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/08.04.06.0085.01

 Input Form

 EllipticE[ArcSin[z], m] == ((m z Sqrt[-z^2])/Sqrt[(-m) z^2]) Sum[(Pochhammer[-(1/2), k]/(z^(2 k) k!)) Hypergeometric2F1[-k, -(1/2), 3/2 - k, 1/m], {k, 0, Infinity}] + ((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])) + (Sqrt[-z^2]/(z Sqrt[(-m) z^2])) Sum[((Pochhammer[-(1/2), k]/((2 k + 1) k!)) Hypergeometric2F1[1/2, -k, 1/2 - k, 1/m])/z^(2 k), {k, 0, Infinity}] /; Abs[z] > 1

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["EllipticE", "[", RowBox[List[RowBox[List["ArcSin", "[", "z", "]"]], ",", "m"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[FractionBox[RowBox[List["m", " ", "z", " ", SqrtBox[RowBox[List["-", SuperscriptBox["z", "2"]]]]]], SqrtBox[RowBox[List[RowBox[List["-", "m"]], " ", SuperscriptBox["z", "2"]]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox["z", RowBox[List[RowBox[List["-", "2"]], " ", "k"]]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], ",", "k"]], "]"]]]], RowBox[List["k", "!"]]], RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["-", "k"]], ",", RowBox[List["-", FractionBox["1", "2"]]], ",", RowBox[List[FractionBox["3", "2"], "-", "k"]], ",", FractionBox["1", "m"]]], "]"]]]]]]]], "+", 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[SqrtBox[RowBox[List["-", SuperscriptBox["z", "2"]]]], " "]], RowBox[List[" ", RowBox[List["z", SqrtBox[RowBox[List[RowBox[List["-", "m"]], " ", SuperscriptBox["z", "2"]]]]]]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], ",", "k"]], "]"]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]], ")"]], " ", RowBox[List["k", "!"]]]]], RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox["1", "2"], ",", RowBox[List["-", "k"]], ",", RowBox[List[FractionBox["1", "2"], "-", "k"]], ",", FractionBox["1", "m"]]], "]"]], SuperscriptBox["z", RowBox[List[RowBox[List["-", "2"]], "k"]]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], ">", "1"]]]]]]

 MathML Form

 E ( sin - 1 ( z ) m ) - 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 ) ) + - z 2 z - m z 2 k = 0 ( - 1 2 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["1", "2"]]], ")"]], "k"], Pochhammer] ( 2 k + 1 ) k ! 2 F 1 ( 1 2 , - k ; 1 2 - k ; 1 m ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "2"], SubscriptBox["F", "1"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[FractionBox["1", "2"], Hypergeometric2F1, Rule[Editable, True]], ",", TagBox[RowBox[List["-", "k"]], Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List[FractionBox["1", "2"], "-", "k"]], Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[FractionBox["1", "m"], Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] z - 2 k + m z - z 2 - m z 2 k = 0 z - 2 k ( - 1 2 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["1", "2"]]], ")"]], "k"], Pochhammer] k ! 2 F 1 ( - k , - 1 2 ; 3 2 - k ; 1 m ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "2"], SubscriptBox["F", "1"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["-", "k"]], Hypergeometric2F1, Rule[Editable, True]], ",", TagBox[RowBox[List["-", FractionBox["1", "2"]]], Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List[FractionBox["3", "2"], "-", "k"]], Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[FractionBox["1", "m"], Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] /; "\[LeftBracketingBar]" z "\[RightBracketingBar]" > 1 Condition EllipticE z m -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 z 2 1 2 z -1 m z 2 1 2 -1 k 0 Pochhammer -1 1 2 k 2 k 1 k -1 Hypergeometric2F1 1 2 -1 k 1 2 -1 k 1 m -1 z -2 k m z -1 z 2 1 2 -1 m z 2 1 2 -1 k 0 z -2 k Pochhammer -1 1 2 k k -1 Hypergeometric2F1 -1 k -1 1 2 3 2 -1 k 1 m -1 z 1 [/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[RowBox[List["(", RowBox[List["m", " ", "z", " ", SqrtBox[RowBox[List["-", SuperscriptBox["z", "2"]]]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["z", RowBox[List[RowBox[List["-", "2"]], " ", "k"]]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], ",", "k"]], "]"]]]], ")"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["-", "k"]], ",", RowBox[List["-", FractionBox["1", "2"]]], ",", RowBox[List[FractionBox["3", "2"], "-", "k"]], ",", FractionBox["1", "m"]]], "]"]]]], RowBox[List["k", "!"]]]]]]], 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[SqrtBox[RowBox[List["-", SuperscriptBox["z", "2"]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], ",", "k"]], "]"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox["1", "2"], ",", RowBox[List["-", "k"]], ",", RowBox[List[FractionBox["1", "2"], "-", "k"]], ",", FractionBox["1", "m"]]], "]"]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "2"]], " ", "k"]]]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]], ")"]], " ", RowBox[List["k", "!"]]]]]]]]], RowBox[List["z", " ", SqrtBox[RowBox[List[RowBox[List["-", "m"]], " ", SuperscriptBox["z", "2"]]]]]]]]], "/;", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], ">", "1"]]]]]]]]

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

 2007-05-02