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variants of this functions
EllipticE




Mathematica Notation

Traditional Notation









Elliptic Integrals > EllipticE[z] > Series representations > Generalized power series > Expansions at z==infinity > For the function itself





http://functions.wolfram.com/08.01.06.0010.02









  


  








Input Form





EllipticE[z] == Sqrt[-z] + (Log[-z]/Sqrt[-z]) Sum[((1/(k! (1 + k)!)) Pochhammer[-(1/2), 1 + k]^2)/z^k, {k, 0, Infinity}] + (1/Sqrt[-z]) Sum[((1/(k! (1 + k)!)) Pochhammer[-(1/2), 1 + k]^2 (1/(1 + k) - 2 PolyGamma[1/2 + k] + 2 PolyGamma[k + 1]))/z^k, {k, 0, Infinity}] /; Abs[z] > 1








Standard Form





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MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <mi> E </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <msqrt> <mrow> <mo> - </mo> <mi> z </mi> </mrow> </msqrt> <mo> + </mo> <mrow> <mfrac> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <msqrt> <mrow> <mo> - </mo> <mi> z </mi> </mrow> </msqrt> </mfrac> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mfrac> <mrow> <msup> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[&quot;-&quot;, FractionBox[&quot;1&quot;, &quot;2&quot;]]], &quot;)&quot;]], RowBox[List[&quot;k&quot;, &quot;+&quot;, &quot;1&quot;]]], Pochhammer] </annotation> </semantics> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mo> - </mo> <mi> k </mi> </mrow> </msup> </mrow> <mrow> <mrow> <mi> k </mi> <mo> ! 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Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["EllipticE", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SqrtBox[RowBox[List["-", "z"]]], "+", FractionBox[RowBox[List[RowBox[List["Log", "[", RowBox[List["-", "z"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], ",", RowBox[List["1", "+", "k"]]]], "]"]], "2"], " ", SuperscriptBox["z", RowBox[List["-", "k"]]]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["1", "+", "k"]], ")"]], "!"]]]]]]]]], SqrtBox[RowBox[List["-", "z"]]]], "+", FractionBox[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], ",", RowBox[List["1", "+", "k"]]]], "]"]], "2"], " ", RowBox[List["(", RowBox[List[FractionBox["1", RowBox[List["1", "+", "k"]]], "-", RowBox[List["2", " ", RowBox[List["PolyGamma", "[", RowBox[List[FractionBox["1", "2"], "+", "k"]], "]"]]]], "+", RowBox[List["2", " ", RowBox[List["PolyGamma", "[", RowBox[List["k", "+", "1"]], "]"]]]]]], ")"]], " ", SuperscriptBox["z", RowBox[List["-", "k"]]]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["1", "+", "k"]], ")"]], "!"]]]]]]], SqrtBox[RowBox[List["-", "z"]]]]]], "/;", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], ">", "1"]]]]]]]]


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





2001-10-29