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JacobiZeta






Mathematica Notation

Traditional Notation









Elliptic Integrals > JacobiZeta[z,m] > Representations through more general functions > Through other functions > Involving some elliptic-type functions





http://functions.wolfram.com/08.07.26.0006.01









  


  










Input Form





JacobiZeta[z, m] == (Subscript[\[Omega], 1]/EllipticK[m]^2) (EllipticK[m] (WeierstrassZeta[(Subscript[\[Omega], 1] EllipticF[z, m])/ EllipticK[m] - Subscript[\[Omega], 3], {Subscript[g, 2], Subscript[g, 3]}] + Subscript[\[Eta], 3]) - EllipticF[z, m] Subscript[\[Eta], 1]) /; m == InverseEllipticNomeQ[E^((I Pi Subscript[\[Omega], 3])/ Subscript[\[Omega], 1])] && {Subscript[\[Omega], 1], Subscript[\[Omega], 3]} == WeierstrassHalfPeriods[{Subscript[g, 2], Subscript[g, 3]}] && {Subscript[\[Eta], 1], Subscript[\[Eta], 3]} == {WeierstrassZeta[Subscript[\[Omega], 1], {Subscript[g, 2], Subscript[g, 3]}], WeierstrassZeta[Subscript[\[Omega], 3], {Subscript[g, 2], Subscript[g, 3]}]}










Standard Form





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MathML Form







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Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["JacobiZeta", "[", RowBox[List["z_", ",", "m_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[SubscriptBox["\[Omega]", "1"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["EllipticK", "[", "m", "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["WeierstrassZeta", "[", RowBox[List[RowBox[List[FractionBox[RowBox[List[SubscriptBox["\[Omega]", "1"], " ", RowBox[List["EllipticF", "[", RowBox[List["z", ",", "m"]], "]"]]]], RowBox[List["EllipticK", "[", "m", "]"]]], "-", SubscriptBox["\[Omega]", "3"]]], ",", RowBox[List["{", RowBox[List[SubscriptBox["g", "2"], ",", SubscriptBox["g", "3"]]], "}"]]]], "]"]], "+", SubscriptBox["\[Eta]", "3"]]], ")"]]]], "-", RowBox[List[RowBox[List["EllipticF", "[", RowBox[List["z", ",", "m"]], "]"]], " ", SubscriptBox["\[Eta]", "1"]]]]], ")"]]]], SuperscriptBox[RowBox[List["EllipticK", "[", "m", "]"]], "2"]], "/;", RowBox[List[RowBox[List["m", "\[Equal]", RowBox[List["InverseEllipticNomeQ", "[", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", SubscriptBox["\[Omega]", "3"]]], SubscriptBox["\[Omega]", "1"]]], "]"]]]], "&&", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["\[Omega]", "1"], ",", SubscriptBox["\[Omega]", "3"]]], "}"]], "\[Equal]", RowBox[List["WeierstrassHalfPeriods", "[", RowBox[List["{", RowBox[List[SubscriptBox["g", "2"], ",", SubscriptBox["g", "3"]]], "}"]], "]"]]]], "&&", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["\[Eta]", "1"], ",", SubscriptBox["\[Eta]", "3"]]], "}"]], "\[Equal]", RowBox[List["{", RowBox[List[RowBox[List["WeierstrassZeta", "[", RowBox[List[SubscriptBox["\[Omega]", "1"], ",", RowBox[List["{", RowBox[List[SubscriptBox["g", "2"], ",", SubscriptBox["g", "3"]]], "}"]]]], "]"]], ",", RowBox[List["WeierstrassZeta", "[", RowBox[List[SubscriptBox["\[Omega]", "3"], ",", RowBox[List["{", RowBox[List[SubscriptBox["g", "2"], ",", SubscriptBox["g", "3"]]], "}"]]]], "]"]]]], "}"]]]]]]]]]]]]










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





2001-10-29





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