Jacobi zeta function: Series representations (formula 08.07.06.0022)

JacobiZeta

$Mathematica Notation$

Elliptic Integrals

JacobiZeta[z,m]

Series representations

Generalized power series

Expansions at z==csc^-1(m^1/2)+Pi u/;u ∈ Z

http://functions.wolfram.com/08.07.06.0022.01

Input Form

JacobiZeta[z, m] \[Proportional] JacobiZeta[ArcCsc[Sqrt[m]], m] + ((Sqrt[2] EllipticE[m])/(Sqrt[-1 + m] EllipticK[m])) Sqrt[(-Sqrt[-1 + m]) (z - Subscript[z, 0])] + Sqrt[2] Sqrt[(-Sqrt[-1 + m]) (z - Subscript[z, 0])] (z - Subscript[z, 0]) (2/3 - ((-2 + m) EllipticE[m])/(12 (-1 + m) EllipticK[m]) + ((-2 + m)/(10 Sqrt[-1 + m]) + ((4 - 4 m + 9 m^2) EllipticE[m])/ (480 (-1 + m)^(3/2) EllipticK[m])) (z - Subscript[z, 0]) - ((-20 + 20 m + 3 m^2)/(336 (-1 + m)) + ((8 - 12 m - 26 m^2 + 15 m^3) EllipticE[m])/(2688 (-1 + m)^2 EllipticK[m])) (z - Subscript[z, 0])^2 + \[Ellipsis]) /; (z -> Subscript[z, 0]) && Subscript[z, 0] == ArcCsc[Sqrt[m]] + Pi u && Element[u, Integers]

Standard Form

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MathML Form

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<mrow> <mrow> <msup> <mi> csc </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <msqrt> <mi> m </mi> </msqrt> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> u </mi> </mrow> </mrow> </mrow> <mo> ∧ </mo> <mrow> <mi> u </mi> <mo> ∈ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[List[], Integers]] </annotation> </semantics> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <ci> Proportional </ci> <apply> <ci> JacobiZeta </ci> <ci> z </ci> <ci> m </ci> </apply> <apply> <plus /> <apply> <ci> JacobiZeta </ci> <apply> <arccsc /> <apply> <power /> <ci> m </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <ci> m </ci> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <ci> EllipticE </ci> <ci> m </ci> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <ci> EllipticK </ci> <ci> m </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> <apply> <plus /> <cn type='rational'> 2 <sep /> 3 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <ci> m </ci> <cn type='integer'> -2 </cn> </apply> <apply> <ci> EllipticE </ci> <ci> m </ci> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 12 </cn> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> <apply> <ci> EllipticK </ci> <ci> m </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <plus /> <ci> m </ci> <cn type='integer'> -2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 10 </cn> <apply> <power /> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 9 </cn> <apply> <power /> <ci> m </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 4 </cn> <ci> m </ci> </apply> </apply> <cn type='integer'> 4 </cn> </apply> <apply> <ci> EllipticE </ci> <ci> m </ci> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 480 </cn> <apply> <power /> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> <cn type='rational'> 3 <sep /> 2 </cn> </apply> <apply> <ci> EllipticK </ci> <ci> m </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <power /> <ci> m </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 20 </cn> <ci> m </ci> </apply> <cn type='integer'> -20 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 336 </cn> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 15 </cn> <apply> <power /> <ci> m </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 26 </cn> <apply> <power /> <ci> m </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 12 </cn> <ci> m </ci> </apply> </apply> <cn type='integer'> 8 </cn> </apply> <apply> <ci> EllipticE </ci> <ci> m </ci> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2688 </cn> <apply> <power /> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <ci> EllipticK </ci> <ci> m </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <ci> … </ci> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <ci> Rule </ci> <ci> z </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> <apply> <eq /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> <apply> <plus /> <apply> <arccsc /> <apply> <power /> <ci> m </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <pi /> <ci> u </ci> </apply> </apply> </apply> <apply> <in /> <ci> u </ci> <integers /> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["JacobiZeta", "[", RowBox[List["z_", ",", "m_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["JacobiZeta", "[", RowBox[List[RowBox[List["ArcCsc", "[", SqrtBox["m"], "]"]], ",", "m"]], "]"]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SqrtBox["2"], " ", RowBox[List["EllipticE", "[", "m", "]"]]]], ")"]], " ", SqrtBox[RowBox[List[RowBox[List["-", SqrtBox[RowBox[List[RowBox[List["-", "1"]], "+", "m"]]]]], " ", RowBox[List["(", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], ")"]]]]]]], RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "1"]], "+", "m"]]], " ", RowBox[List["EllipticK", "[", "m", "]"]]]]], "+", RowBox[List[SqrtBox["2"], " ", SqrtBox[RowBox[List[RowBox[List["-", SqrtBox[RowBox[List[RowBox[List["-", "1"]], "+", "m"]]]]], " ", RowBox[List["(", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], ")"]]]]], " ", RowBox[List["(", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], ")"]], " ", RowBox[List["(", RowBox[List[FractionBox["2", "3"], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "+", "m"]], ")"]], " ", RowBox[List["EllipticE", "[", "m", "]"]]]], RowBox[List["12", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "m"]], ")"]], " ", RowBox[List["EllipticK", "[", "m", "]"]]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[FractionBox[RowBox[List[RowBox[List["-", "2"]], "+", "m"]], RowBox[List["10", " ", SqrtBox[RowBox[List[RowBox[List["-", "1"]], "+", "m"]]]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["4", "-", RowBox[List["4", " ", "m"]], "+", RowBox[List["9", " ", SuperscriptBox["m", "2"]]]]], ")"]], " ", RowBox[List["EllipticE", "[", "m", "]"]]]], RowBox[List["480", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "m"]], ")"]], RowBox[List["3", "/", "2"]]], " ", RowBox[List["EllipticK", "[", "m", "]"]]]]]]], ")"]], " ", RowBox[List["(", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], ")"]]]], "-", RowBox[List[RowBox[List["(", RowBox[List[FractionBox[RowBox[List[RowBox[List["-", "20"]], "+", RowBox[List["20", " ", "m"]], "+", RowBox[List["3", " ", SuperscriptBox["m", "2"]]]]], RowBox[List["336", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "m"]], ")"]]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["8", "-", RowBox[List["12", " ", "m"]], "-", RowBox[List["26", " ", SuperscriptBox["m", "2"]]], "+", RowBox[List["15", " ", SuperscriptBox["m", "3"]]]]], ")"]], " ", RowBox[List["EllipticE", "[", "m", "]"]]]], RowBox[List["2688", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "m"]], ")"]], "2"], " ", RowBox[List["EllipticK", "[", "m", "]"]]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], ")"]], "2"]]], "+", "\[Ellipsis]"]], ")"]]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List["z", "\[Rule]", SubscriptBox["zz", "0"]]], ")"]], "&&", RowBox[List[SubscriptBox["zz", "0"], "\[Equal]", RowBox[List[RowBox[List["ArcCsc", "[", SqrtBox["m"], "]"]], "+", RowBox[List["\[Pi]", " ", "u"]]]]]], "&&", RowBox[List["u", "\[Element]", "Integers"]]]]]]]]]]

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

2007-05-02