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/08.04.06.0041.01

 Input Form

 EllipticE[z, m] == (-EllipticE[ArcCsc[Sqrt[m]], m]) (-(-1)^Floor[Arg[z - Subscript[z, 0]]/Pi] + (-1)^Floor[1/2 - Arg[z - Subscript[z, 0]]/Pi]) + EllipticE[m] (-(-1)^Floor[Arg[z - Subscript[z, 0]]/Pi] + (-1)^Floor[1/2 - Arg[z - Subscript[z, 0]]/Pi] - (-1)^(Floor[3/4 - Arg[z - Subscript[z, 0]]/(2 Pi)] + Floor[1/4 + Arg[z - Subscript[z, 0]]/(2 Pi)]) + 2 Round[Re[Subscript[z, 0]]/Pi]) + Sqrt[1 - m] (z - Subscript[z, 0]) + Sqrt[1 - m] Sum[((I^(k - 1) Binomial[k - 3/2, k - 1])/k!) Sum[((((-1)^q Binomial[k - 1, q])/(1 - 2 q)) Sum[Binomial[q, j] (-m)^j (2 - m)^(q - j) 2^(k - j - q - 1) Sum[Binomial[j, i] (2 i - j)^(k - 1) (z - Subscript[z, 0])^k, {i, 0, j}], {j, 0, q}])/(1 - m)^q, {q, 1, k - 1}], {k, 2, Infinity}] /; (z -> Subscript[z, 0]) && Subscript[z, 0] == (3 Pi)/2 + 2 Pi u && Element[u, Integers] && Element[m, Reals] && m > 1

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["EllipticE", "[", RowBox[List["z", ",", "m"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["-", RowBox[List["EllipticE", "[", RowBox[List[RowBox[List["ArcCsc", "[", SqrtBox["m"], "]"]], ",", "m"]], "]"]]]], RowBox[List["(", RowBox[List[RowBox[List["-", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", SubscriptBox["z", "0"]]], "]"]], "\[Pi]"], "]"]]]]], "+", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], "-", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", SubscriptBox["z", "0"]]], "]"]], "\[Pi]"]]], "]"]]]]], ")"]]]], " ", "+", RowBox[List[RowBox[List["EllipticE", "[", "m", "]"]], RowBox[List["(", RowBox[List[RowBox[List["-", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", SubscriptBox["z", "0"]]], "]"]], "\[Pi]"], "]"]]]]], "+", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], "-", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", SubscriptBox["z", "0"]]], "]"]], "\[Pi]"]]], "]"]]], "-", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["Floor", "[", RowBox[List[FractionBox["3", "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["2", " ", RowBox[List["Round", "[", FractionBox[RowBox[List["Re", "[", SubscriptBox["z", "0"], "]"]], "\[Pi]"], "]"]]]]]], ")"]]]], "+", RowBox[List[SqrtBox[RowBox[List["1", "-", "m"]]], RowBox[List["(", RowBox[List["z", "-", SubscriptBox["z", "0"]]], ")"]]]], "+", RowBox[List[SqrtBox[RowBox[List["1", "-", "m"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "2"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox["\[ImaginaryI]", RowBox[List["k", "-", "1"]]], RowBox[List["Binomial", "[", RowBox[List[RowBox[List["k", "-", RowBox[List["3", "/", "2"]]]], ",", RowBox[List["k", "-", "1"]]]], "]"]], " "]], RowBox[List["k", "!"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["q", "=", "1"]], RowBox[List["k", "-", "1"]]], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "q"], " ", RowBox[List["Binomial", "[", RowBox[List[RowBox[List["k", "-", "1"]], ",", "q"]], "]"]]]], RowBox[List["1", "-", RowBox[List["2", "q"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "m"]], ")"]], RowBox[List["-", "q"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "q"], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["q", ",", "j"]], "]"]], SuperscriptBox[RowBox[List["(", RowBox[List["-", "m"]], ")"]], "j"], SuperscriptBox[RowBox[List["(", RowBox[List["2", "-", "m"]], " ", ")"]], RowBox[List["q", "-", "j"]]], SuperscriptBox["2", RowBox[List["k", "-", "j", "-", "q", "-", "1"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "0"]], "j"], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["j", ",", "i"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "i"]], "-", "j"]], ")"]], RowBox[List["k", "-", "1"]]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", SubscriptBox["z", "0"]]], ")"]], "k"]]]]]]]]]]]]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List["z", "\[Rule]", SubscriptBox["z", "0"]]], ")"]], "\[And]", RowBox[List[SubscriptBox["z", "0"], "\[Equal]", RowBox[List[FractionBox[RowBox[List["3", "\[Pi]"]], "2"], "+", RowBox[List["2", "\[Pi]", " ", "u"]]]]]], "\[And]", RowBox[List["u", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", "\[Element]", "Reals"]], "\[And]", RowBox[List["m", ">", "1"]]]]]]]]

 MathML Form

 E ( z m ) ( 2 Re ( z 0 ) π + ( - 1 ) 1 2 - arg ( z - z 0 ) π - ( - 1 ) arg ( z - z 0 ) π - ( - 1 ) arg ( z - z 0 ) 2 π + 1 4 + 3 4 - arg ( z - z 0 ) 2 π ) E ( m ) - ( - ( - 1 ) arg ( z - z 0 ) π + ( - 1 ) 1 2 - arg ( z - z 0 ) π ) E ( csc - 1 ( m ) m ) + 1 - m ( z - z 0 ) + 1 - m k = 2 k - 1 k ! ( k - 3 2 k - 1 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List["k", "-", FractionBox["3", "2"]]], Identity, Rule[Editable, True]]], List[TagBox[RowBox[List["k", "-", "1"]], Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] q = 1 k - 1 ( - 1 ) q 1 - 2 q ( k - 1 q ) TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List["k", "-", "1"]], Identity, Rule[Editable, True]]], List[TagBox["q", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( 1 - m ) - q j = 0 q ( q j ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["q", Identity, Rule[Editable, True]]], List[TagBox["j", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( - m ) j ( 2 - m ) q - j 2 k - j - q - 1 i = 0 j ( j i ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["j", Identity, Rule[Editable, True]]], List[TagBox["i", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( 2 i - j ) k - 1 ( z - z 0 ) k /; ( z "\[Rule]" z 0 ) z 0 3 π 2 + 2 π u u TagBox["\[DoubleStruckCapitalZ]", Function[List[], Integers]] m TagBox["\[DoubleStruckCapitalR]", Function[List[], Reals]] m > 1 E ( z m ) ( 2 Re ( z 0 ) π + ( - 1 ) 1 2 - arg ( z - z 0 ) π - ( - 1 ) arg ( z - z 0 ) π - ( - 1 ) arg ( z - z 0 ) 2 π + 1 4 + 3 4 - arg ( z - z 0 ) 2 π ) E ( m ) - ( - ( - 1 ) arg ( z - z 0 ) π + ( - 1 ) 1 2 - arg ( z - z 0 ) π ) E ( csc - 1 ( m ) m ) + 1 - m ( z - z 0 ) + 1 - m k = 2 k - 1 k ! ( k - 3 2 k - 1 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List["k", "-", FractionBox["3", "2"]]], Identity, Rule[Editable, True]]], List[TagBox[RowBox[List["k", "-", "1"]], Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] q = 1 k - 1 ( - 1 ) q 1 - 2 q ( k - 1 q ) TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List["k", "-", "1"]], Identity, Rule[Editable, True]]], List[TagBox["q", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( 1 - m ) - q j = 0 q ( q j ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["q", Identity, Rule[Editable, True]]], List[TagBox["j", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( - m ) j ( 2 - m ) q - j 2 k - j - q - 1 i = 0 j ( j i ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["j", Identity, Rule[Editable, True]]], List[TagBox["i", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( 2 i - j ) k - 1 ( z - z 0 ) k /; ( z "\[Rule]" z 0 ) z 0 3 π 2 + 2 π u u TagBox["\[DoubleStruckCapitalZ]", Function[List[], Integers]] m TagBox["\[DoubleStruckCapitalR]", Function[List[], Reals]] m > 1 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["EllipticE", "[", RowBox[List["z_", ",", "m_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[RowBox[List["-", RowBox[List["EllipticE", "[", RowBox[List[RowBox[List["ArcCsc", "[", SqrtBox["m"], "]"]], ",", "m"]], "]"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], "]"]], "\[Pi]"], "]"]]]]], "+", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], "-", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], "]"]], "\[Pi]"]]], "]"]]]]], ")"]]]], "+", RowBox[List[RowBox[List["EllipticE", "[", "m", "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], "]"]], "\[Pi]"], "]"]]]]], "+", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], "-", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], "]"]], "\[Pi]"]]], "]"]]], "-", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["Floor", "[", RowBox[List[FractionBox["3", "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["2", " ", RowBox[List["Round", "[", FractionBox[RowBox[List["Re", "[", SubscriptBox["zz", "0"], "]"]], "\[Pi]"], "]"]]]]]], ")"]]]], "+", RowBox[List[SqrtBox[RowBox[List["1", "-", "m"]]], " ", RowBox[List["(", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], ")"]]]], "+", RowBox[List[SqrtBox[RowBox[List["1", "-", "m"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "2"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["\[ImaginaryI]", RowBox[List["k", "-", "1"]]], " ", RowBox[List["Binomial", "[", RowBox[List[RowBox[List["k", "-", FractionBox["3", "2"]]], ",", RowBox[List["k", "-", "1"]]]], "]"]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["q", "=", "1"]], RowBox[List["k", "-", "1"]]], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "q"], " ", RowBox[List["Binomial", "[", RowBox[List[RowBox[List["k", "-", "1"]], ",", "q"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "m"]], ")"]], RowBox[List["-", "q"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "q"], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["q", ",", "j"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "m"]], ")"]], "j"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["2", "-", "m"]], ")"]], RowBox[List["q", "-", "j"]]], " ", SuperscriptBox["2", RowBox[List["k", "-", "j", "-", "q", "-", "1"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "0"]], "j"], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["j", ",", "i"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "i"]], "-", "j"]], ")"]], RowBox[List["k", "-", "1"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], ")"]], "k"]]]]]]]]]]], RowBox[List["1", "-", RowBox[List["2", " ", "q"]]]]]]]]], RowBox[List["k", "!"]]]]]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List["z", "\[Rule]", SubscriptBox["zz", "0"]]], ")"]], "&&", RowBox[List[SubscriptBox["zz", "0"], "\[Equal]", RowBox[List[FractionBox[RowBox[List["3", " ", "\[Pi]"]], "2"], "+", RowBox[List["2", " ", "\[Pi]", " ", "u"]]]]]], "&&", RowBox[List["u", "\[Element]", "Integers"]], "&&", RowBox[List["m", "\[Element]", "Reals"]], "&&", RowBox[List["m", ">", "1"]]]]]]]]]]

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

 2007-05-02