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JacobiZeta






Mathematica Notation

Traditional Notation









Elliptic Integrals > JacobiZeta[z,m] > Series representations > Generalized power series > Expansions on branch cuts > Formulas on real axis for real m > For m>1,csc-1(m1/2)+Pi u<xu+1/2)/;uZ





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





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MathML Form







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</ci> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <ci> Rule </ci> <ci> z </ci> <ci> x </ci> </apply> <apply> <in /> <ci> x </ci> <reals /> </apply> <apply> <in /> <ci> m </ci> <reals /> </apply> <apply> <gt /> <ci> m </ci> <cn type='integer'> 1 </cn> </apply> <apply> <lt /> <apply> <plus /> <apply> <times /> <pi /> <ci> u </ci> </apply> <apply> <arccsc /> <apply> <power /> <ci> m </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <ci> x </ci> <apply> <plus /> <apply> <times /> <pi /> <ci> u </ci> </apply> <apply> <times /> <pi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </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[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





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