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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 m==infinity





http://functions.wolfram.com/08.04.06.0064.01









  


  








Input Form





EllipticE[z, m] == (-1)^Round[Re[z]/Pi] Sin[z] Sum[((Pochhammer[-(1/2), j] Pochhammer[j, i] Csc[z]^(2 i))/ ((-1)^j ((1 - 2 i) j! i!))) (m - 1)^j, {j, 0, Infinity}, {i, 0, Infinity}] + 2 Round[Re[z]/Pi] (1 + Log[1 - m] ((m - 1)/4) Sum[((Pochhammer[1/2, k] Pochhammer[3/2, k])/ (k! (1 + k)!)) (1 - m)^k, {k, 0, Infinity}] + ((m - 1)/4) Sum[((Pochhammer[1/2, k] Pochhammer[3/2, k])/(k! (1 + k)!)) (1/(1 + 3 k + 2 k^2) + 2 PolyGamma[1/2 + k] - 2 PolyGamma[1 + k]) (1 - m)^k, {k, 0, Infinity}])








Standard Form





Cell[BoxData[RowBox[List[RowBox[List["EllipticE", "[", RowBox[List["z", ",", "m"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["Round", "[", FractionBox[RowBox[List["Re", "[", "z", "]"]], "\[Pi]"], "]"]]], RowBox[List["Sin", "[", "z", "]"]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "\[Infinity]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["-", "j"]]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], ",", "j"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List["j", ",", "i"]], "]"]], " ", SuperscriptBox[RowBox[List["Csc", "[", "z", "]"]], RowBox[List["2", " ", "i"]]]]], RowBox[List[RowBox[List["(", RowBox[List["1", "-", RowBox[List["2", "i"]]]], ")"]], " ", RowBox[List["j", "!"]], " ", RowBox[List["i", "!"]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["m", "-", "1"]], ")"]], "j"]]]]]]]]], "+", RowBox[List["2", " ", RowBox[List["Round", "[", FractionBox[RowBox[List["Re", "[", "z", "]"]], "\[Pi]"], "]"]], RowBox[List["(", RowBox[List["1", "+", RowBox[List[RowBox[List["Log", "[", RowBox[List["1", "-", "m"]], "]"]], FractionBox[RowBox[List["m", "-", "1"]], "4"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["3", "2"], ",", "k"]], "]"]]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["1", "+", "k"]], ")"]], "!"]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "m"]], ")"]], "k"]]]]]]], "+", RowBox[List[FractionBox[RowBox[List["m", "-", "1"]], "4"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["3", "2"], ",", "k"]], "]"]]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["1", "+", "k"]], ")"]], "!"]]]]], RowBox[List["(", RowBox[List[FractionBox["1", RowBox[List["1", "+", RowBox[List["3", " ", "k"]], "+", RowBox[List["2", " ", SuperscriptBox["k", "2"]]]]]], "+", RowBox[List["2", " ", RowBox[List["PolyGamma", "[", RowBox[List[FractionBox["1", "2"], "+", "k"]], "]"]]]], "-", RowBox[List["2", " ", RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "k"]], "]"]]]]]], ")"]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "m"]], ")"]], "k"]]]]]]]]], ")"]]]]]]]]]]








MathML Form







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</ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> i </ms> <ms> ! </ms> </list> </apply> </list> </apply> </apply> <apply> <ci> SuperscriptBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> m </ms> <ms> - </ms> <ms> 1 </ms> </list> </apply> <ms> ) </ms> </list> </apply> <ms> j </ms> </apply> </list> </apply> </list> </apply> </list> </apply> </list> </apply> <ms> + </ms> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <apply> <ci> RowBox </ci> <list> <ms> &#8970; </ms> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <ms> Re </ms> <ms> ( </ms> <ms> z </ms> <ms> ) </ms> </list> </apply> <ms> &#960; </ms> </apply> <ms> &#8969; </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> 1 </ms> <ms> + </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <ms> m </ms> <ms> - </ms> <ms> 1 </ms> </list> </apply> <ms> 4 </ms> </apply> <apply> <ci> RowBox </ci> <list> <ms> log </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> 1 </ms> <ms> - </ms> <ms> m </ms> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> UnderoverscriptBox </ci> <ms> &#8721; </ms> <apply> <ci> RowBox </ci> <list> <ms> k </ms> <ms> = </ms> <ms> 0 </ms> </list> </apply> <ms> &#8734; </ms> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> TagBox </ci> <apply> <ci> SubscriptBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> FractionBox </ci> <ms> 1 </ms> <ms> 2 </ms> </apply> <ms> ) </ms> </list> </apply> <ms> k </ms> </apply> <ci> Pochhammer </ci> </apply> <apply> <ci> TagBox </ci> <apply> <ci> SubscriptBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> FractionBox </ci> <ms> 3 </ms> <ms> 2 </ms> </apply> <ms> ) </ms> </list> </apply> <ms> k </ms> </apply> <ci> Pochhammer </ci> </apply> </list> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> k </ms> <ms> ! </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> k </ms> <ms> + </ms> <ms> 1 </ms> </list> </apply> <ms> ) </ms> </list> </apply> <ms> ! </ms> </list> </apply> </list> </apply> </apply> <apply> <ci> SuperscriptBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> 1 </ms> <ms> - </ms> <ms> m </ms> </list> </apply> <ms> ) </ms> </list> </apply> <ms> k </ms> </apply> </list> </apply> </list> </apply> </list> </apply> <ms> + </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <ms> 1 </ms> <ms> 4 </ms> </apply> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> m </ms> <ms> - </ms> <ms> 1 </ms> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> UnderoverscriptBox </ci> <ms> &#8721; </ms> <apply> <ci> RowBox </ci> <list> <ms> k </ms> <ms> = </ms> <ms> 0 </ms> </list> </apply> <ms> &#8734; </ms> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> TagBox </ci> <apply> <ci> SubscriptBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> FractionBox </ci> <ms> 1 </ms> <ms> 2 </ms> </apply> <ms> ) </ms> </list> </apply> <ms> k </ms> </apply> <ci> Pochhammer </ci> </apply> <apply> <ci> TagBox </ci> <apply> <ci> SubscriptBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> FractionBox </ci> <ms> 3 </ms> <ms> 2 </ms> </apply> <ms> ) </ms> </list> </apply> <ms> k </ms> </apply> <ci> Pochhammer </ci> </apply> </list> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> k </ms> <ms> ! </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> k </ms> <ms> + </ms> <ms> 1 </ms> </list> </apply> <ms> ) </ms> </list> </apply> <ms> ! </ms> </list> </apply> </list> </apply> </apply> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> - </ms> <ms> 2 </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> TagBox </ci> <ms> &#968; </ms> <ci> PolyGamma </ci> </apply> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> k </ms> <ms> + </ms> <ms> 1 </ms> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> <ms> + </ms> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> TagBox </ci> <ms> &#968; </ms> <ci> PolyGamma </ci> </apply> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> k </ms> <ms> + </ms> <apply> <ci> FractionBox </ci> <ms> 1 </ms> <ms> 2 </ms> </apply> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> <ms> + </ms> <apply> <ci> FractionBox </ci> <ms> 1 </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <apply> <ci> SuperscriptBox </ci> <ms> k </ms> <ms> 2 </ms> </apply> </list> </apply> <ms> + </ms> <apply> <ci> RowBox </ci> <list> <ms> 3 </ms> <ms> k </ms> </list> </apply> <ms> + </ms> <ms> 1 </ms> </list> </apply> </apply> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> SuperscriptBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> 1 </ms> <ms> - </ms> <ms> m </ms> </list> </apply> <ms> ) </ms> </list> </apply> <ms> k </ms> </apply> </list> </apply> </list> </apply> </list> </apply> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> </list> </apply> </list> </apply> <ci> TraditionalForm </ci> </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[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["Round", "[", FractionBox[RowBox[List["Re", "[", "z", "]"]], "\[Pi]"], "]"]]], " ", RowBox[List["Sin", "[", "z", "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "\[Infinity]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["-", "j"]]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], ",", "j"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List["j", ",", "i"]], "]"]], " ", SuperscriptBox[RowBox[List["Csc", "[", "z", "]"]], RowBox[List["2", " ", "i"]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["m", "-", "1"]], ")"]], "j"]]], RowBox[List[RowBox[List["(", RowBox[List["1", "-", RowBox[List["2", " ", "i"]]]], ")"]], " ", RowBox[List["j", "!"]], " ", RowBox[List["i", "!"]]]]]]]]]]], "+", RowBox[List["2", " ", RowBox[List["Round", "[", FractionBox[RowBox[List["Re", "[", "z", "]"]], "\[Pi]"], "]"]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["Log", "[", RowBox[List["1", "-", "m"]], "]"]], " ", RowBox[List["(", RowBox[List["m", "-", "1"]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["3", "2"], ",", "k"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "m"]], ")"]], "k"]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["1", "+", "k"]], ")"]], "!"]]]]]]]]], "+", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["m", "-", "1"]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["3", "2"], ",", "k"]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[FractionBox["1", RowBox[List["1", "+", RowBox[List["3", " ", "k"]], "+", RowBox[List["2", " ", SuperscriptBox["k", "2"]]]]]], "+", RowBox[List["2", " ", RowBox[List["PolyGamma", "[", RowBox[List[FractionBox["1", "2"], "+", "k"]], "]"]]]], "-", RowBox[List["2", " ", RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "k"]], "]"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "m"]], ")"]], "k"]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["1", "+", "k"]], ")"]], "!"]]]]]]]]]]], ")"]]]]]]]]]]


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





2007-05-02