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variants of this functions
EllipticE




Mathematica Notation

Traditional Notation









Elliptic Integrals > EllipticE[z,m] > Series representations > Generalized power series > Expansions at z==csc-1(m1/2)+Pi u/;u ∈ Z





http://functions.wolfram.com/08.04.06.0031.01









  


  








Input Form





EllipticE[z, m] == Sqrt[m] (EllipticE[1/m] + (-1 + 1/m) EllipticK[1/m]) + 2 u EllipticE[m] + 2 Sqrt[2] Sqrt[(-Sqrt[-1 + m]) (z - Subscript[z, 0])] (z - Subscript[z, 0]) Sum[(Binomial[k - 1/2, k]/(3 + 2 k)) Sum[(((-1)^j Binomial[k, j])/(1 - 2 j)) Subscript[p, j, k] (z - Subscript[z, 0])^k, {j, 0, k}], {k, 0, Infinity}] /; Subscript[z, 0] == ArcCsc[Sqrt[m]] + Pi u && Element[u, Integers] && Subscript[a, 0] == 1 && Subscript[a, 2 k] == ((-1)^k 2^(2 k))/(2 k + 1)! && Subscript[a, 2 k + 1] == ((-1)^(k - 1) 2^(2 k) (2 - m))/ (Sqrt[m - 1] (2 k + 2)!) && Element[k, Integers] && k >= 0 && Subscript[p, u, 0] == 1 && Subscript[p, u, v] == (1/v) Sum[(u j - v + j) Subscript[a, j] Subscript[p, u, v - j], {j, 1, v}]








Standard Form





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MathML Form







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</apply> </apply> </apply> <apply> <eq /> <apply> <ci> Subscript </ci> <ci> a </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <plus /> <ci> k </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> </apply> <apply> <plus /> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <factorial /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> 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</apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["EllipticE", "[", RowBox[List["z_", ",", "m_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[SqrtBox["m"], " ", RowBox[List["(", RowBox[List[RowBox[List["EllipticE", "[", FractionBox["1", "m"], "]"]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", FractionBox["1", "m"]]], ")"]], " ", RowBox[List["EllipticK", "[", FractionBox["1", "m"], "]"]]]]]], ")"]]]], "+", RowBox[List["2", " ", "u", " ", RowBox[List["EllipticE", "[", "m", "]"]]]], "+", RowBox[List["2", " ", SqrtBox["2"], " ", SqrtBox[RowBox[List[RowBox[List["-", SqrtBox[RowBox[List[RowBox[List["-", "1"]], "+", "m"]]]]], " ", RowBox[List["(", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], ")"]]]]], " ", RowBox[List["(", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["Binomial", "[", RowBox[List[RowBox[List["k", "-", FractionBox["1", "2"]]], ",", "k"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "k"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], " ", RowBox[List["Binomial", "[", RowBox[List["k", ",", "j"]], "]"]]]], ")"]], " ", SubscriptBox["p", RowBox[List["j", ",", "k"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], ")"]], "k"]]], RowBox[List["1", "-", RowBox[List["2", " ", "j"]]]]]]]]], RowBox[List["3", "+", RowBox[List["2", " ", "k"]]]]]]]]]]], "/;", RowBox[List[RowBox[List[SubscriptBox["zz", "0"], "\[Equal]", RowBox[List[RowBox[List["ArcCsc", "[", SqrtBox["m"], "]"]], "+", RowBox[List["\[Pi]", " ", "u"]]]]]], "&&", RowBox[List["u", "\[Element]", "Integers"]], "&&", RowBox[List[SubscriptBox["a", "0"], "\[Equal]", "1"]], "&&", RowBox[List[SubscriptBox["a", RowBox[List["2", " ", "k"]]], "\[Equal]", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", SuperscriptBox["2", RowBox[List["2", " ", "k"]]]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]], ")"]], "!"]]]]], "&&", RowBox[List[SubscriptBox["a", RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]]], "\[Equal]", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["k", "-", "1"]]], " ", SuperscriptBox["2", RowBox[List["2", " ", "k"]]], " ", RowBox[List["(", RowBox[List["2", "-", "m"]], ")"]]]], RowBox[List[SqrtBox[RowBox[List["m", "-", "1"]]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "+", "2"]], ")"]], "!"]]]]]]], "&&", RowBox[List["k", "\[Element]", "Integers"]], "&&", RowBox[List["k", "\[GreaterEqual]", "0"]], "&&", RowBox[List[SubscriptBox["p", RowBox[List["u", ",", "0"]]], "\[Equal]", "1"]], "&&", RowBox[List[SubscriptBox["p", RowBox[List["u", ",", "v"]]], "\[Equal]", FractionBox[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], "v"], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["u", " ", "j"]], "-", "v", "+", "j"]], ")"]], " ", SubscriptBox["a", "j"], " ", SubscriptBox["p", RowBox[List["u", ",", RowBox[List["v", "-", "j"]]]]]]]]], "v"]]]]]]]]]]]


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





2007-05-02