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 JacobiZeta

 http://functions.wolfram.com/08.07.06.0028.01

 Input Form

 JacobiZeta[z, m] \[Proportional] (-JacobiZeta[ArcCsc[Sqrt[m]], m]) (I Sqrt[-(1/(z - Subscript[z, 0])^2)] (z - Subscript[z, 0]) + Sqrt[(z - Subscript[z, 0])^2]/(z - Subscript[z, 0])) + (Sqrt[1 - m] - EllipticE[m]/(Sqrt[1 - m] EllipticK[m])) (z - Subscript[z, 0]) + (m/(6 Sqrt[1 - m]) + (m EllipticE[m])/ (6 (1 - m)^(3/2) EllipticK[m])) (z - Subscript[z, 0])^3 + (((-4 + m) m)/(120 (1 - m)^(3/2)) - (m (4 + 5 m) EllipticE[m])/ (120 (1 - m)^(5/2) EllipticK[m])) (z - Subscript[z, 0])^5 + \[Ellipsis] /; (z -> Subscript[z, 0]) && Subscript[z, 0] == Pi/2 + Pi u && Element[u, Integers] && Element[m, Reals] && m > 1

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["JacobiZeta", "[", RowBox[List["z", ",", "m"]], "]"]], "\[Proportional]", RowBox[List[RowBox[List[RowBox[List["-", RowBox[List["JacobiZeta", "[", RowBox[List[RowBox[List["ArcCsc", "[", SqrtBox["m"], "]"]], ",", "m"]], "]"]]]], RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", SqrtBox[RowBox[List["-", FractionBox["1", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", SubscriptBox["z", "0"]]], ")"]], "2"]]]]], " ", RowBox[List["(", RowBox[List["z", "-", SubscriptBox["z", "0"]]], ")"]]]], "+", FractionBox[SqrtBox[SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", SubscriptBox["z", "0"]]], ")"]], "2"]], RowBox[List["z", "-", SubscriptBox["z", "0"]]]]]], ")"]]]], " ", "+", RowBox[List[RowBox[List["(", RowBox[List[SqrtBox[RowBox[List["1", "-", "m"]]], "-", FractionBox[RowBox[List["EllipticE", "[", "m", "]"]], RowBox[List[SqrtBox[RowBox[List["1", "-", "m"]]], RowBox[List["EllipticK", "[", "m", "]"]]]]]]], ")"]], RowBox[List["(", RowBox[List["z", "-", SubscriptBox["z", "0"]]], ")"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[FractionBox["m", RowBox[List["6", " ", SqrtBox[RowBox[List["1", "-", "m"]]]]]], "+", FractionBox[RowBox[List["m", " ", RowBox[List["EllipticE", "[", "m", "]"]]]], RowBox[List["6", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "m"]], ")"]], RowBox[List["3", "/", "2"]]], RowBox[List["EllipticK", "[", "m", "]"]]]]]]], ")"]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", SubscriptBox["z", "0"]]], ")"]], "3"]]], "+", RowBox[List[RowBox[List["(", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "4"]], "+", "m"]], ")"]], " ", "m"]], RowBox[List["120", " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "m"]], ")"]], RowBox[List["3", "/", "2"]]]]]], "-", FractionBox[RowBox[List["m", " ", RowBox[List["(", RowBox[List["4", "+", RowBox[List["5", " ", "m"]]]], ")"]], " ", RowBox[List["EllipticE", "[", "m", "]"]]]], RowBox[List["120", " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "m"]], ")"]], RowBox[List["5", "/", "2"]]], " ", RowBox[List["EllipticK", "[", "m", "]"]]]]]]], ")"]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", SubscriptBox["z", "0"]]], ")"]], "5"]]], "+", "\[Ellipsis]"]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List["z", "\[Rule]", SubscriptBox["z", "0"]]], ")"]], "\[And]", RowBox[List[SubscriptBox["z", "0"], "\[Equal]", RowBox[List[FractionBox["\[Pi]", "2"], "+", RowBox[List["\[Pi]", " ", "u"]]]]]], "\[And]", RowBox[List["u", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", "\[Element]", "Reals"]], "\[And]", RowBox[List["m", ">", "1"]]]]]]]]

 MathML Form

 Ζ ( z m ) - Ζ ( csc - 1 ( m ) m ) ( - 1 ( z - z 0 ) 2 ( z - z 0 ) + ( z - z 0 ) 2 z - z 0 ) + ( 1 - m - E ( m ) 1 - m K ( m ) ) ( z - z 0 ) + ( m 6 1 - m + E ( m ) m 6 ( 1 - m ) 3 / 2 K ( m ) ) ( z - z 0 ) 3 + ( ( m - 4 ) m 120 ( 1 - m ) 3 / 2 - m ( 5 m + 4 ) E ( m ) 120 ( 1 - m ) 5 / 2 K ( m ) ) ( z - z 0 ) 5 + /; ( z "\[Rule]" z 0 ) z 0 π 2 + π u u TagBox["\[DoubleStruckCapitalZ]", Function[List[], Integers]] m TagBox["\[DoubleStruckCapitalR]", Function[List[], Reals]] m > 1 Condition Proportional JacobiZeta z m -1 Ζ VerticalSeparator m 1 2 m -1 1 z -1 Subscript z 0 2 -1 1 2 z -1 Subscript z 0 z -1 Subscript z 0 2 1 2 z -1 Subscript z 0 -1 1 -1 m 1 2 -1 EllipticE m 1 -1 m 1 2 EllipticK m -1 z -1 Subscript z 0 m 6 1 -1 m 1 2 -1 EllipticE m m 6 1 -1 m 3 2 EllipticK m -1 z -1 Subscript z 0 3 m -4 m 120 1 -1 m 3 2 -1 -1 m 5 m 4 EllipticE m 120 1 -1 m 5 2 EllipticK m -1 z -1 Subscript z 0 5 Rule z Subscript z 0 Subscript z 0 2 -1 u u m m 1 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["JacobiZeta", "[", RowBox[List["z_", ",", "m_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[RowBox[List["-", RowBox[List["JacobiZeta", "[", RowBox[List[RowBox[List["ArcCsc", "[", SqrtBox["m"], "]"]], ",", "m"]], "]"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", SqrtBox[RowBox[List["-", FractionBox["1", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], ")"]], "2"]]]]], " ", RowBox[List["(", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], ")"]]]], "+", FractionBox[SqrtBox[SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], ")"]], "2"]], RowBox[List["z", "-", SubscriptBox["zz", "0"]]]]]], ")"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[SqrtBox[RowBox[List["1", "-", "m"]]], "-", FractionBox[RowBox[List["EllipticE", "[", "m", "]"]], RowBox[List[SqrtBox[RowBox[List["1", "-", "m"]]], " ", RowBox[List["EllipticK", "[", "m", "]"]]]]]]], ")"]], " ", RowBox[List["(", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], ")"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[FractionBox["m", RowBox[List["6", " ", SqrtBox[RowBox[List["1", "-", "m"]]]]]], "+", FractionBox[RowBox[List["m", " ", RowBox[List["EllipticE", "[", "m", "]"]]]], RowBox[List["6", " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "m"]], ")"]], RowBox[List["3", "/", "2"]]], " ", RowBox[List["EllipticK", "[", "m", "]"]]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], ")"]], "3"]]], "+", RowBox[List[RowBox[List["(", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "4"]], "+", "m"]], ")"]], " ", "m"]], RowBox[List["120", " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "m"]], ")"]], RowBox[List["3", "/", "2"]]]]]], "-", FractionBox[RowBox[List["m", " ", RowBox[List["(", RowBox[List["4", "+", RowBox[List["5", " ", "m"]]]], ")"]], " ", RowBox[List["EllipticE", "[", "m", "]"]]]], RowBox[List["120", " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "m"]], ")"]], RowBox[List["5", "/", "2"]]], " ", RowBox[List["EllipticK", "[", "m", "]"]]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], ")"]], "5"]]], "+", "\[Ellipsis]"]], "/;", RowBox[List[RowBox[List["(", RowBox[List["z", "\[Rule]", SubscriptBox["zz", "0"]]], ")"]], "&&", RowBox[List[SubscriptBox["zz", "0"], "\[Equal]", RowBox[List[FractionBox["\[Pi]", "2"], "+", RowBox[List["\[Pi]", " ", "u"]]]]]], "&&", RowBox[List["u", "\[Element]", "Integers"]], "&&", RowBox[List["m", "\[Element]", "Reals"]], "&&", RowBox[List["m", ">", "1"]]]]]]]]]]

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

 2007-05-02