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/08.01.06.0027.01

 Input Form

 EllipticE[z] == 1 + Log[1 - z] ((z - 1)/4) Sum[((Pochhammer[1/2, k] Pochhammer[3/2, k])/(k! (1 + k)!)) (1 - z)^k, {k, 0, Infinity}] + ((z - 1)/4) ((1/16) (29 - 13 z) - 4 Log[2] Sum[((Pochhammer[1/2, k] Pochhammer[3/2, k])/(k! (1 + k)!)) (1 - z)^k, {k, 0, Infinity}]) + (45/64) (z - 1) Sum[((Pochhammer[5/2, k] Pochhammer[7/2, k])/((3 + k) (k + 2)!^2)) (-(2/(1 + k)) - 2/(k + 2) - 1/(3 + k) + 4/(3 + 2 k) + 2/(5 + 2 k) + 4 Sum[1/i, {i, k + 1, 2 k + 1}]) (1 - z)^(k + 2), {k, 0, Infinity}] /; Abs[z - 1] < 1

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["EllipticE", "[", "z", "]"]], "\[Equal]", RowBox[List["1", "+", RowBox[List[RowBox[List["Log", "[", RowBox[List["1", "-", "z"]], "]"]], FractionBox[RowBox[List["z", "-", "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", "-", "z"]], ")"]], "k"]]]]]]], "+", " ", RowBox[List[FractionBox[RowBox[List["z", "-", "1"]], "4"], " ", RowBox[List["(", RowBox[List[RowBox[List[FractionBox["1", "16"], " ", RowBox[List["(", RowBox[List["29", "-", RowBox[List["13", " ", "z"]]]], ")"]]]], "-", RowBox[List["4", RowBox[List["Log", "[", "2", "]"]], 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", "-", "z"]], ")"]], "k"]]]]]]]]], ")"]]]], "+", RowBox[List[FractionBox["45", "64"], RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["5", "2"], ",", "k"]], "]"]], RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["7", "2"], ",", "k"]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List["3", "+", "k"]], ")"]], SuperscriptBox[RowBox[List[RowBox[List["(", RowBox[List["k", "+", "2"]], ")"]], "!"]], "2"]]]], RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox["2", RowBox[List["1", "+", "k"]]]]], "-", FractionBox["2", RowBox[List["k", "+", "2"]]], "-", FractionBox["1", RowBox[List["3", "+", "k"]]], "+", FractionBox["4", RowBox[List["3", "+", RowBox[List["2", " ", "k"]]]]], "+", FractionBox["2", RowBox[List["5", "+", RowBox[List["2", " ", "k"]]]]], "+", 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[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["k", "+", "2"]]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["Abs", "[", RowBox[List["z", "-", "1"]], "]"]], "<", "1"]]]]]]

 MathML Form

 E ( z ) 1 + 1 4 ( z - 1 ) log ( 1 - z ) 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 - z ) k + z - 1 4 ( 29 - 13 z 16 - 4 log ( 2 ) 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 - z ) k ) + 45 64 ( z - 1 ) k = 0 ( 5 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["5", "2"], ")"]], "k"], Pochhammer] ( 7 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["7", "2"], ")"]], "k"], Pochhammer] ( k + 3 ) ( ( k + 2 ) ! ) 2 ( 4 i = k + 1 2 k + 1 1 i - 2 k + 2 - 2 k + 1 - 1 k + 3 + 4 2 k + 3 + 2 2 k + 5 ) ( 1 - z ) k + 2 /; "\[LeftBracketingBar]" z - 1 "\[RightBracketingBar]" < 1 Condition EllipticE z 1 1 4 z -1 1 -1 z k 0 Pochhammer 1 2 k Pochhammer 3 2 k k k 1 -1 1 -1 z k z -1 4 -1 29 -1 13 z 16 -1 -1 4 2 k 0 Pochhammer 1 2 k Pochhammer 3 2 k k k 1 -1 1 -1 z k 45 64 z -1 k 0 Pochhammer 5 2 k Pochhammer 7 2 k k 3 k 2 2 -1 4 i k 1 2 k 1 1 i -1 -1 2 k 2 -1 -1 2 k 1 -1 -1 1 k 3 -1 4 2 k 3 -1 2 2 k 5 -1 1 -1 z k 2 z -1 1 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["EllipticE", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["1", "+", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["Log", "[", RowBox[List["1", "-", "z"]], "]"]], " ", RowBox[List["(", RowBox[List["z", "-", "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", "-", "z"]], ")"]], "k"]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["1", "+", "k"]], ")"]], "!"]]]]]]]]], "+", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[FractionBox["1", "16"], " ", RowBox[List["(", RowBox[List["29", "-", RowBox[List["13", " ", "z"]]]], ")"]]]], "-", RowBox[List["4", " ", RowBox[List["Log", "[", "2", "]"]], " ", 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", "-", "z"]], ")"]], "k"]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["1", "+", "k"]], ")"]], "!"]]]]]]]]]]], ")"]]]], "+", RowBox[List[FractionBox["45", "64"], " ", RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["5", "2"], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["7", "2"], ",", "k"]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox["2", RowBox[List["1", "+", "k"]]]]], "-", FractionBox["2", RowBox[List["k", "+", "2"]]], "-", FractionBox["1", RowBox[List["3", "+", "k"]]], "+", FractionBox["4", RowBox[List["3", "+", RowBox[List["2", " ", "k"]]]]], "+", FractionBox["2", RowBox[List["5", "+", RowBox[List["2", " ", "k"]]]]], "+", 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[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["k", "+", "2"]]]]], RowBox[List[RowBox[List["(", RowBox[List["3", "+", "k"]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["k", "+", "2"]], ")"]], "!"]], ")"]], "2"]]]]]]]]]], "/;", RowBox[List[RowBox[List["Abs", "[", RowBox[List["z", "-", "1"]], "]"]], "<", "1"]]]]]]]]

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

 2007-05-02