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; }

 JacobiSC

 http://functions.wolfram.com/09.34.20.0006.01

 Input Form

 D[JacobiSC[z, m], {z, \[Alpha]}] == (1/(2 Sqrt[1 - m])) Sum[((-1)^(k - 1) (2^(2 k) - 1) Pi^(2 k) z^(2 k - \[Alpha] - 1) BernoulliB[2 k])/(EllipticK[m]^(2 k) (k Gamma[2 k - \[Alpha]])), {k, 1, Infinity}] + ((2^\[Alpha] Pi^(5/2) z^(1 - \[Alpha]))/(Sqrt[1 - m] EllipticK[m]^2)) Sum[(((-1)^k k EllipticNomeQ[m]^(2 k))/(EllipticNomeQ[m]^(2 k) + 1)) HypergeometricPFQRegularized[{1}, {1 - \[Alpha]/2, (3 - \[Alpha])/\[Alpha]}, -((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["JacobiSC", "[", RowBox[List["z", ",", "m"]], "]"]]]], "\[Equal]", RowBox[List[RowBox[List[FractionBox["1", RowBox[List["2", " ", SqrtBox[RowBox[List["1", "-", "m"]]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["k", "-", "1"]]], " ", RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List["2", "k"]]], "-", "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["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]"]]]]], RowBox[List[SqrtBox[RowBox[List["1", "-", "m"]]], " ", SuperscriptBox[RowBox[List["EllipticK", "[", "m", "]"]], "2"]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], "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]"]], "\[Alpha]"]]], "}"]], ",", 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 sc ( z m ) z α 1 2 1 - m k = 1 ( - 1 ) k - 1 ( 2 2 k - 1 ) π 2 k z 2 k - α - 1 B TagBox["B", BernoulliB] 2 k K ( m ) - 2 k k Γ ( 2 k - α ) + 2 α π 5 / 2 z 1 - α 1 - m K ( m ) 2 k = 1 ( - 1 ) k 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 JacobiSC z m 1 2 1 -1 m 1 2 -1 k 1 -1 k -1 2 2 k -1 2 k z 2 k -1 α -1 BernoulliB 2 k EllipticK m -2 k k Gamma 2 k -1 α -1 2 α 5 2 z 1 -1 α 1 -1 m 1 2 EllipticK m 2 -1 k 1 -1 k 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["JacobiSC", "[", RowBox[List["z_", ",", "m_"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["k", "-", "1"]]], " ", RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List["2", " ", "k"]]], "-", "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["k", " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["2", " ", "k"]], "-", "\[Alpha]"]], "]"]]]]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["1", "-", "m"]]]]]], "+", 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[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", "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]"]], "\[Alpha]"]]], "}"]], ",", 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"]]]]]]], RowBox[List[SqrtBox[RowBox[List["1", "-", "m"]]], " ", SuperscriptBox[RowBox[List["EllipticK", "[", "m", "]"]], "2"]]]]]]]]]]

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

 2001-10-29