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/08.01.06.0031.01

 Input Form

 EllipticE[z] == Sqrt[-z] + (Log[-z]/Sqrt[-z]) Sum[((1/(k! (1 + k)!)) Pochhammer[-(1/2), 1 + k]^2)/z^k, {k, 0, Infinity}] + (1/Sqrt[-z]) (1/4 - (3/(64 z)) Sum[KroneckerDelta[i - 1], {i, 0, Infinity}] + Log[2] Sum[((1/(k! (1 + k)!)) Pochhammer[1/2, k]^2)/z^k, {k, 0, Infinity}] + (9/64) Sum[(1/((k + 2)!^2 (k + 3))) Pochhammer[5/2, k]^2 (1/(3 + k) + 2/(k + 1) + 2/(k + 2) - 4/(2 k + 3) - 4 Sum[1/i, {i, k + 1, 2 k + 1}]) z^(-k - 2), {k, 0, Infinity}]) /; Abs[z] > 1

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["EllipticE", "[", "z", "]"]], "\[Equal]", RowBox[List[SqrtBox[RowBox[List["-", "z"]]], "+", RowBox[List[FractionBox[RowBox[List["Log", "[", RowBox[List["-", "z"]], "]"]], RowBox[List[" ", SqrtBox[RowBox[List["-", "z"]]]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox["1", RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["1", "+", "k"]], ")"]], "!"]]]]], SuperscriptBox[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], ",", RowBox[List["1", "+", "k"]]]], "]"]], "2"], SuperscriptBox["z", RowBox[List["-", "k"]]]]]]]]], "+", RowBox[List[FractionBox["1", RowBox[List[" ", SqrtBox[RowBox[List["-", "z"]]]]]], RowBox[List["(", RowBox[List[FractionBox["1", "4"], "-", RowBox[List[FractionBox["3", RowBox[List["64", " ", "z"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "0"]], "\[Infinity]"], RowBox[List["KroneckerDelta", "[", RowBox[List["i", "-", "1"]], "]"]]]]]], "+", RowBox[List[RowBox[List["Log", "[", "2", "]"]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox["1", RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["1", "+", "k"]], ")"]], "!"]]]]], SuperscriptBox[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "k"]], "]"]], "2"], SuperscriptBox["z", RowBox[List["-", "k"]]]]]]]]], "+", RowBox[List[FractionBox["9", "64"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox["1", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["k", "+", "2"]], ")"]], "!"]], ")"]], "2"], " ", RowBox[List["(", RowBox[List["k", "+", "3"]], ")"]]]]], SuperscriptBox[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["5", "2"], ",", "k"]], "]"]], "2"], " ", RowBox[List["(", RowBox[List[FractionBox["1", RowBox[List["3", "+", "k"]]], "+", FractionBox["2", RowBox[List["k", "+", "1"]]], "+", FractionBox["2", RowBox[List["k", "+", "2"]]], "-", FractionBox["4", RowBox[List[RowBox[List["2", "k"]], "+", "3"]]], "-", RowBox[List["4", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", RowBox[List["k", "+", "1"]]]], RowBox[List[RowBox[List["2", "k"]], "+", "1"]]], FractionBox["1", "i"]]]]]]], ")"]], SuperscriptBox["z", RowBox[List[RowBox[List["-", "k"]], "-", "2"]]]]]]]]]]], ")"]]]]]]]], "/;", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], ">", "1"]]]]]]

 MathML Form

 E ( z ) - z + log ( - z ) - z k = 0 ( - 1 2 ) k + 1 TagBox[SubscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["1", "2"]]], ")"]], RowBox[List["k", "+", "1"]]], Pochhammer] 2 z - k k ! ( k + 1 ) ! + 1 - z ( log ( 2 ) k = 0 ( 1 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], "k"], Pochhammer] 2 z - k k ! ( k + 1 ) ! + 1 4 - 3 64 z i = 0 δ KroneckerDelta i - 1 + 9 64 k = 0 ( 5 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["5", "2"], ")"]], "k"], Pochhammer] 2 ( ( k + 2 ) ! ) 2 ( k + 3 ) ( 2 k + 1 + 2 k + 2 - 4 2 k + 3 + 1 k + 3 - 4 i = k + 1 2 k + 1 1 i ) z - k - 2 ) /; "\[LeftBracketingBar]" z "\[RightBracketingBar]" > 1 Condition EllipticE z -1 z 1 2 -1 z -1 z 1 2 -1 k 0 Pochhammer -1 1 2 k 1 2 z -1 k k k 1 -1 1 -1 z 1 2 -1 2 k 0 Pochhammer 1 2 k 2 z -1 k k k 1 -1 1 4 -1 3 64 z -1 i 0 KroneckerDelta i -1 9 64 k 0 Pochhammer 5 2 k 2 k 2 2 k 3 -1 2 k 1 -1 2 k 2 -1 -1 4 2 k 3 -1 1 k 3 -1 -1 4 i k 1 2 k 1 1 i -1 z -1 k -2 z 1 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["EllipticE", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SqrtBox[RowBox[List["-", "z"]]], "+", FractionBox[RowBox[List[RowBox[List["Log", "[", RowBox[List["-", "z"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], ",", RowBox[List["1", "+", "k"]]]], "]"]], "2"], " ", SuperscriptBox["z", RowBox[List["-", "k"]]]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["1", "+", "k"]], ")"]], "!"]]]]]]]]], SqrtBox[RowBox[List["-", "z"]]]], "+", FractionBox[RowBox[List[FractionBox["1", "4"], "-", FractionBox[RowBox[List["3", " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "0"]], "\[Infinity]"], RowBox[List["KroneckerDelta", "[", RowBox[List["i", "-", "1"]], "]"]]]]]], RowBox[List["64", " ", "z"]]], "+", RowBox[List[RowBox[List["Log", "[", "2", "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "k"]], "]"]], "2"], " ", SuperscriptBox["z", RowBox[List["-", "k"]]]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["1", "+", "k"]], ")"]], "!"]]]]]]]]], "+", RowBox[List[FractionBox["9", "64"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["5", "2"], ",", "k"]], "]"]], "2"], " ", RowBox[List["(", RowBox[List[FractionBox["1", RowBox[List["3", "+", "k"]]], "+", FractionBox["2", RowBox[List["k", "+", "1"]]], "+", FractionBox["2", RowBox[List["k", "+", "2"]]], "-", FractionBox["4", RowBox[List[RowBox[List["2", " ", "k"]], "+", "3"]]], "-", RowBox[List["4", " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", RowBox[List["k", "+", "1"]]]], RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]]], FractionBox["1", "i"]]]]]]], ")"]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "k"]], "-", "2"]]]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["k", "+", "2"]], ")"]], "!"]], ")"]], "2"], " ", RowBox[List["(", RowBox[List["k", "+", "3"]], ")"]]]]]]]]]]], SqrtBox[RowBox[List["-", "z"]]]]]], "/;", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], ">", "1"]]]]]]]]

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

 2007-05-02