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InverseJacobiSC






Mathematica Notation

Traditional Notation









Elliptic Functions > InverseJacobiSC[z,m] > Series representations > Generalized power series > Expansions at generic point z==z0





http://functions.wolfram.com/09.46.06.0010.01









  


  










Input Form





InverseJacobiSC[z, m] == InverseJacobiSC[Subscript[z, 0], m] + (1/(JacobiDC[InverseJacobiSC[Subscript[z, 0], m], m] JacobiNC[InverseJacobiSC[Subscript[z, 0], m], m])) Sum[(1/k!) Sum[((((-1)^j 2^(1 + 2 j - k) (1 - m)^j Gamma[1/2 + j] Pochhammer[1 - k, -2 + 2 (-j + k)] Subscript[z, 0]^(1 + 2 j - k))/ (Sqrt[Pi] (-1 - j + k)! (1 + (1 - m) Subscript[z, 0]^2)^j)) Hypergeometric2F1[1/2, -j, 1/2 - j, (1 + Subscript[z, 0]^2 - m Subscript[z, 0]^2)/((1 - m) (1 + Subscript[z, 0]^2))]) (z - Subscript[z, 0])^k, {j, 0, k - 1}], {k, 1, Infinity}]










Standard Form





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MathML Form







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Rule Form





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










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





2007-05-02





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