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 JacobiNC

 http://functions.wolfram.com/09.31.20.0006.01

 Input Form

 D[JacobiNC[z, m], {z, \[Alpha]}] == (Pi/(2 Sqrt[1 - m] EllipticK[m])) Sum[(((-1)^k EulerE[2 k])/(EllipticK[m]^(2 k) Gamma[2 k - \[Alpha] + 1])) (Pi/2)^(2 k) z^(2 k - \[Alpha]), {k, 0, Infinity}] - ((2^(\[Alpha] + 1) Pi^(3/2))/(z^\[Alpha] (Sqrt[1 - m] EllipticK[m]))) Sum[(((-1)^k EllipticNomeQ[m]^(2 k + 1))/(EllipticNomeQ[m]^(2 k + 1) + 1)) HypergeometricPFQRegularized[{1}, {(1 - \[Alpha])/2, 1 - \[Alpha]/2}, -(((2 k + 1)^2 Pi^2 z^2)/(16 EllipticK[m]^2))], {k, 0, Infinity}]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List["z", ",", "\[Alpha]"]], "}"]]], RowBox[List["JacobiNC", "[", RowBox[List["z", ",", "m"]], "]"]]]], "\[Equal]", RowBox[List[RowBox[List[FractionBox["\[Pi]", RowBox[List["2", SqrtBox[RowBox[List["1", "-", "m"]]], " ", RowBox[List["EllipticK", "[", "m", "]"]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", SuperscriptBox[RowBox[List["EllipticK", "[", "m", "]"]], RowBox[List[RowBox[List["-", "2"]], "k"]]], " ", RowBox[List["EulerE", "[", RowBox[List["2", "k"]], "]"]]]], RowBox[List["Gamma", "[", RowBox[List[RowBox[List["2", "k"]], "-", "\[Alpha]", "+", "1"]], "]"]]], " ", SuperscriptBox[RowBox[List["(", FractionBox["\[Pi]", "2"], ")"]], RowBox[List["2", "k"]]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["2", "k"]], "-", "\[Alpha]"]]]]]]]]], "-", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List["\[Alpha]", "+", "1"]]], SuperscriptBox["\[Pi]", RowBox[List["3", "/", "2"]]], SuperscriptBox["z", RowBox[List["-", "\[Alpha]"]]]]], RowBox[List[SqrtBox[RowBox[List["1", "-", "m"]]], " ", RowBox[List["EllipticK", "[", "m", "]"]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", SuperscriptBox[RowBox[List["EllipticNomeQ", "[", "m", "]"]], RowBox[List[RowBox[List["2", "k"]], "+", "1"]]]]], RowBox[List[SuperscriptBox[RowBox[List["EllipticNomeQ", "[", "m", "]"]], RowBox[List[RowBox[List["2", "k"]], "+", "1"]]], "+", "1"]]], " ", 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[RowBox[List["2", "k"]], "+", "1"]], ")"]], "2"], " ", SuperscriptBox["\[Pi]", "2"], " ", SuperscriptBox["z", "2"]]], RowBox[List["16", " ", SuperscriptBox[RowBox[List["EllipticK", "[", "m", "]"]], "2"]]]]]]]], "]"]]]]]]]]]]]]]]

 MathML Form

 α nc ( z m ) z α π 2 1 - m K ( m ) k = 0 ( - 1 ) k K ( m ) - 2 k E TagBox["E", EulerE] 2 k Γ ( 2 k - α + 1 ) ( π 2 ) 2 k z 2 k - α - 2 α + 1 π 3 / 2 z - α 1 - m K ( m ) k = 0 ( - 1 ) k q EllipticNomeQ ( m ) 2 k + 1 q EllipticNomeQ ( m ) 2 k + 1 + 1 1 F ~ 2 ( 1 ; 1 - α 2 , 1 - α 2 ; - ( 2 k + 1 ) 2 π 2 z 2 16 K ( m ) 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["-", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]], ")"]], "2"], " ", SuperscriptBox["\[Pi]", "2"], " ", SuperscriptBox["z", "2"]]], RowBox[List["16", " ", SuperscriptBox[RowBox[List["K", "(", "m", ")"]], "2"]]]]]], HypergeometricPFQRegularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQRegularized] z α JacobiNC z m 2 1 -1 m 1 2 EllipticK m -1 k 0 -1 k EllipticK m -2 k EulerE 2 k Gamma 2 k -1 α 1 -1 2 -1 2 k z 2 k -1 α -1 2 α 1 3 2 z -1 α 1 -1 m 1 2 EllipticK m -1 k 0 -1 k EllipticNomeQ m 2 k 1 EllipticNomeQ m 2 k 1 1 -1 HypergeometricPFQRegularized 1 1 -1 α 2 -1 1 -1 α 2 -1 -1 2 k 1 2 2 z 2 16 EllipticK m 2 -1 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["z_", ",", "\[Alpha]_"]], "}"]]]]], RowBox[List["JacobiNC", "[", RowBox[List["z_", ",", "m_"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List["\[Pi]", " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", SuperscriptBox[RowBox[List["EllipticK", "[", "m", "]"]], RowBox[List[RowBox[List["-", "2"]], " ", "k"]]], " ", RowBox[List["EulerE", "[", RowBox[List["2", " ", "k"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", FractionBox["\[Pi]", "2"], ")"]], RowBox[List["2", " ", "k"]]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["2", " ", "k"]], "-", "\[Alpha]"]]]]], RowBox[List["Gamma", "[", RowBox[List[RowBox[List["2", " ", "k"]], "-", "\[Alpha]", "+", "1"]], "]"]]]]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["1", "-", "m"]]], " ", RowBox[List["EllipticK", "[", "m", "]"]]]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List["\[Alpha]", "+", "1"]]], " ", SuperscriptBox["\[Pi]", RowBox[List["3", "/", "2"]]], " ", SuperscriptBox["z", RowBox[List["-", "\[Alpha]"]]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", SuperscriptBox[RowBox[List["EllipticNomeQ", "[", "m", "]"]], RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]]]]], ")"]], " ", 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[RowBox[List["2", " ", "k"]], "+", "1"]], ")"]], "2"], " ", SuperscriptBox["\[Pi]", "2"], " ", SuperscriptBox["z", "2"]]], RowBox[List["16", " ", SuperscriptBox[RowBox[List["EllipticK", "[", "m", "]"]], "2"]]]]]]]], "]"]]]], RowBox[List[SuperscriptBox[RowBox[List["EllipticNomeQ", "[", "m", "]"]], RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]]], "+", "1"]]]]]]], RowBox[List[SqrtBox[RowBox[List["1", "-", "m"]]], " ", RowBox[List["EllipticK", "[", "m", "]"]]]]]]]]]]]

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

 2001-10-29

© 1998-2013 Wolfram Research, Inc.