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/08.04.06.0059.01

 Input Form

 EllipticE[z, m] == (-1)^Round[Re[z]/Pi] Sin[z] (1 + Sum[(Pochhammer[-(1/2), k]/(k! (2 k - Sin[z]^2 - 1))) (Cos[z]^2 Tan[z]^(2 k) + ((2 (-1)^k Pochhammer[1/2, k])/(-1 + k)!) (1 - ArcTanh[Sin[z]] Csc[z] (2 k - Sin[z]^2 - 1) + Csc[z]^2 Sum[(((-1)^j (j - 1)! (j - k + Sin[z]^2))/Pochhammer[1/ 2, j]) Tan[z]^(2 j), {j, 1, k - 1}])) (m - 1)^k, {k, 1, Infinity}]) + 2 Round[Re[z]/Pi] (1 + Log[1 - m] ((m - 1)/4) Sum[((Pochhammer[1/2, k] Pochhammer[3/2, k])/ (k! (1 + k)!)) (1 - m)^k, {k, 0, Infinity}] + ((m - 1)/4) Sum[((Pochhammer[1/2, k] Pochhammer[3/2, k])/(k! (1 + k)!)) (1/(1 + 3 k + 2 k^2) + 2 PolyGamma[1/2 + k] - 2 PolyGamma[1 + k]) (1 - m)^k, {k, 0, Infinity}]) /; !Element[(Pi + 2 Re[z])/(4 Pi), Integers]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["EllipticE", "[", RowBox[List["z", ",", "m"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["Round", "[", FractionBox[RowBox[List["Re", "[", "z", "]"]], "\[Pi]"], "]"]]], RowBox[List["Sin", "[", "z", "]"]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[" ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], ",", "k"]], "]"]], " "]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"], "-", "1"]], ")"]]]]], RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["Cos", "[", "z", "]"]], "2"], " ", SuperscriptBox[RowBox[List["Tan", "[", "z", "]"]], RowBox[List["2", " ", "k"]]]]], "+", RowBox[List[FractionBox[RowBox[List["2", " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "k"]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "k"]], ")"]], "!"]]], RowBox[List["(", RowBox[List["1", "-", RowBox[List[RowBox[List["ArcTanh", "[", RowBox[List["Sin", "[", "z", "]"]], "]"]], " ", RowBox[List["Csc", "[", "z", "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"], "-", "1"]], ")"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["Csc", "[", "z", "]"]], "2"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], RowBox[List["k", "-", "1"]]], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], " ", RowBox[List[RowBox[List["(", RowBox[List["j", "-", "1"]], ")"]], "!"]], " ", RowBox[List["(", RowBox[List["j", "-", "k", "+", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]], ")"]], " "]], RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "j"]], "]"]]], SuperscriptBox[RowBox[List["Tan", "[", "z", "]"]], RowBox[List["2", " ", "j"]]]]]]]]]]], ")"]]]]]], ")"]], SuperscriptBox[RowBox[List["(", RowBox[List["m", "-", "1"]], ")"]], "k"]]]]]]], ")"]]]], "+", RowBox[List["2", " ", RowBox[List["Round", "[", FractionBox[RowBox[List["Re", "[", "z", "]"]], "\[Pi]"], "]"]], RowBox[List["(", RowBox[List["1", "+", RowBox[List[RowBox[List["Log", "[", RowBox[List["1", "-", "m"]], "]"]], FractionBox[RowBox[List["m", "-", "1"]], "4"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["3", "2"], ",", "k"]], "]"]]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["1", "+", "k"]], ")"]], "!"]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "m"]], ")"]], "k"]]]]]]], "+", RowBox[List[FractionBox[RowBox[List["m", "-", "1"]], "4"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["3", "2"], ",", "k"]], "]"]]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["1", "+", "k"]], ")"]], "!"]]]]], RowBox[List["(", RowBox[List[FractionBox["1", RowBox[List["1", "+", RowBox[List["3", " ", "k"]], "+", RowBox[List["2", " ", SuperscriptBox["k", "2"]]]]]], "+", RowBox[List["2", " ", RowBox[List["PolyGamma", "[", RowBox[List[FractionBox["1", "2"], "+", "k"]], "]"]]]], "-", RowBox[List["2", " ", RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "k"]], "]"]]]]]], ")"]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "m"]], ")"]], "k"]]]]]]]]], ")"]]]]]]]], "/;", RowBox[List["Not", "[", RowBox[List[FractionBox[RowBox[List["\[Pi]", "+", RowBox[List["2", " ", RowBox[List["Re", "[", "z", "]"]]]]]], RowBox[List["4", " ", "\[Pi]"]]], "\[Element]", "Integers"]], "]"]]]]]]

 MathML Form

 E ( z m ) ( - 1 ) Re ( z ) π sin ( z ) ( 1 + k = 1 ( - 1 2 ) k k ! ( 2 k - sin 2 ( z ) - 1 ) ( cos 2 ( z ) tan 2 k ( z ) + 2 ( - 1 ) k ( 1 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], "k"], Pochhammer] ( k - 1 ) ! ( csc 2 ( z ) j = 1 k - 1 ( - 1 ) j ( j - 1 ) ! ( sin 2 ( z ) + j - k ) ( 1 2 ) j TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], "j"], Pochhammer] tan 2 j ( z ) - tanh - 1 ( sin ( z ) ) csc ( z ) ( 2 k - sin 2 ( z ) - 1 ) + 1 ) ) ( m - 1 ) k ) + 2 Re ( z ) π ( 1 + m - 1 4 log ( 1 - m ) k = 0 ( 1 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], "k"], Pochhammer] ( 3 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["3", "2"], ")"]], "k"], Pochhammer] k ! ( k + 1 ) ! ( 1 - m ) k + 1 4 ( m - 1 ) k = 0 ( 1 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], "k"], Pochhammer] ( 3 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["3", "2"], ")"]], "k"], Pochhammer] k ! ( k + 1 ) ! ( - 2 ψ TagBox["\[Psi]", PolyGamma] ( k + 1 ) + 2 ψ TagBox["\[Psi]", PolyGamma] ( k + 1 2 ) + 1 2 k 2 + 3 k + 1 ) ( 1 - m ) k ) /; ¬ 2 Re ( z ) + π 4 π TagBox["\[DoubleStruckCapitalZ]", Function[List[], Integers]] FormBox RowBox RowBox RowBox E ( RowBox z m ) RowBox RowBox SuperscriptBox RowBox ( RowBox - 1 ) ErrorBox RowBox FractionBox RowBox Re ( z ) π RowBox sin ( z ) RowBox ( RowBox 1 + RowBox UnderoverscriptBox RowBox k = 1 RowBox FractionBox SubscriptBox RowBox ( RowBox - FractionBox 1 2 ) k RowBox RowBox k ! RowBox ( RowBox RowBox 2 k - RowBox SuperscriptBox sin 2 ( z ) - 1 ) RowBox ( RowBox RowBox RowBox SuperscriptBox cos 2 ( z ) RowBox SuperscriptBox tan RowBox 2 k ( z ) + RowBox FractionBox RowBox 2 SuperscriptBox RowBox ( RowBox - 1 ) k TagBox SubscriptBox RowBox ( FractionBox 1 2 ) k Pochhammer RowBox RowBox ( RowBox k - 1 ) ! RowBox ( RowBox RowBox RowBox SuperscriptBox csc 2 ( z ) RowBox UnderoverscriptBox RowBox j = 1 RowBox k - 1 RowBox FractionBox RowBox SuperscriptBox RowBox ( RowBox - 1 ) j RowBox RowBox ( RowBox j - 1 ) ! RowBox ( RowBox RowBox SuperscriptBox sin 2 ( z ) + j - k ) TagBox SubscriptBox RowBox ( FractionBox 1 2 ) j Pochhammer RowBox SuperscriptBox tan RowBox 2 j ( z ) - RowBox RowBox SuperscriptBox tanh RowBox - 1 ( RowBox sin ( z ) ) RowBox csc ( z ) RowBox ( RowBox RowBox 2 k - RowBox SuperscriptBox sin 2 ( z ) - 1 ) + 1 ) ) SuperscriptBox RowBox ( RowBox m - 1 ) k ) + RowBox 2 RowBox FractionBox RowBox Re ( z ) π RowBox ( RowBox 1 + RowBox FractionBox RowBox m - 1 4 RowBox log ( RowBox 1 - m ) RowBox UnderoverscriptBox RowBox k = 0 RowBox FractionBox RowBox TagBox SubscriptBox RowBox ( FractionBox 1 2 ) k Pochhammer TagBox SubscriptBox RowBox ( FractionBox 3 2 ) k Pochhammer RowBox RowBox k ! RowBox RowBox ( RowBox k + 1 ) ! SuperscriptBox RowBox ( RowBox 1 - m ) k + RowBox FractionBox 1 4 RowBox ( RowBox m - 1 ) RowBox UnderoverscriptBox RowBox k = 0 RowBox FractionBox RowBox TagBox SubscriptBox RowBox ( FractionBox 1 2 ) k Pochhammer TagBox SubscriptBox RowBox ( FractionBox 3 2 ) k Pochhammer RowBox RowBox k ! RowBox RowBox ( RowBox k + 1 ) ! RowBox ( RowBox RowBox RowBox - 2 RowBox TagBox ψ PolyGamma ( RowBox k + 1 ) + RowBox 2 RowBox TagBox ψ PolyGamma ( RowBox k + FractionBox 1 2 ) + FractionBox 1 RowBox RowBox 2 SuperscriptBox k 2 + RowBox 3 k + 1 ) SuperscriptBox RowBox ( RowBox 1 - m ) k ) /; RowBox ¬ RowBox FractionBox RowBox RowBox 2 RowBox Re ( z ) + π RowBox 4 π TagBox Function TraditionalForm [/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[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["Round", "[", FractionBox[RowBox[List["Re", "[", "z", "]"]], "\[Pi]"], "]"]]], " ", RowBox[List["Sin", "[", "z", "]"]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], ",", "k"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["Cos", "[", "z", "]"]], "2"], " ", SuperscriptBox[RowBox[List["Tan", "[", "z", "]"]], RowBox[List["2", " ", "k"]]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["2", " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "k"]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List[RowBox[List["ArcTanh", "[", RowBox[List["Sin", "[", "z", "]"]], "]"]], " ", RowBox[List["Csc", "[", "z", "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"], "-", "1"]], ")"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["Csc", "[", "z", "]"]], "2"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], RowBox[List["k", "-", "1"]]], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], " ", RowBox[List[RowBox[List["(", RowBox[List["j", "-", "1"]], ")"]], "!"]], " ", RowBox[List["(", RowBox[List["j", "-", "k", "+", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]], ")"]]]], ")"]], " ", SuperscriptBox[RowBox[List["Tan", "[", "z", "]"]], RowBox[List["2", " ", "j"]]]]], RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "j"]], "]"]]]]]]]]], ")"]]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "k"]], ")"]], "!"]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["m", "-", "1"]], ")"]], "k"]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"], "-", "1"]], ")"]]]]]]]]], ")"]]]], "+", RowBox[List["2", " ", RowBox[List["Round", "[", FractionBox[RowBox[List["Re", "[", "z", "]"]], "\[Pi]"], "]"]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["Log", "[", RowBox[List["1", "-", "m"]], "]"]], " ", RowBox[List["(", RowBox[List["m", "-", "1"]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["3", "2"], ",", "k"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "m"]], ")"]], "k"]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["1", "+", "k"]], ")"]], "!"]]]]]]]]], "+", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["m", "-", "1"]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["3", "2"], ",", "k"]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[FractionBox["1", RowBox[List["1", "+", RowBox[List["3", " ", "k"]], "+", RowBox[List["2", " ", SuperscriptBox["k", "2"]]]]]], "+", RowBox[List["2", " ", RowBox[List["PolyGamma", "[", RowBox[List[FractionBox["1", "2"], "+", "k"]], "]"]]]], "-", RowBox[List["2", " ", RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "k"]], "]"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "m"]], ")"]], "k"]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["1", "+", "k"]], ")"]], "!"]]]]]]]]]]], ")"]]]]]], "/;", RowBox[List["!", RowBox[List[FractionBox[RowBox[List["\[Pi]", "+", RowBox[List["2", " ", RowBox[List["Re", "[", "z", "]"]]]]]], RowBox[List["4", " ", "\[Pi]"]]], "\[Element]", "Integers"]]]]]]]]]]

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

 2007-05-02