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EllipticK




Mathematica Notation

Traditional Notation









Elliptic Integrals > EllipticK[z] > Series representations > Generalized power series > Expansions at z==1 > For the function itself





http://functions.wolfram.com/08.02.06.0028.01









  


  








Input Form





EllipticK[z] \[Proportional] (-(1/2)) Log[1 - z] (1 - (z - 1)/4 + (9/64) (z - 1)^2 + O[(z - 1)^3]) + Log[4] + ((1 - Log[4])/4) (z - 1) + (3/128) (-7 + 6 Log[4]) (z - 1)^2 + O[(z - 1)^3]








Standard Form





Cell[BoxData[RowBox[List[RowBox[List["EllipticK", "[", "z", "]"]], "\[Proportional]", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", RowBox[List["Log", "[", RowBox[List["1", "-", "z"]], "]"]], RowBox[List["(", RowBox[List["1", "-", FractionBox[RowBox[List["z", "-", "1"]], "4"], "+", RowBox[List[FractionBox["9", "64"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], "2"]]], "+", RowBox[List["O", "[", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], "3"], "]"]]]], ")"]]]], "+", RowBox[List["Log", "[", "4", "]"]], "+", RowBox[List[FractionBox[RowBox[List["1", "-", RowBox[List["Log", "[", "4", "]"]]]], "4"], RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]]]], "+", RowBox[List[FractionBox["3", "128"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "7"]], "+", RowBox[List["6", " ", RowBox[List["Log", "[", "4", "]"]]]]]], ")"]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], "2"]]], "+", RowBox[List["O", "[", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], "3"], "]"]]]]]]]]








MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mi> K </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#8733; </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> &#8290; </mo> <mtext> </mtext> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <mrow> <mi> z </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mn> 4 </mn> </mfrac> <mo> + </mo> <mrow> <mfrac> <mn> 9 </mn> <mn> 64 </mn> </mfrac> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mi> O </mi> <mo> &#8289; </mo> <mo> ( </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mn> 4 </mn> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mn> 4 </mn> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mn> 3 </mn> <mn> 128 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 6 </mn> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mn> 4 </mn> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mn> 7 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mi> O </mi> <mo> &#8289; </mo> <mo> ( </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Proportional </ci> <apply> <ci> EllipticK </ci> <ci> z </ci> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <ln /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='rational'> 9 <sep /> 64 </cn> <apply> <power /> <apply> <plus /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <ci> O </ci> <apply> <power /> <apply> <plus /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> 3 </cn> </apply> </apply> </apply> </apply> <apply> <ln /> <cn type='integer'> 4 </cn> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 4 </cn> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ln /> <cn type='integer'> 4 </cn> </apply> </apply> </apply> <apply> <plus /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='rational'> 3 <sep /> 128 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 6 </cn> <apply> <ln /> <cn type='integer'> 4 </cn> </apply> </apply> <cn type='integer'> -7 </cn> </apply> <apply> <power /> <apply> <plus /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <ci> O </ci> <apply> <power /> <apply> <plus /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> 3 </cn> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["EllipticK", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", RowBox[List["Log", "[", RowBox[List["1", "-", "z"]], "]"]], " ", RowBox[List["(", RowBox[List["1", "-", FractionBox[RowBox[List["z", "-", "1"]], "4"], "+", RowBox[List[FractionBox["9", "64"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], "2"]]], "+", SuperscriptBox[RowBox[List["O", "[", RowBox[List["z", "-", "1"]], "]"]], "3"]]], ")"]]]], "+", RowBox[List["Log", "[", "4", "]"]], "+", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["Log", "[", "4", "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]]]], "+", RowBox[List[FractionBox["3", "128"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "7"]], "+", RowBox[List["6", " ", RowBox[List["Log", "[", "4", "]"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], "2"]]], "+", SuperscriptBox[RowBox[List["O", "[", RowBox[List["z", "-", "1"]], "]"]], "3"]]]]]]]


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





2007-05-02