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InverseJacobiNC






Mathematica Notation

Traditional Notation









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





http://functions.wolfram.com/09.43.06.0005.01









  


  










Input Form





InverseJacobiNC[z, m] == EllipticK[m] - (1/z) (Sum[((-1)^k z^(-2 j - 2 k) Pochhammer[1/2, j] Pochhammer[1/2, k]^2 m^k)/((1 + 2 j + 2 k) k! (j + k)!), {k, 0, Infinity}, {j, 0, Infinity}] + (1/2) Sum[((-1)^k Pochhammer[3/2, j + k]^2 m^(j + k + 1))/(z^(2 k) ((1 + 2 k) Pochhammer[3/2, j] k! (j + k + 1)!)), {k, 0, Infinity}, {j, 0, Infinity}])










Standard Form





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MathML Form







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Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["InverseJacobiNC", "[", RowBox[List["z_", ",", "m_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["EllipticK", "[", "m", "]"]], "-", FractionBox[RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", SuperscriptBox["z", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "j"]], "-", RowBox[List["2", " ", "k"]]]]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "j"]], "]"]], " ", SuperscriptBox[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "k"]], "]"]], "2"], " ", SuperscriptBox["m", "k"]]], RowBox[List[RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "j"]], "+", RowBox[List["2", " ", "k"]]]], ")"]], " ", RowBox[List["k", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["j", "+", "k"]], ")"]], "!"]]]]]]]]], "+", RowBox[List[FractionBox["1", "2"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "2"]], " ", "k"]]], " ", SuperscriptBox[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["3", "2"], ",", RowBox[List["j", "+", "k"]]]], "]"]], "2"], " ", SuperscriptBox["m", RowBox[List["j", "+", "k", "+", "1"]]]]], RowBox[List[RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "k"]]]], ")"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["3", "2"], ",", "j"]], "]"]], " ", RowBox[List["k", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["j", "+", "k", "+", "1"]], ")"]], "!"]]]]]]]]]]]]], "z"]]]]]]]










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





2001-10-29