Incomplete elliptic integral of the second kind: Series representations (formula 08.04.06.0045)

EllipticE

$Mathematica Notation$

Elliptic Integrals

EllipticE[z,m]

Series representations

Generalized power series

Expansions at z==infinity

http://functions.wolfram.com/08.04.06.0045.01

Input Form

EllipticE[z, m] == (-1)^Round[Re[z]/Pi] (((Sqrt[(-m) Sin[z]^2] Sqrt[-Sin[z]^2])/(-Sin[z])) Sum[(Pochhammer[-(1/2), k]/(Sin[z]^(2 k) k!)) Hypergeometric2F1[-k, -(1/2), 3/2 - k, 1/m], {k, 0, Infinity}] + ((Sqrt[-Sin[z]^2] Sqrt[-m])/(2 Sin[z])) ((((I Sqrt[-Sin[z]^2])/Sqrt[Sin[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[-Sin[z]^2]/(Sin[z] Sqrt[(-m) Sin[z]^2])) Sum[((Pochhammer[-(1/2), k]/((2 k + 1) k!)) Hypergeometric2F1[1/2, -k, 1/2 - k, 1/m])/Sin[z]^(2 k), {k, 0, Infinity}]) + 2 Round[Re[z]/Pi] EllipticE[m]

Standard Form

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MathML Form

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</mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mo> ⌊ </mo> <mfrac> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mi> π </mi> </mfrac> <mo> ⌉ </mo> </mrow> </msup> <mo> ⁢ </mo> <mtext> </mtext> <mrow> <mo> ( </mo> <mrow> <mrow> <mfrac> <mrow> <msqrt> <mrow> <mo> - </mo> <mrow> <msup> <mi> sin </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </msqrt> <mo> ⁢ </mo> <msqrt> <mrow> <mo> - </mo> <mi> m </mi> </mrow> </msqrt> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mi> E </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mn> 1 </mn> <mi> m </mi> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mfrac> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <msqrt> <mrow> <mo> - </mo> <mrow> <msup> <mi> sin </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </msqrt> </mrow> <msqrt> <mrow> <msup> <mi> sin </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </msqrt> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msqrt> <mfrac> <mn> 1 </mn> <mi> m </mi> </mfrac> </msqrt> <mo> - </mo> <mfrac> <mn> 1 </mn> <msqrt> <mi> m </mi> </msqrt> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <msqrt> <mfrac> <mn> 1 </mn> <mi> m </mi> </mfrac> </msqrt> <mo> - </mo> <mfrac> <mn> 1 </mn> <msqrt> <mi> m </mi> </msqrt> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> E </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <msqrt> <mfrac> <mi> m </mi> <mrow> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </mfrac> </msqrt> <mo> ⁢ </mo> <msqrt> <mfrac> <mrow> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mi> m </mi> </mfrac> </msqrt> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mfrac> <mn> 1 </mn> <mi> m </mi> </mfrac> <mo> ⁢ </mo> <mrow> <mi> K </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <mn> 1 </mn> <mi> m </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mi> E </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <mn> 1 </mn> <mi> m </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mi> m </mi> </mfrac> <mo> ⁢ </mo> <mrow> <mi> K </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mn> 1 </mn> <mi> m </mi> </mfrac> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mfrac> <mrow> <msqrt> <mrow> <mrow> <mo> - </mo> <mi> m </mi> </mrow> <mo> ⁢ </mo> <mrow> <msup> <mi> sin </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </msqrt> <mo> ⁢ </mo> <msqrt> <mrow> <mo> - </mo> <mrow> <msup> <mi> sin </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </msqrt> </mrow> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> ∞ </mi> </munderover> <mrow> <mfrac> <mrow> <mrow> <msup> <mi> sin </mi> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ⁢ </mo> <mi> k </mi> </mrow> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["1", "2"]]], ")"]], "k"], Pochhammer] </annotation> </semantics> </mrow> <mrow> <mi> k </mi> <mo> ! </mo> </mrow> </mfrac> <mo> ⁢ </mo> <semantics> <mrow> <mrow> <msub> <mo>   </mo> <mn> 2 </mn> </msub> <msub> <mi> F </mi> <mn> 1 </mn> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mi> k </mi> </mrow> <mo> , </mo> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </mrow> <mo> ; </mo> <mrow> <mfrac> <mn> 3 </mn> <mn> 2 </mn> </mfrac> <mo> - </mo> <mi> k </mi> </mrow> <mo> ; </mo> <mfrac> <mn> 1 </mn> <mi> m </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> 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] </annotation> </semantics> </mrow> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mrow> <msqrt> <mrow> <mo> - </mo> <mrow> <msup> <mi> sin </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </msqrt> <mtext> </mtext> </mrow> <mrow> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msqrt> <mrow> <mrow> <mo> - </mo> <mi> m </mi> </mrow> <mo> ⁢ </mo> <mrow> <msup> <mi> sin </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </msqrt> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> ∞ </mi> </munderover> <mrow> <mfrac> <mrow> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["1", "2"]]], ")"]], "k"], Pochhammer] </annotation> </semantics> <mo> ⁢ </mo> <mrow> <msup> <mi> sin </mi> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ⁢ </mo> <mi> k </mi> </mrow> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> k </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> k </mi> <mo> ! </mo> </mrow> </mrow> </mfrac> <mo> ⁢ </mo> <semantics> <mrow> <mrow> <msub> <mo>   </mo> <mn> 2 </mn> </msub> <msub> <mi> F </mi> <mn> 1 </mn> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> , </mo> <mrow> <mo> - </mo> <mi> k </mi> </mrow> </mrow> <mo> ; </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> - </mo> <mi> k </mi> </mrow> <mo> ; </mo> <mfrac> <mn> 1 </mn> <mi> m </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> 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] </annotation> </semantics> </mrow> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["EllipticE", "[", RowBox[List["z_", ",", "m_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["Round", "[", FractionBox[RowBox[List["Re", "[", "z", "]"]], "\[Pi]"], "]"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "m"]], " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]]], " ", SqrtBox[RowBox[List["-", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["Sin", "[", "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", "!"]]]]]]], RowBox[List["Sin", "[", "z", "]"]]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SqrtBox[RowBox[List["-", SuperscriptBox[RowBox[List["Sin", "[", "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[RowBox[List["Sin", "[", "z", "]"]], "2"]]]]]], ")"]], " ", RowBox[List["(", RowBox[List[SqrtBox[FractionBox["1", "m"]], "-", FractionBox["1", SqrtBox["m"]]]], ")"]]]], SqrtBox[SuperscriptBox[RowBox[List["Sin", "[", "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", " ", RowBox[List["Sin", "[", "z", "]"]]]]], "+", FractionBox[RowBox[List[SqrtBox[RowBox[List["-", SuperscriptBox[RowBox[List["Sin", "[", "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[RowBox[List["Sin", "[", "z", "]"]], RowBox[List[RowBox[List["-", "2"]], " ", "k"]]]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]], ")"]], " ", RowBox[List["k", "!"]]]]]]]]], RowBox[List[RowBox[List["Sin", "[", "z", "]"]], " ", SqrtBox[RowBox[List[RowBox[List["-", "m"]], " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]]]]]]]], ")"]]]], "+", RowBox[List["2", " ", RowBox[List["Round", "[", FractionBox[RowBox[List["Re", "[", "z", "]"]], "\[Pi]"], "]"]], " ", RowBox[List["EllipticE", "[", "m", "]"]]]]]]]]]]

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

2007-05-02

EllipticE[z]