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/08.04.20.0009.01

 Input Form

 D[EllipticE[z, m], {z, \[Alpha]}] == EllipticE[m]/(z^\[Alpha] Gamma[1 - \[Alpha]]) - (2^\[Alpha] Sqrt[Pi] (Sum[(((-m)^j 2^(-2 j - 2 n))/((n + j)! j! (2 j + 2 n + 1))) Pochhammer[-(1/2), j] Pochhammer[1/2, j] Pochhammer[1/2, n] Sum[Binomial[2 j + 2 n + 1, k] HypergeometricPFQRegularized[{1}, {(1 - \[Alpha])/2, 1 - \[Alpha]/2}, -(((1 + 2 j + 2 n - 2 k)^2 z^2)/ 4)], {k, 0, j + n}], {n, 0, Infinity}, {j, 0, Infinity}] - (m/2) Sum[(((-1)^j m^(j + n))/(2^(2 j) ((2 j + 1) j! Pochhammer[2, n + j] Pochhammer[3/2, n]))) Pochhammer[1/2, n + j] Pochhammer[3/2, n + j] Sum[Binomial[2 j + 1, k] HypergeometricPFQRegularized[{1}, {(1 - \[Alpha])/2, 1 - \[Alpha]/2}, -(((1 + 2 j - 2 k)^2 z^2)/4)], {k, 0, j}], {n, 0, Infinity}, {j, 0, Infinity}]))/z^\[Alpha]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List["z", ",", "\[Alpha]"]], "}"]]], RowBox[List["EllipticE", "[", RowBox[List["z", ",", "m"]], "]"]]]], "\[Equal]", " ", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["z", RowBox[List["-", "\[Alpha]"]]], " ", RowBox[List["EllipticE", "[", "m", "]"]]]], RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Alpha]"]], "]"]]], "-", RowBox[List[SuperscriptBox["2", "\[Alpha]"], " ", SqrtBox["\[Pi]"], " ", SuperscriptBox["z", RowBox[List["-", "\[Alpha]"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["n", "=", "0"]], "\[Infinity]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "m"]], ")"]], "j"], " ", SuperscriptBox["2", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "j"]], "-", RowBox[List["2", " ", "n"]]]]]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["n", "+", "j"]], ")"]], "!"]], " ", RowBox[List["j", "!"]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "j"]], "+", RowBox[List["2", " ", "n"]], "+", "1"]], ")"]]]]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], ",", "j"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "j"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "n"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["j", "+", "n"]]], RowBox[List[RowBox[List["Binomial", "[", RowBox[List[RowBox[List[RowBox[List["2", " ", "j"]], "+", RowBox[List["2", " ", "n"]], "+", "1"]], ",", "k"]], "]"]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", "1", "}"]], ",", RowBox[List["{", RowBox[List[FractionBox[RowBox[List["1", "-", "\[Alpha]"]], "2"], ",", RowBox[List["1", "-", FractionBox["\[Alpha]", "2"]]]]], "}"]], ",", RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", "j"]], "+", RowBox[List["2", " ", "n"]], "-", RowBox[List["2", " ", "k"]]]], ")"]], "2"], " ", SuperscriptBox["z", "2"]]], "4"]]]]], "]"]]]]]]]]]]]], "-", RowBox[List[FractionBox["m", "2"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["n", "=", "0"]], "\[Infinity]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], " ", SuperscriptBox["m", RowBox[List["j", "+", "n"]]], " ", SuperscriptBox["2", RowBox[List[RowBox[List["-", "2"]], " ", "j"]]]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "j"]], "+", "1"]], ")"]], " ", RowBox[List["j", "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List["2", ",", RowBox[List["n", "+", "j"]]]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["3", "2"], ",", "n"]], "]"]]]]], RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", RowBox[List["n", "+", "j"]]]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["3", "2"], ",", RowBox[List["n", "+", "j"]]]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "j"], RowBox[List[RowBox[List["Binomial", "[", RowBox[List[RowBox[List[RowBox[List["2", " ", "j"]], "+", "1"]], ",", "k"]], "]"]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", "1", "}"]], ",", RowBox[List["{", RowBox[List[FractionBox[RowBox[List["1", "-", "\[Alpha]"]], "2"], ",", RowBox[List["1", "-", FractionBox["\[Alpha]", "2"]]]]], "}"]], ",", RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", "j"]], "-", RowBox[List["2", " ", "k"]]]], ")"]], "2"], " ", SuperscriptBox["z", "2"]]], "4"]]]]], "]"]]]]]]]]]]]]]]]], ")"]]]]]]]]]]

 MathML Form

 α E ( z m ) z α z - α Γ ( 1 - α ) E ( m ) - 2 α π z - α ( n = 0 j = 0 ( - m ) j 2 - 2 j - 2 n ( j + n ) ! j ! ( 2 j + 2 n + 1 ) ( - 1 2 ) j TagBox[SubscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["1", "2"]]], ")"]], "j"], Pochhammer] ( 1 2 ) j TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], "j"], Pochhammer] ( 1 2 ) n TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], "n"], Pochhammer] k = 0 j + n ( 2 j + 2 n + 1 k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List[RowBox[List["2", " ", "j"]], "+", RowBox[List["2", " ", "n"]], "+", "1"]], Identity, Rule[Editable, True]]], List[TagBox["k", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] 1 F ~ 2 ( 1 ; 1 - α 2 , 1 - α 2 ; - 1 4 ( 2 j - 2 k + 2 n + 1 ) 2 z 2 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["1", TraditionalForm]], SubscriptBox[OverscriptBox["F", "~"], FormBox["2", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[TagBox["1", HypergeometricPFQRegularized, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[FractionBox[RowBox[List["1", "-", "\[Alpha]"]], "2"], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", FractionBox["\[Alpha]", "2"]]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox[RowBox[List[RowBox[List["-", FractionBox["1", "4"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "j"]], "-", RowBox[List["2", " ", "k"]], "+", RowBox[List["2", " ", "n"]], "+", "1"]], ")"]], "2"], " ", SuperscriptBox["z", "2"]]], HypergeometricPFQRegularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQRegularized] - m 2 n = 0 j = 0 ( - 1 ) j m j + n 2 - 2 j ( 2 j + 1 ) j ! ( 2 ) j + n TagBox[SubscriptBox[RowBox[List["(", "2", ")"]], RowBox[List["j", "+", "n"]]], Pochhammer] ( 3 2 ) n TagBox[SubscriptBox[RowBox[List["(", FractionBox["3", "2"], ")"]], "n"], Pochhammer] ( 1 2 ) j + n TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], RowBox[List["j", "+", "n"]]], Pochhammer] ( 3 2 ) j + n TagBox[SubscriptBox[RowBox[List["(", FractionBox["3", "2"], ")"]], RowBox[List["j", "+", "n"]]], Pochhammer] k = 0 j ( 2 j + 1 k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List[RowBox[List["2", " ", "j"]], "+", "1"]], Identity, Rule[Editable, True]]], List[TagBox["k", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] 1 F ~ 2 ( 1 ; 1 - α 2 , 1 - α 2 ; - 1 4 ( 2 j - 2 k + 1 ) 2 z 2 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["1", TraditionalForm]], SubscriptBox[OverscriptBox["F", "~"], FormBox["2", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[TagBox["1", HypergeometricPFQRegularized, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[FractionBox[RowBox[List["1", "-", "\[Alpha]"]], "2"], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", FractionBox["\[Alpha]", "2"]]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox[RowBox[List[RowBox[List["-", FractionBox["1", "4"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "j"]], "-", RowBox[List["2", " ", "k"]], "+", "1"]], ")"]], "2"], " ", SuperscriptBox["z", "2"]]], HypergeometricPFQRegularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQRegularized] ) z α EllipticE z m z -1 α Gamma 1 -1 α -1 EllipticE m -1 2 α 1 2 z -1 α j 0 n 0 -1 m j 2 -2 j -1 2 n j n j 2 j 2 n 1 -1 Pochhammer -1 1 2 j Pochhammer 1 2 j Pochhammer 1 2 n k 0 j n Binomial 2 j 2 n 1 k HypergeometricPFQRegularized 1 1 -1 α 2 -1 1 -1 α 2 -1 -1 1 4 2 j -1 2 k 2 n 1 2 z 2 -1 m 2 -1 j 0 n 0 -1 j m j n 2 -2 j 2 j 1 j Pochhammer 2 j n Pochhammer 3 2 n -1 Pochhammer 1 2 j n Pochhammer 3 2 j n k 0 j Binomial 2 j 1 k HypergeometricPFQRegularized 1 1 -1 α 2 -1 1 -1 α 2 -1 -1 1 4 2 j -1 2 k 1 2 z 2 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["z_", ",", "\[Alpha]_"]], "}"]]]]], RowBox[List["EllipticE", "[", RowBox[List["z_", ",", "m_"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["z", RowBox[List["-", "\[Alpha]"]]], " ", RowBox[List["EllipticE", "[", "m", "]"]]]], RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Alpha]"]], "]"]]], "-", RowBox[List[SuperscriptBox["2", "\[Alpha]"], " ", SqrtBox["\[Pi]"], " ", SuperscriptBox["z", RowBox[List["-", "\[Alpha]"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["n", "=", "0"]], "\[Infinity]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "m"]], ")"]], "j"], " ", SuperscriptBox["2", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "j"]], "-", RowBox[List["2", " ", "n"]]]]]]], ")"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], ",", "j"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "j"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "n"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["j", "+", "n"]]], RowBox[List[RowBox[List["Binomial", "[", RowBox[List[RowBox[List[RowBox[List["2", " ", "j"]], "+", RowBox[List["2", " ", "n"]], "+", "1"]], ",", "k"]], "]"]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", "1", "}"]], ",", RowBox[List["{", RowBox[List[FractionBox[RowBox[List["1", "-", "\[Alpha]"]], "2"], ",", RowBox[List["1", "-", FractionBox["\[Alpha]", "2"]]]]], "}"]], ",", RowBox[List[RowBox[List["-", FractionBox["1", "4"]]], " ", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "j"]], "+", RowBox[List["2", " ", "n"]], "-", RowBox[List["2", " ", "k"]]]], ")"]], "2"], " ", SuperscriptBox["z", "2"]]], ")"]]]]]], "]"]]]]]]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["n", "+", "j"]], ")"]], "!"]], " ", RowBox[List["j", "!"]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "j"]], "+", RowBox[List["2", " ", "n"]], "+", "1"]], ")"]]]]]]]]], "-", RowBox[List[FractionBox["1", "2"], " ", "m", " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["n", "=", "0"]], "\[Infinity]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], " ", SuperscriptBox["m", RowBox[List["j", "+", "n"]]], " ", SuperscriptBox["2", RowBox[List[RowBox[List["-", "2"]], " ", "j"]]]]], ")"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", RowBox[List["n", "+", "j"]]]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["3", "2"], ",", RowBox[List["n", "+", "j"]]]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "j"], RowBox[List[RowBox[List["Binomial", "[", RowBox[List[RowBox[List[RowBox[List["2", " ", "j"]], "+", "1"]], ",", "k"]], "]"]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", "1", "}"]], ",", RowBox[List["{", RowBox[List[FractionBox[RowBox[List["1", "-", "\[Alpha]"]], "2"], ",", RowBox[List["1", "-", FractionBox["\[Alpha]", "2"]]]]], "}"]], ",", RowBox[List[RowBox[List["-", FractionBox["1", "4"]]], " ", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "j"]], "-", RowBox[List["2", " ", "k"]]]], ")"]], "2"], " ", SuperscriptBox["z", "2"]]], ")"]]]]]], "]"]]]]]]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "j"]], "+", "1"]], ")"]], " ", RowBox[List["j", "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List["2", ",", RowBox[List["n", "+", "j"]]]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["3", "2"], ",", "n"]], "]"]]]]]]]]]]]]], ")"]]]]]]]]]]

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

 2001-10-29