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/08.04.06.0016.01

 Input Form

 EllipticE[z, m] == Exp[(-Pi) I Floor[Arg[x - z]/(2 Pi)]] EllipticE[x, m] + (Sqrt[m] (EllipticE[1/m] + (-1 + 1/m) EllipticK[1/m]) + 2 (Floor[x/Pi - 1/2] + 1) EllipticE[m]) (1 - Exp[(-Pi) I Floor[Arg[x - z]/(2 Pi)]]) + Exp[(-Pi) I Floor[Arg[x - z]/(2 Pi)]] Sum[(1/k!) Sum[(1/j!) Sum[Binomial[j, q] Sum[((-1)^q 2^(q - j) Sin[x]^q (2 p + q - j)^k Binomial[j - q, p] Sum[(Pochhammer[1 - j, 2 (j - i) - 2]/((j - i - 1)! (2 Sin[x])^ (j - 2 i - 1))) Sum[Binomial[i, s] Pochhammer[1/2, s] Pochhammer[-(1/2), i - s] m^(i - s) Cos[x]^(-1 - 2 s) (1 - m Sin[x]^2)^(1/2 - i + s), {s, 0, i}], {i, 0, j - 1}])/ E^((1/2) I ((k - 2 p - q + j) Pi + 2 (2 p + q - j) x)), {p, 0, j - q}], {q, 0, j - 1}], {j, 1, k}] (z - x)^k, {k, 1, Infinity}] /; Element[x, Reals] && Element[m, Reals] && m > 1 && ArcCsc[Sqrt[m]] + Pi u < x < Pi/2 + Pi u && Element[u, Integers]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["EllipticE", "[", RowBox[List["z", ",", "m"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["Exp", "[", RowBox[List[RowBox[List["-", "\[Pi]"]], " ", "\[ImaginaryI]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["x", "-", "z"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]], "]"]], RowBox[List["EllipticE", "[", RowBox[List["x", ",", "m"]], "]"]]]], "+", RowBox[List[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", RowBox[List["(", RowBox[List[RowBox[List["Floor", "[", RowBox[List[FractionBox["x", "\[Pi]"], "-", FractionBox["1", "2"]]], "]"]], "+", "1"]], ")"]], RowBox[List["EllipticE", "[", "m", "]"]]]]]], ")"]], RowBox[List["(", RowBox[List["1", "-", RowBox[List["Exp", "[", RowBox[List[RowBox[List["-", "\[Pi]"]], " ", "\[ImaginaryI]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["x", "-", "z"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]], "]"]]]], ")"]]]], "+", RowBox[List[RowBox[List["Exp", "[", RowBox[List[RowBox[List["-", "\[Pi]"]], " ", "\[ImaginaryI]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["x", "-", "z"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]], "]"]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], RowBox[List[FractionBox["1", RowBox[List["k", "!"]]], RowBox[List["(", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], "k"], RowBox[List[FractionBox["1", RowBox[List["j", "!"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["q", "=", "0"]], RowBox[List["j", "-", "1"]]], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["j", ",", "q"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["p", "=", "0"]], RowBox[List["j", "-", "q"]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "q"], SuperscriptBox["2", RowBox[List["q", "-", "j"]]], " ", SuperscriptBox[RowBox[List["Sin", "[", "x", "]"]], "q"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "p"]], "+", "q", "-", "j"]], ")"]], "k"], SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["k", "-", RowBox[List["2", " ", "p"]], "-", "q", "+", "j"]], ")"]], " ", "\[Pi]"]], "+", RowBox[List["2", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "p"]], "+", "q", "-", "j"]], ")"]], "x"]]]], ")"]]]]], RowBox[List["Binomial", "[", RowBox[List[RowBox[List["j", "-", "q"]], ",", "p"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "0"]], RowBox[List["j", "-", "1"]]], RowBox[List["(", RowBox[List[FractionBox[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "j"]], ",", RowBox[List[RowBox[List["2", " ", RowBox[List["(", RowBox[List["j", "-", "i"]], ")"]]]], "-", "2"]]]], "]"]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["j", "-", "i", "-", "1"]], ")"]], "!"]], SuperscriptBox[RowBox[List["(", RowBox[List["2", " ", RowBox[List["Sin", "[", "x", "]"]]]], ")"]], RowBox[List["j", "-", RowBox[List["2", " ", "i"]], "-", "1"]]]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["s", "=", "0"]], "i"], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["i", ",", "s"]], "]"]], RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "s"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], ",", RowBox[List["i", "-", "s"]]]], "]"]], " ", SuperscriptBox["m", RowBox[List["i", "-", "s"]]], SuperscriptBox[RowBox[List["Cos", "[", "x", "]"]], RowBox[List[RowBox[List["-", "1"]], "-", RowBox[List["2", "s"]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", RowBox[List["m", " ", SuperscriptBox[RowBox[List["Sin", "[", "x", "]"]], "2"]]]]], ")"]], RowBox[List[FractionBox["1", "2"], "-", "i", "+", "s"]]]]]]]]], ")"]]]]]]]]]]]]]]]], ")"]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "x"]], ")"]], "k"]]]]]]]]]]], "/;", RowBox[List[RowBox[List["x", "\[Element]", "Reals"]], "\[And]", RowBox[List["m", "\[Element]", "Reals"]], "\[And]", RowBox[List["m", ">", "1"]], "\[And]", RowBox[List[RowBox[List[RowBox[List["ArcCsc", "[", SqrtBox["m"], "]"]], "+", RowBox[List["\[Pi]", " ", "u"]]]], "<", "x", "<", RowBox[List[FractionBox["\[Pi]", "2"], "+", RowBox[List["\[Pi]", " ", "u"]]]]]], "\[And]", RowBox[List["u", "\[Element]", "Integers"]]]]]]]]

 MathML Form

 E ( z m ) ( 2 ( x π - 1 2 + 1 ) E ( m ) + m ( E ( 1 m ) + ( 1 m - 1 ) K ( 1 m ) ) ) ( 1 - - π arg ( x - z ) 2 π ) + E ( x m ) - π arg ( x - z ) 2 π + - π arg ( x - z ) 2 π k = 1 1 k ! j = 1 k 1 j ! ( q = 0 j - 1 ( j q ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["j", Identity, Rule[Editable, True]]], List[TagBox["q", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] p = 0 j - q ( - 1 ) q 2 q - j sin q ( x ) ( 2 p + q - j ) k - 1 2 ( π ( j + k - 2 p - q ) + 2 ( 2 p + q - j ) x ) ( j - q p ) TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List["j", "-", "q"]], Identity, Rule[Editable, True]]], List[TagBox["p", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] i = 0 j - 1 ( 1 - j ) 2 ( j - i ) - 2 TagBox[SubscriptBox[RowBox[List["(", RowBox[List["1", "-", "j"]], ")"]], RowBox[List[RowBox[List["2", " ", RowBox[List["(", RowBox[List["j", "-", "i"]], ")"]]]], "-", "2"]]], Pochhammer] ( j - i - 1 ) ! ( 2 sin ( x ) ) j - 2 i - 1 s = 0 i ( i s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["i", Identity, Rule[Editable, True]]], List[TagBox["s", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( 1 2 ) s TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], "s"], Pochhammer] ( - 1 2 ) i - s TagBox[SubscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["1", "2"]]], ")"]], RowBox[List["i", "-", "s"]]], Pochhammer] m i - s cos - 2 s - 1 ( x ) ( 1 - m sin 2 ( x ) ) s - i + 1 2 ) ( z - x ) k /; x TagBox["\[DoubleStruckCapitalR]", Function[List[], Reals]] m TagBox["\[DoubleStruckCapitalR]", Function[List[], Reals]] m > 1 π u + csc - 1 ( m ) < x < π u + π 2 u TagBox["\[DoubleStruckCapitalZ]", Function[List[], Integers]] Condition EllipticE z m 2 x -1 -1 1 2 1 EllipticE m m 1 2 EllipticE 1 m -1 1 m -1 -1 EllipticK 1 m -1 1 -1 -1 x -1 z 2 -1 EllipticE x m -1 x -1 z 2 -1 -1 x -1 z 2 -1 k 1 1 k -1 j 1 k 1 j -1 q 0 j -1 Binomial j q p 0 j -1 q -1 q 2 q -1 j x q 2 p q -1 j k -1 1 2 j k -1 2 p -1 q 2 2 p q -1 j x Binomial j -1 q p i 0 j -1 Pochhammer 1 -1 j 2 j -1 i -2 j -1 i -1 2 x j -1 2 i -1 -1 s 0 i Binomial i s Pochhammer 1 2 s Pochhammer -1 1 2 i -1 s m i -1 s x -2 s -1 1 -1 m x 2 s -1 i 1 2 z -1 x k x m m 1 u m 1 2 x u 2 -1 u [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["EllipticE", "[", RowBox[List["z_", ",", "m_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "\[Pi]"]], " ", "\[ImaginaryI]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["x", "-", "z"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], " ", RowBox[List["EllipticE", "[", RowBox[List["x", ",", "m"]], "]"]]]], "+", RowBox[List[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", " ", RowBox[List["(", RowBox[List[RowBox[List["Floor", "[", RowBox[List[FractionBox["x", "\[Pi]"], "-", FractionBox["1", "2"]]], "]"]], "+", "1"]], ")"]], " ", RowBox[List["EllipticE", "[", "m", "]"]]]]]], ")"]], " ", RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "\[Pi]"]], " ", "\[ImaginaryI]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["x", "-", "z"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]]]], ")"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "\[Pi]"]], " ", "\[ImaginaryI]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["x", "-", "z"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], "k"], FractionBox[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["q", "=", "0"]], RowBox[List["j", "-", "1"]]], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["j", ",", "q"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["p", "=", "0"]], RowBox[List["j", "-", "q"]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "q"], " ", SuperscriptBox["2", RowBox[List["q", "-", "j"]]], " ", SuperscriptBox[RowBox[List["Sin", "[", "x", "]"]], "q"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "p"]], "+", "q", "-", "j"]], ")"]], "k"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["k", "-", RowBox[List["2", " ", "p"]], "-", "q", "+", "j"]], ")"]], " ", "\[Pi]"]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "p"]], "+", "q", "-", "j"]], ")"]], " ", "x"]]]], ")"]]]]], " ", RowBox[List["Binomial", "[", RowBox[List[RowBox[List["j", "-", "q"]], ",", "p"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "0"]], RowBox[List["j", "-", "1"]]], FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "j"]], ",", RowBox[List[RowBox[List["2", " ", RowBox[List["(", RowBox[List["j", "-", "i"]], ")"]]]], "-", "2"]]]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["s", "=", "0"]], "i"], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["i", ",", "s"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "s"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], ",", RowBox[List["i", "-", "s"]]]], "]"]], " ", SuperscriptBox["m", RowBox[List["i", "-", "s"]]], " ", SuperscriptBox[RowBox[List["Cos", "[", "x", "]"]], RowBox[List[RowBox[List["-", "1"]], "-", RowBox[List["2", " ", "s"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", RowBox[List["m", " ", SuperscriptBox[RowBox[List["Sin", "[", "x", "]"]], "2"]]]]], ")"]], RowBox[List[FractionBox["1", "2"], "-", "i", "+", "s"]]]]]]]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["j", "-", "i", "-", "1"]], ")"]], "!"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["2", " ", RowBox[List["Sin", "[", "x", "]"]]]], ")"]], RowBox[List["j", "-", RowBox[List["2", " ", "i"]], "-", "1"]]]]]]]]]]]]]]]], RowBox[List["j", "!"]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "x"]], ")"]], "k"]]], RowBox[List["k", "!"]]]]]]]]], "/;", RowBox[List[RowBox[List["x", "\[Element]", "Reals"]], "&&", RowBox[List["m", "\[Element]", "Reals"]], "&&", RowBox[List["m", ">", "1"]], "&&", RowBox[List[RowBox[List[RowBox[List["ArcCsc", "[", SqrtBox["m"], "]"]], "+", RowBox[List["\[Pi]", " ", "u"]]]], "<", "x", "<", RowBox[List[FractionBox["\[Pi]", "2"], "+", RowBox[List["\[Pi]", " ", "u"]]]]]], "&&", RowBox[List["u", "\[Element]", "Integers"]]]]]]]]]]

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

 2007-05-02

© 1998-2013 Wolfram Research, Inc.