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InverseJacobiCN






Mathematica Notation

Traditional Notation









Elliptic Functions > InverseJacobiCN[z,m] > Series representations > Generalized power series > Expansions at generic point z==z0 > For the function itself





http://functions.wolfram.com/09.38.06.0012.01









  


  










Input Form





InverseJacobiCN[z, m] == InverseJacobiCN[Subscript[z, 0], m] - (Pi/(JacobiDN[InverseJacobiCN[Subscript[z, 0], m], m] JacobiSN[InverseJacobiCN[Subscript[z, 0], m], m])) Sum[((2 Subscript[z, 0])^(k - 1)/k) Sum[(m^(k - j - 1) ((1 - m + m Subscript[z, 0]^2)^(1 + j - k)/ (j! (k - j - 1)! Gamma[1/2 - j] Gamma[3/2 + j - k])) Hypergeometric2F1[(1 - j)/2, -(j/2), 1/2 - j, 1 - 1/Subscript[z, 0]^2] Hypergeometric2F1[(2 + j - k)/2, (1 + j - k)/2, 3/2 + j - k, 1 + (1 - m)/(m Subscript[z, 0]^2)] (z - Subscript[z, 0])^k)/(Subscript[z, 0]^2 - 1)^j, {j, 0, k - 1}], {k, 1, Infinity}]










Standard Form





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MathML Form







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</mi> </msup> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> InverseJacobiCN </ci> <ci> z </ci> <ci> m </ci> </apply> <apply> <plus /> <apply> <ci> InverseJacobiCN </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> <ci> m </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <pi /> <apply> <power /> <apply> <times /> <apply> <ci> JacobiDN </ci> <apply> <ci> InverseJacobiCN </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> <ci> m </ci> </apply> <ci> m </ci> </apply> <apply> <ci> JacobiSN </ci> <apply> <ci> InverseJacobiCN </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> <ci> m </ci> </apply> <ci> m </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> 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0 </cn> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> <ci> k </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["InverseJacobiCN", "[", RowBox[List["z_", ",", "m_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["InverseJacobiCN", "[", RowBox[List[SubscriptBox["zz", "0"], ",", "m"]], "]"]], "-", FractionBox[RowBox[List["\[Pi]", " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["2", " ", SubscriptBox["zz", "0"]]], ")"]], RowBox[List["k", "-", "1"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List["k", "-", "1"]]], FractionBox[RowBox[List[SuperscriptBox["m", RowBox[List["k", "-", "j", "-", "1"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[SubsuperscriptBox["zz", "0", "2"], "-", "1"]], ")"]], RowBox[List["-", "j"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "m", "+", RowBox[List["m", " ", SubsuperscriptBox["zz", "0", "2"]]]]], ")"]], RowBox[List["1", "+", "j", "-", "k"]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List["1", "-", "j"]], "2"], ",", RowBox[List["-", FractionBox["j", "2"]]], ",", RowBox[List[FractionBox["1", "2"], "-", "j"]], ",", RowBox[List["1", "-", FractionBox["1", SubsuperscriptBox["zz", "0", "2"]]]]]], "]"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "+", "j", "-", "k"]], ")"]]]], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "j", "-", "k"]], ")"]]]], ",", RowBox[List[FractionBox["3", "2"], "+", "j", "-", "k"]], ",", RowBox[List["1", "+", FractionBox[RowBox[List["1", "-", "m"]], RowBox[List["m", " ", SubsuperscriptBox["zz", "0", "2"]]]]]]]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], ")"]], "k"]]], RowBox[List[RowBox[List["j", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["k", "-", "j", "-", "1"]], ")"]], "!"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "-", "j"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["3", "2"], "+", "j", "-", "k"]], "]"]]]]]]]]], "k"]]]]], RowBox[List[RowBox[List["JacobiDN", "[", RowBox[List[RowBox[List["InverseJacobiCN", "[", RowBox[List[SubscriptBox["zz", "0"], ",", "m"]], "]"]], ",", "m"]], "]"]], " ", RowBox[List["JacobiSN", "[", RowBox[List[RowBox[List["InverseJacobiCN", "[", RowBox[List[SubscriptBox["zz", "0"], ",", "m"]], "]"]], ",", "m"]], "]"]]]]]]]]]]]










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





2007-05-02





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