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 JacobiZeta

 http://functions.wolfram.com/08.07.06.0013.01

 Input Form

 JacobiZeta[z, m] \[Proportional] Exp[(-Pi) I Floor[Arg[x - z]/(2 Pi)]] JacobiZeta[x, m] + JacobiZeta[ArcCsc[Sqrt[m]], m] (1 - Exp[(-Pi) I Floor[Arg[x - z]/(2 Pi)]]) - Exp[(-Pi) I Floor[Arg[x - z]/(2 Pi)]] (((2 EllipticE[m] + (-2 + m - m Cos[2 x]) EllipticK[m])/ (2 EllipticK[m] Sqrt[1 - m Sin[x]^2])) (z - x) + ((m (2 EllipticE[m] + (2 - m + m Cos[2 x]) EllipticK[m]) Sin[2 x])/ (8 EllipticK[m] (1 - m Sin[x]^2)^(3/2))) (z - x)^2 + \[Ellipsis]) /; (z -> x) && Element[x, Reals] && Element[m, Reals] && m > 1 && ArcCsc[Sqrt[m]] + Pi u < x < Pi/2 + Pi u && Element[u, Integers]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["JacobiZeta", "[", RowBox[List["z", ",", "m"]], "]"]], "\[Proportional]", RowBox[List[RowBox[List[RowBox[List["Exp", "[", RowBox[List[RowBox[List["-", "\[Pi]"]], " ", "\[ImaginaryI]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["x", "-", "z"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]], "]"]], RowBox[List["JacobiZeta", "[", RowBox[List["x", ",", "m"]], "]"]]]], "+", RowBox[List[RowBox[List["JacobiZeta", "[", RowBox[List[RowBox[List["ArcCsc", "[", SqrtBox["m"], "]"]], ",", "m"]], "]"]], RowBox[List["(", RowBox[List["1", "-", RowBox[List["Exp", "[", RowBox[List[RowBox[List["-", "\[Pi]"]], " ", "\[ImaginaryI]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["x", "-", "z"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]], "]"]]]], ")"]]]], "-", RowBox[List[RowBox[List["Exp", "[", RowBox[List[RowBox[List["-", "\[Pi]"]], " ", "\[ImaginaryI]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["x", "-", "z"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]], "]"]], RowBox[List["(", RowBox[List[RowBox[List[FractionBox[RowBox[List[RowBox[List["2", " ", RowBox[List["EllipticE", "[", "m", "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "+", "m", "-", RowBox[List["m", " ", RowBox[List["Cos", "[", RowBox[List["2", " ", "x"]], "]"]]]]]], ")"]], " ", RowBox[List["EllipticK", "[", "m", "]"]]]]]], RowBox[List["2", " ", RowBox[List["EllipticK", "[", "m", "]"]], " ", SqrtBox[RowBox[List["1", "-", RowBox[List["m", " ", SuperscriptBox[RowBox[List["Sin", "[", "x", "]"]], "2"]]]]]]]]], RowBox[List["(", RowBox[List["z", "-", "x"]], ")"]]]], "+", RowBox[List[FractionBox[RowBox[List["m", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", RowBox[List["EllipticE", "[", "m", "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["2", "-", "m", "+", RowBox[List["m", " ", RowBox[List["Cos", "[", RowBox[List["2", " ", "x"]], "]"]]]]]], ")"]], " ", RowBox[List["EllipticK", "[", "m", "]"]]]]]], ")"]], " ", RowBox[List["Sin", "[", RowBox[List["2", " ", "x"]], "]"]]]], RowBox[List["8", " ", RowBox[List["EllipticK", "[", "m", "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", RowBox[List["m", " ", SuperscriptBox[RowBox[List["Sin", "[", "x", "]"]], "2"]]]]], ")"]], RowBox[List["3", "/", "2"]]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "x"]], ")"]], "2"]]], "+", "\[Ellipsis]"]], ")"]]]]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List["z", "\[Rule]", "x"]], ")"]], "\[And]", RowBox[List["x", "\[Element]", "Reals"]], "\[And]", RowBox[List["m", "\[Element]", "Reals"]], "\[And]", RowBox[List["m", ">", "1"]], "\[And]", RowBox[List[RowBox[List[RowBox[List["ArcCsc", "[", SqrtBox["m"], "]"]], "+", RowBox[List["\[Pi]", " ", "u"]]]], "<", "x", "<", RowBox[List[FractionBox["\[Pi]", "2"], "+", RowBox[List["\[Pi]", " ", "u"]]]]]], "\[And]", RowBox[List["u", "\[Element]", "Integers"]]]]]]]]

 MathML Form

 Ζ ( z m ) Ζ ( x m ) - π arg ( x - z ) 2 π + ( 1 - - π arg ( x - z ) 2 π ) Ζ ( csc - 1 ( m ) m ) - - π arg ( x - z ) 2 π ( 2 E ( m ) + ( - cos ( 2 x ) m + m - 2 ) K ( m ) 2 K ( m ) 1 - m sin 2 ( x ) ( z - x ) + m ( 2 E ( m ) + ( cos ( 2 x ) m - m + 2 ) K ( m ) ) sin ( 2 x ) 8 K ( m ) ( 1 - m sin 2 ( x ) ) 3 / 2 ( z - x ) 2 + ) /; ( z "\[Rule]" x ) x TagBox["\[DoubleStruckCapitalR]", Function[List[], Reals]] m TagBox["\[DoubleStruckCapitalR]", Function[List[], Reals]] m > 1 π u + csc - 1 ( m ) < x < π u + π 2 u TagBox["\[DoubleStruckCapitalZ]", Function[List[], Integers]] Condition Proportional JacobiZeta z m Ζ VerticalSeparator x m -1 x -1 z 2 -1 1 -1 -1 x -1 z 2 -1 JacobiZeta m 1 2 m -1 -1 x -1 z 2 -1 2 EllipticE m -1 2 x m m -2 EllipticK m 2 EllipticK m 1 -1 m x 2 1 2 -1 z -1 x m 2 EllipticE m 2 x m -1 m 2 EllipticK m 2 x 8 EllipticK m 1 -1 m x 2 3 2 -1 z -1 x 2 Rule z x x m m 1 u m 1 2 x u 2 -1 u [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["JacobiZeta", "[", RowBox[List["z_", ",", "m_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "\[Pi]"]], " ", "\[ImaginaryI]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["x", "-", "z"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], " ", RowBox[List["JacobiZeta", "[", RowBox[List["x", ",", "m"]], "]"]]]], "+", RowBox[List[RowBox[List["JacobiZeta", "[", RowBox[List[RowBox[List["ArcCsc", "[", SqrtBox["m"], "]"]], ",", "m"]], "]"]], " ", RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "\[Pi]"]], " ", "\[ImaginaryI]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["x", "-", "z"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]]]], ")"]]]], "-", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "\[Pi]"]], " ", "\[ImaginaryI]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["x", "-", "z"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", RowBox[List["EllipticE", "[", "m", "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "+", "m", "-", RowBox[List["m", " ", RowBox[List["Cos", "[", RowBox[List["2", " ", "x"]], "]"]]]]]], ")"]], " ", RowBox[List["EllipticK", "[", "m", "]"]]]]]], ")"]], " ", RowBox[List["(", RowBox[List["z", "-", "x"]], ")"]]]], RowBox[List["2", " ", RowBox[List["EllipticK", "[", "m", "]"]], " ", SqrtBox[RowBox[List["1", "-", RowBox[List["m", " ", SuperscriptBox[RowBox[List["Sin", "[", "x", "]"]], "2"]]]]]]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["m", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", RowBox[List["EllipticE", "[", "m", "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["2", "-", "m", "+", RowBox[List["m", " ", RowBox[List["Cos", "[", RowBox[List["2", " ", "x"]], "]"]]]]]], ")"]], " ", RowBox[List["EllipticK", "[", "m", "]"]]]]]], ")"]], " ", RowBox[List["Sin", "[", RowBox[List["2", " ", "x"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "x"]], ")"]], "2"]]], RowBox[List["8", " ", RowBox[List["EllipticK", "[", "m", "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", RowBox[List["m", " ", SuperscriptBox[RowBox[List["Sin", "[", "x", "]"]], "2"]]]]], ")"]], RowBox[List["3", "/", "2"]]]]]], "+", "\[Ellipsis]"]], ")"]]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List["z", "\[Rule]", "x"]], ")"]], "&&", RowBox[List["x", "\[Element]", "Reals"]], "&&", RowBox[List["m", "\[Element]", "Reals"]], "&&", RowBox[List["m", ">", "1"]], "&&", RowBox[List[RowBox[List[RowBox[List["ArcCsc", "[", SqrtBox["m"], "]"]], "+", RowBox[List["\[Pi]", " ", "u"]]]], "<", "x", "<", RowBox[List[FractionBox["\[Pi]", "2"], "+", RowBox[List["\[Pi]", " ", "u"]]]]]], "&&", RowBox[List["u", "\[Element]", "Integers"]]]]]]]]]]

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

 2007-05-02