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InverseJacobiDN






Mathematica Notation

Traditional Notation









Elliptic Functions > InverseJacobiDN[z,m] > Differentiation > Symbolic differentiation > With respect to z





http://functions.wolfram.com/09.41.20.0013.01









  


  










Input Form





D[InverseJacobiDN[z, m], {z, n}] == KroneckerDelta[n] InverseJacobiDN[z, m] - (JacobiCS[InverseJacobiDN[z, m], m]/(-1 + m + z^2)) Sum[(Pochhammer[1 - n, 2 (n - j) - 2]/((n - j - 1)! (2 z)^(n - 2 j - 1))) Sum[((-1)^(k + j) Binomial[j, k] Pochhammer[1/2, k] Pochhammer[1/2, j - k] (m - 1 + z^2)^(-j + k))/(1 - z^2)^k, {k, 0, j}], {j, 0, n - 1}] /; Element[n, Integers] && n >= 0










Standard Form





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MathML Form







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</ci> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["z_", ",", "n_"]], "}"]]]]], RowBox[List["InverseJacobiDN", "[", RowBox[List["z_", ",", "m_"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[RowBox[List["KroneckerDelta", "[", "n", "]"]], " ", RowBox[List["InverseJacobiDN", "[", RowBox[List["z", ",", "m"]], "]"]]]], "-", FractionBox[RowBox[List[RowBox[List["JacobiCS", "[", RowBox[List[RowBox[List["InverseJacobiDN", "[", RowBox[List["z", ",", "m"]], "]"]], ",", "m"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List["n", "-", "1"]]], FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "n"]], ",", RowBox[List[RowBox[List["2", " ", RowBox[List["(", RowBox[List["n", "-", "j"]], ")"]]]], "-", "2"]]]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "j"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["k", "+", "j"]]], " ", RowBox[List["Binomial", "[", RowBox[List["j", ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", RowBox[List["j", "-", "k"]]]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["z", "2"]]], ")"]], RowBox[List["-", "k"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["m", "-", "1", "+", SuperscriptBox["z", "2"]]], ")"]], RowBox[List[RowBox[List["-", "j"]], "+", "k"]]]]]]]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["n", "-", "j", "-", "1"]], ")"]], "!"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["2", " ", "z"]], ")"]], RowBox[List["n", "-", RowBox[List["2", " ", "j"]], "-", "1"]]]]]]]]]], RowBox[List[RowBox[List["-", "1"]], "+", "m", "+", SuperscriptBox["z", "2"]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]










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





2007-05-02