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 JacobiCS

 http://functions.wolfram.com/09.27.20.0006.01

 Input Form

 D[JacobiCS[z, m], {z, \[Alpha]}] == FDPowerConstant[z, -1, \[Alpha]] z^(-\[Alpha] - 1) + (1/2) Sum[((-1)^k Pi^(2 k) BernoulliB[2 k] z^(2 k - \[Alpha] - 1))/ (EllipticK[m]^(2 k) (k Gamma[2 k - \[Alpha]])), {k, 1, Infinity}] - ((2^\[Alpha] Pi^(5/2) z^(1 - \[Alpha]))/EllipticK[m]^2) Sum[((k EllipticNomeQ[m]^(2 k))/(EllipticNomeQ[m]^(2 k) + 1)) HypergeometricPFQRegularized[{1}, {1 - \[Alpha]/2, (3 - \[Alpha])/2}, -((k^2 Pi^2 z^2)/(4 EllipticK[m]^2))], {k, 1, Infinity}]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List["z", ",", "\[Alpha]"]], "}"]]], RowBox[List["JacobiCS", "[", RowBox[List["z", ",", "m"]], "]"]]]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["FDPowerConstant", "[", RowBox[List["z", ",", RowBox[List["-", "1"]], ",", "\[Alpha]"]], "]"]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "\[Alpha]"]], "-", "1"]]]]], "+", RowBox[List[FractionBox["1", "2"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], SuperscriptBox["\[Pi]", RowBox[List["2", "k"]]], " ", RowBox[List["BernoulliB", "[", RowBox[List["2", "k"]], "]"]], " ", SuperscriptBox[RowBox[List["EllipticK", "[", "m", "]"]], RowBox[List[RowBox[List["-", "2"]], "k"]]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["2", "k"]], "-", "\[Alpha]", "-", "1"]]]]], RowBox[List["k", " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["2", "k"]], "-", "\[Alpha]"]], "]"]]]]]]]]], "-", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", "\[Alpha]"], " ", SuperscriptBox["\[Pi]", RowBox[List["5", "/", "2"]]], " ", SuperscriptBox["z", RowBox[List["1", "-", "\[Alpha]"]]]]], SuperscriptBox[RowBox[List["EllipticK", "[", "m", "]"]], "2"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List["k", " ", SuperscriptBox[RowBox[List["EllipticNomeQ", "[", "m", "]"]], RowBox[List["2", "k"]]]]], RowBox[List[SuperscriptBox[RowBox[List["EllipticNomeQ", "[", "m", "]"]], 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["k", "2"], " ", SuperscriptBox["\[Pi]", "2"], " ", SuperscriptBox["z", "2"]]], RowBox[List["4", SuperscriptBox[RowBox[List["EllipticK", "[", "m", "]"]], "2"]]]]]]]], "]"]]]]]]]]]]]]]]

 MathML Form

 α cs ( z m ) z α ℱ𝒞 exp ( α ) ( z , - 1 ) z - α - 1 + 1 2 k = 1 ( - 1 ) k π 2 k B TagBox["B", BernoulliB] 2 k K ( m ) - 2 k z 2 k - α - 1 k Γ ( 2 k - α ) - 2 α π 5 / 2 z 1 - α K ( m ) 2 k = 1 k q EllipticNomeQ ( m ) 2 k q EllipticNomeQ ( m ) 2 k + 1 1 F ~ 2 ( 1 ; 1 - α 2 , 3 - α 2 ; - k 2 π 2 z 2 4 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["k", "2"], " ", SuperscriptBox["\[Pi]", "2"], " ", SuperscriptBox["z", "2"]]], RowBox[List["4", " ", 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 JacobiCS z m Subscript ℱ𝒞 exp α z -1 z -1 α -1 1 2 k 1 -1 k 2 k BernoulliB 2 k EllipticK m -2 k z 2 k -1 α -1 k Gamma 2 k -1 α -1 -1 2 α 5 2 z 1 -1 α EllipticK m 2 -1 k 1 k EllipticNomeQ m 2 k EllipticNomeQ m 2 k 1 -1 HypergeometricPFQRegularized 1 1 -1 α 2 -1 3 -1 α 2 -1 -1 k 2 2 z 2 4 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["JacobiCS", "[", 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[FractionBox["1", "2"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", SuperscriptBox["\[Pi]", RowBox[List["2", " ", "k"]]], " ", RowBox[List["BernoulliB", "[", RowBox[List["2", " ", "k"]], "]"]], " ", SuperscriptBox[RowBox[List["EllipticK", "[", "m", "]"]], RowBox[List[RowBox[List["-", "2"]], " ", "k"]]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["2", " ", "k"]], "-", "\[Alpha]", "-", "1"]]]]], RowBox[List["k", " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["2", " ", "k"]], "-", "\[Alpha]"]], "]"]]]]]]]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", "\[Alpha]"], " ", SuperscriptBox["\[Pi]", RowBox[List["5", "/", "2"]]], " ", SuperscriptBox["z", RowBox[List["1", "-", "\[Alpha]"]]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List["k", " ", SuperscriptBox[RowBox[List["EllipticNomeQ", "[", "m", "]"]], RowBox[List["2", " ", "k"]]]]], ")"]], " ", 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["k", "2"], " ", SuperscriptBox["\[Pi]", "2"], " ", SuperscriptBox["z", "2"]]], RowBox[List["4", " ", SuperscriptBox[RowBox[List["EllipticK", "[", "m", "]"]], "2"]]]]]]]], "]"]]]], RowBox[List[SuperscriptBox[RowBox[List["EllipticNomeQ", "[", "m", "]"]], RowBox[List["2", " ", "k"]]], "+", "1"]]]]]]], SuperscriptBox[RowBox[List["EllipticK", "[", "m", "]"]], "2"]]]]]]]]

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

 2001-10-29