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JacobiZeta






Mathematica Notation

Traditional Notation









Elliptic Integrals > JacobiZeta[z,m] > Series representations > Generalized power series > Expansions at z==0





http://functions.wolfram.com/08.07.06.0001.02









  


  










Input Form





JacobiZeta[z, m] \[Proportional] (1 - EllipticE[m]/EllipticK[m]) z - (m/6) (1 + EllipticE[m]/EllipticK[m]) z^3 + (m/120) (4 - 3 m - ((9 m - 4) EllipticE[m])/EllipticK[m]) z^5 + (m/5040) (-16 + 60 m - 45 m^2 - ((16 - 180 m + 225 m^2) EllipticE[m])/ EllipticK[m]) z^7 + (m/362880) (64 - 1008 m + 2520 m^2 - 1575 m^3 - (-64 + 3024 m - 12600 m^2 + 11025 m^3) (EllipticE[m]/EllipticK[m])) z^9 + O[z^11]










Standard Form





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MathML Form







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</ci> <apply> <power /> <cn type='integer'> 120 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <cn type='integer'> 4 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 3 </cn> <ci> m </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 9 </cn> <ci> m </ci> </apply> <cn type='integer'> -4 </cn> </apply> <apply> <ci> EllipticE </ci> <ci> m </ci> </apply> <apply> <power /> <apply> <ci> EllipticK </ci> <ci> m </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 5 </cn> </apply> </apply> <apply> <times /> <apply> <times /> <ci> m </ci> <apply> <power /> <cn type='integer'> 5040 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -45 </cn> <apply> <power /> <ci> m </ci> <cn type='integer'> 2 </cn> </apply> 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</apply> </apply> </apply> </apply> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 9 </cn> </apply> </apply> <apply> <ci> O </ci> <apply> <power /> <ci> z </ci> <cn type='integer'> 11 </cn> </apply> </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["(", RowBox[List["1", "-", FractionBox[RowBox[List["EllipticE", "[", "m", "]"]], RowBox[List["EllipticK", "[", "m", "]"]]]]], ")"]], " ", "z"]], "-", RowBox[List[FractionBox["1", "6"], " ", "m", " ", RowBox[List["(", RowBox[List["1", "+", FractionBox[RowBox[List["EllipticE", "[", "m", "]"]], RowBox[List["EllipticK", "[", "m", "]"]]]]], ")"]], " ", SuperscriptBox["z", "3"]]], "+", RowBox[List[FractionBox["1", "120"], " ", "m", " ", RowBox[List["(", RowBox[List["4", "-", RowBox[List["3", " ", "m"]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["9", " ", "m"]], "-", "4"]], ")"]], " ", RowBox[List["EllipticE", "[", "m", "]"]]]], RowBox[List["EllipticK", "[", "m", "]"]]]]], ")"]], " ", SuperscriptBox["z", "5"]]], "+", FractionBox[RowBox[List["m", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "16"]], "+", RowBox[List["60", " ", "m"]], "-", RowBox[List["45", " ", SuperscriptBox["m", "2"]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["16", "-", RowBox[List["180", " ", "m"]], "+", RowBox[List["225", " ", SuperscriptBox["m", "2"]]]]], ")"]], " ", RowBox[List["EllipticE", "[", "m", "]"]]]], RowBox[List["EllipticK", "[", "m", "]"]]]]], ")"]], " ", SuperscriptBox["z", "7"]]], "5040"], "+", FractionBox[RowBox[List["m", " ", RowBox[List["(", RowBox[List["64", "-", RowBox[List["1008", " ", "m"]], "+", RowBox[List["2520", " ", SuperscriptBox["m", "2"]]], "-", RowBox[List["1575", " ", SuperscriptBox["m", "3"]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "64"]], "+", RowBox[List["3024", " ", "m"]], "-", RowBox[List["12600", " ", SuperscriptBox["m", "2"]]], "+", RowBox[List["11025", " ", SuperscriptBox["m", "3"]]]]], ")"]], " ", RowBox[List["EllipticE", "[", "m", "]"]]]], RowBox[List["EllipticK", "[", "m", "]"]]]]], ")"]], " ", SuperscriptBox["z", "9"]]], "362880"], "+", SuperscriptBox[RowBox[List["O", "[", "z", "]"]], "11"]]]]]]]










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





2001-10-29





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