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 JacobiNS

 http://functions.wolfram.com/09.33.20.0006.01

 Input Form

 D[JacobiNS[z, m], {z, \[Alpha]}] == FDPowerConstant[z, -1, \[Alpha]] z^(-\[Alpha] - 1) + Sum[((-1)^(k - 1) 2^(1 - 2 k) (2^(2 k - 1) - 1) Pi^(2 k) z^(2 k - \[Alpha] - 1) BernoulliB[2 k])/EllipticK[m]^(2 k)/ ((2 k + 1) Gamma[2 k - \[Alpha]]), {k, 1, Infinity}] + ((2^(\[Alpha] - 1) Pi^(5/2) z^(1 - \[Alpha]))/EllipticK[m]^2) Sum[(((2 k + 1) EllipticNomeQ[m]^(2 k + 1))/ (1 - EllipticNomeQ[m]^(2 k + 1))) HypergeometricPFQRegularized[{1}, {1 - \[Alpha]/2, (3 - \[Alpha])/2}, -(((2 k + 1)^2 Pi^2 z^2)/(16 EllipticK[m]^2))], {k, 0, Infinity}]

 Standard Form

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 MathML Form

 α ns ( z m ) z α ℱ𝒞 exp ( α ) ( z , - 1 ) z - α - 1 + k = 1 ( - 1 ) k - 1 2 1 - 2 k ( 2 2 k - 1 - 1 ) π 2 k z 2 k - α - 1 B TagBox["B", BernoulliB] 2 k K ( m ) - 2 k ( 2 k + 1 ) Γ ( 2 k - α ) + 2 α - 1 π 5 / 2 z 1 - α K ( m ) 2 k = 0 ( 2 k + 1 ) q EllipticNomeQ ( m ) 2 k + 1 1 - q EllipticNomeQ ( m ) 2 k + 1 1 F ~ 2 ( 1 ; 1 - α 2 , 3 - 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[RowBox[List["1", "-", FractionBox["\[Alpha]", "2"]]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["3", "-", "\[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 JacobiNS z m Subscript ℱ𝒞 exp α z -1 z -1 α -1 k 1 -1 k -1 2 1 -1 2 k 2 2 k -1 -1 2 k z 2 k -1 α -1 BernoulliB 2 k EllipticK m -2 k 2 k 1 Gamma 2 k -1 α -1 2 α -1 5 2 z 1 -1 α EllipticK m 2 -1 k 0 2 k 1 EllipticNomeQ m 2 k 1 1 -1 EllipticNomeQ m 2 k 1 -1 HypergeometricPFQRegularized 1 1 -1 α 2 -1 3 -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["JacobiNS", "[", RowBox[List["z_", ",", "m_"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["FDPowerConstant", "[", RowBox[List["z", ",", RowBox[List["-", "1"]], ",", "\[Alpha]"]], "]"]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "\[Alpha]"]], "-", "1"]]]]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["k", "-", "1"]]], " ", SuperscriptBox["2", RowBox[List["1", "-", RowBox[List["2", " ", "k"]]]]], " ", RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["2", " ", "k"]], "-", "1"]]], "-", "1"]], ")"]], " ", SuperscriptBox["\[Pi]", RowBox[List["2", " ", "k"]]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["2", " ", "k"]], "-", "\[Alpha]", "-", "1"]]], " ", RowBox[List["BernoulliB", "[", RowBox[List["2", " ", "k"]], "]"]], " ", SuperscriptBox[RowBox[List["EllipticK", "[", "m", "]"]], RowBox[List[RowBox[List["-", "2"]], " ", "k"]]]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["2", " ", "k"]], "-", "\[Alpha]"]], "]"]]]]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List["\[Alpha]", "-", "1"]]], " ", SuperscriptBox["\[Pi]", RowBox[List["5", "/", "2"]]], " ", SuperscriptBox["z", RowBox[List["1", "-", "\[Alpha]"]]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]], ")"]], " ", SuperscriptBox[RowBox[List["EllipticNomeQ", "[", "m", "]"]], RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]]]]], ")"]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", "1", "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "-", FractionBox["\[Alpha]", "2"]]], ",", FractionBox[RowBox[List["3", "-", "\[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["1", "-", SuperscriptBox[RowBox[List["EllipticNomeQ", "[", "m", "]"]], RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]]]]]]]]]], SuperscriptBox[RowBox[List["EllipticK", "[", "m", "]"]], "2"]]]]]]]]

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

 2001-10-29