + 2 m - 7 12 ( m - 1 ) m z 2 + 15 m 3 - 9 m 2 - 11 m + 21 120 ( m - 1 ) m 2 z 4 + ) /; ( "\[LeftBracketingBar]" z "\[RightBracketingBar]" "\[Rule]" ) Condition Proportional EllipticE z m m z -1 z 2 1 2 -1 m z 2 1 2 -1 -1 z 2 1 2 -1 m 1 2 2 z -1 2 EllipticE 1 m -1 -1 z 2 1 2 z 2 1 2 -1 1 m -1 1 2 -1 1 m 1 2 -1 -1 1 m -1 1 2 -1 1 m 1 2 -1 EllipticE m 2 1 -1 m m -1 -1 1 2 m -1 m -1 1 2 1 m -1 EllipticK 1 -1 1 m -1 -1 EllipticE 1 -1 1 m -1 2 1 -1 m m -1 EllipticK 1 m -1 1 -1 m -1 z 2 1 2 2 z -1 m z 2 1 2 -1 1 -1 -3 m 2 2 m -7 12 m -1 m z 2 -1 15 m 3 -1 9 m 2 -1 11 m 21 120 m -1 m 2 z 4 -1 Rule z [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["EllipticE", "[", RowBox[List[RowBox[List["ArcSin", "[", "z_", "]"]], ",", "m_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List["m", " ", "z", " ", SqrtBox[RowBox[List["-", SuperscriptBox["z", "2"]]]]]], SqrtBox[RowBox[List[RowBox[List["-", "m"]], " ", SuperscriptBox["z", "2"]]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SqrtBox[RowBox[List["-", SuperscriptBox["z", "2"]]]], " ", SqrtBox[RowBox[List["-", "m"]]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", SqrtBox[RowBox[List["-", SuperscriptBox["z", "2"]]]]]], ")"]], " ", RowBox[List["(", RowBox[List[SqrtBox[FractionBox["1", "m"]], "-", FractionBox["1", SqrtBox["m"]]]], ")"]]]], SqrtBox[SuperscriptBox["z", "2"]]], "-", SqrtBox[FractionBox["1", "m"]], "-", FractionBox["1", SqrtBox["m"]]]], ")"]], " ", RowBox[List["EllipticE", "[", "m", "]"]]]], "+", RowBox[List["2", " ", RowBox[List["EllipticE", "[", FractionBox["1", "m"], "]"]]]], "+", FractionBox[RowBox[List["2", " ", RowBox[List["(", RowBox[List["1", "-", "m"]], ")"]], " ", RowBox[List["EllipticK", "[", FractionBox["1", "m"], "]"]]]], "m"], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List[SqrtBox[FractionBox["m", RowBox[List["m", "-", "1"]]]], " ", SqrtBox[FractionBox[RowBox[List["m", "-", "1"]], "m"]]]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["EllipticE", "[", RowBox[List["1", "-", FractionBox["1", "m"]]], "]"]]]], "+", FractionBox[RowBox[List["EllipticK", "[", RowBox[List["1", "-", FractionBox["1", "m"]]], "]"]], "m"]]], ")"]]]]]], ")"]]]], RowBox[List["2", " ", "z"]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["1", "-", "m"]], ")"]], " ", SqrtBox[RowBox[List["-", SuperscriptBox["z", "2"]]]]]], ")"]], " ", RowBox[List["(", RowBox[List["1", "-", FractionBox[RowBox[List[RowBox[List["-", "7"]], "+", RowBox[List["2", " ", "m"]], "-", RowBox[List["3", " ", SuperscriptBox["m", "2"]]]]], RowBox[List["12", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "m"]], ")"]], " ", "m", " ", SuperscriptBox["z", "2"]]]], "+", FractionBox[RowBox[List["21", "-", RowBox[List["11", " ", "m"]], "-", RowBox[List["9", " ", SuperscriptBox["m", "2"]]], "+", RowBox[List["15", " ", SuperscriptBox["m", "3"]]]]], RowBox[List["120", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "m"]], ")"]], " ", SuperscriptBox["m", "2"], " ", SuperscriptBox["z", "4"]]]], "+", "\[Ellipsis]"]], ")"]]]], RowBox[List["2", " ", "z", " ", SqrtBox[RowBox[List[RowBox[List["-", "m"]], " ", SuperscriptBox["z", "2"]]]]]]]]], "/;", RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "\[Rule]", "\[Infinity]"]], ")"]]]]]]]]

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

 2007-05-02