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 EllipticE

 http://functions.wolfram.com/08.04.06.0031.01

 Input Form

 EllipticE[z, m] == Sqrt[m] (EllipticE[1/m] + (-1 + 1/m) EllipticK[1/m]) + 2 u EllipticE[m] + 2 Sqrt[2] Sqrt[(-Sqrt[-1 + m]) (z - Subscript[z, 0])] (z - Subscript[z, 0]) Sum[(Binomial[k - 1/2, k]/(3 + 2 k)) Sum[(((-1)^j Binomial[k, j])/(1 - 2 j)) Subscript[p, j, k] (z - Subscript[z, 0])^k, {j, 0, k}], {k, 0, Infinity}] /; Subscript[z, 0] == ArcCsc[Sqrt[m]] + Pi u && Element[u, Integers] && Subscript[a, 0] == 1 && Subscript[a, 2 k] == ((-1)^k 2^(2 k))/(2 k + 1)! && Subscript[a, 2 k + 1] == ((-1)^(k - 1) 2^(2 k) (2 - m))/ (Sqrt[m - 1] (2 k + 2)!) && Element[k, Integers] && k >= 0 && Subscript[p, u, 0] == 1 && Subscript[p, u, v] == (1/v) Sum[(u j - v + j) Subscript[a, j] Subscript[p, u, v - j], {j, 1, v}]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["EllipticE", "[", RowBox[List["z", ",", "m"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[SqrtBox["m"], " ", RowBox[List["(", RowBox[List[RowBox[List["EllipticE", "[", FractionBox["1", "m"], "]"]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", FractionBox["1", "m"]]], ")"]], " ", RowBox[List["EllipticK", "[", FractionBox["1", "m"], "]"]]]]]], ")"]]]], "+", RowBox[List["2", "u", " ", RowBox[List["EllipticE", "[", "m", "]"]]]], "+", RowBox[List["2", SqrtBox["2"], SqrtBox[RowBox[List[RowBox[List["-", SqrtBox[RowBox[List[RowBox[List["-", "1"]], "+", "m"]]]]], " ", RowBox[List["(", RowBox[List["z", "-", SubscriptBox["z", "0"]]], ")"]]]]], RowBox[List["(", RowBox[List["z", "-", SubscriptBox["z", "0"]]], ")"]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List["Binomial", "[", RowBox[List[RowBox[List["k", "-", RowBox[List["1", "/", "2"]]]], ",", "k"]], "]"]], RowBox[List["3", "+", RowBox[List["2", " ", "k"]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "k"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], " ", RowBox[List["Binomial", "[", RowBox[List["k", ",", "j"]], "]"]]]], RowBox[List["1", "-", RowBox[List["2", "j"]]]]], SubscriptBox["p", RowBox[List["j", ",", "k"]]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", SubscriptBox["z", "0"]]], ")"]], "k"]]]]]]]]]]]]]]], " ", "/;", RowBox[List[RowBox[List[SubscriptBox["z", "0"], "\[Equal]", RowBox[List[RowBox[List["ArcCsc", "[", SqrtBox["m"], "]"]], "+", RowBox[List["\[Pi]", " ", "u"]]]]]], "\[And]", RowBox[List["u", "\[Element]", "Integers"]], "\[And]", RowBox[List[SubscriptBox["a", "0"], "\[Equal]", "1"]], "\[And]", RowBox[List[SubscriptBox["a", RowBox[List["2", "k"]]], "\[Equal]", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", SuperscriptBox["2", RowBox[List["2", "k"]]]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", "k"]], "+", "1"]], ")"]], "!"]]]]], "\[And]", RowBox[List[SubscriptBox["a", RowBox[List[RowBox[List["2", "k"]], "+", "1"]]], "\[Equal]", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["k", "-", "1"]]], SuperscriptBox["2", RowBox[List["2", "k"]]], " ", RowBox[List["(", RowBox[List["2", "-", "m"]], ")"]]]], RowBox[List[SqrtBox[RowBox[List["m", "-", "1"]]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", "k"]], "+", "2"]], ")"]], "!"]]]]]]], "\[And]", RowBox[List["k", "\[Element]", "Integers"]], "\[And]", RowBox[List["k", "\[GreaterEqual]", "0"]], "\[And]", RowBox[List[SubscriptBox["p", RowBox[List["u", ",", "0"]]], "\[Equal]", "1"]], "\[And]", RowBox[List[SubscriptBox["p", RowBox[List["u", ",", "v"]]], "\[Equal]", RowBox[List[FractionBox["1", "v"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], "v"], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["u", " ", "j"]], "-", "v", "+", "j"]], ")"]], SubscriptBox["a", "j"], " ", SubscriptBox["p", RowBox[List["u", ",", RowBox[List["v", "-", "j"]]]]]]]]]]]]]]]]]]]

 MathML Form

 E ( z m ) 2 u E ( m ) + m ( E ( 1 m ) + ( 1 m - 1 ) K ( 1 m ) ) + 2 2 ( z - z 0 ) - m - 1 ( z - z 0 ) k = 0 1 2 k + 3 ( k - 1 2 k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List["k", "-", FractionBox["1", "2"]]], Identity, Rule[Editable, True]]], List[TagBox["k", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] j = 0 k ( - 1 ) j 1 - 2 j ( k j ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["k", Identity, Rule[Editable, True]]], List[TagBox["j", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] p j , k ( z - z 0 ) k /; z 0 csc - 1 ( m ) + π u u TagBox["\[DoubleStruckCapitalZ]", Function[List[], Integers]] a 0 1 a 2 k ( - 1 ) k 2 2 k ( 2 k + 1 ) ! a 2 k + 1 ( - 1 ) k - 1 2 2 k ( 2 - m ) m - 1 ( 2 k + 2 ) ! k p u , 0 1 p u , v 1 v j = 1 v ( u j + j - v ) a j p u , v - j Condition EllipticE z m 2 u EllipticE m m 1 2 EllipticE 1 m -1 1 m -1 -1 EllipticK 1 m -1 2 2 1 2 z -1 Subscript z 0 -1 m -1 1 2 z -1 Subscript z 0 1 2 k 0 1 2 k 3 -1 Binomial k -1 1 2 k j 0 k -1 j 1 -1 2 j -1 Binomial k j Subscript p j k z -1 Subscript z 0 k Subscript z 0 m 1 2 u u Subscript a 0 1 Subscript a 2 k -1 k 2 2 k 2 k 1 -1 Subscript a 2 k 1 -1 k -1 2 2 k 2 -1 m m -1 1 2 2 k 2 -1 k Subscript p u 0 1 Subscript p u v 1 v -1 j 1 v u j j -1 v Subscript a j Subscript p u v -1 j [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["EllipticE", "[", RowBox[List["z_", ",", "m_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[SqrtBox["m"], " ", RowBox[List["(", RowBox[List[RowBox[List["EllipticE", "[", FractionBox["1", "m"], "]"]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", FractionBox["1", "m"]]], ")"]], " ", RowBox[List["EllipticK", "[", FractionBox["1", "m"], "]"]]]]]], ")"]]]], "+", RowBox[List["2", " ", "u", " ", RowBox[List["EllipticE", "[", "m", "]"]]]], "+", RowBox[List["2", " ", SqrtBox["2"], " ", SqrtBox[RowBox[List[RowBox[List["-", SqrtBox[RowBox[List[RowBox[List["-", "1"]], "+", "m"]]]]], " ", RowBox[List["(", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], ")"]]]]], " ", RowBox[List["(", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["Binomial", "[", RowBox[List[RowBox[List["k", "-", FractionBox["1", "2"]]], ",", "k"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "k"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], " ", RowBox[List["Binomial", "[", RowBox[List["k", ",", "j"]], "]"]]]], ")"]], " ", SubscriptBox["p", RowBox[List["j", ",", "k"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], ")"]], "k"]]], RowBox[List["1", "-", RowBox[List["2", " ", "j"]]]]]]]]], RowBox[List["3", "+", RowBox[List["2", " ", "k"]]]]]]]]]]], "/;", RowBox[List[RowBox[List[SubscriptBox["zz", "0"], "\[Equal]", RowBox[List[RowBox[List["ArcCsc", "[", SqrtBox["m"], "]"]], "+", RowBox[List["\[Pi]", " ", "u"]]]]]], "&&", RowBox[List["u", "\[Element]", "Integers"]], "&&", RowBox[List[SubscriptBox["a", "0"], "\[Equal]", "1"]], "&&", RowBox[List[SubscriptBox["a", RowBox[List["2", " ", "k"]]], "\[Equal]", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", SuperscriptBox["2", RowBox[List["2", " ", "k"]]]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]], ")"]], "!"]]]]], "&&", RowBox[List[SubscriptBox["a", RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]]], "\[Equal]", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["k", "-", "1"]]], " ", SuperscriptBox["2", RowBox[List["2", " ", "k"]]], " ", RowBox[List["(", RowBox[List["2", "-", "m"]], ")"]]]], RowBox[List[SqrtBox[RowBox[List["m", "-", "1"]]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "+", "2"]], ")"]], "!"]]]]]]], "&&", RowBox[List["k", "\[Element]", "Integers"]], "&&", RowBox[List["k", "\[GreaterEqual]", "0"]], "&&", RowBox[List[SubscriptBox["p", RowBox[List["u", ",", "0"]]], "\[Equal]", "1"]], "&&", RowBox[List[SubscriptBox["p", RowBox[List["u", ",", "v"]]], "\[Equal]", FractionBox[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], "v"], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["u", " ", "j"]], "-", "v", "+", "j"]], ")"]], " ", SubscriptBox["a", "j"], " ", SubscriptBox["p", RowBox[List["u", ",", RowBox[List["v", "-", "j"]]]]]]]]], "v"]]]]]]]]]]]

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

 2007-05-02