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EllipticLog






Mathematica Notation

Traditional Notation









Elliptic Functions > EllipticLog[{z1,z2},{a,b}] > Primary definition





http://functions.wolfram.com/09.57.02.0001.01









  


  










Input Form





EllipticLog[{Subscript[z, 1], Subscript[z, 2]}, {a, b}] == (Sqrt[Subscript[z, 2]^2]/(2 Subscript[z, 2])) Integrate[1/Sqrt[t^3 + a t^2 + b t], {t, Infinity, Subscript[z, 1]}] /; Subscript[z, 1]^3 + a Subscript[z, 1]^2 + b Subscript[z, 1] - Subscript[z, 2]^2 == 0 && Element[a, Reals] && Element[b, Reals]










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["EllipticLog", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["z", "1"], ",", SubscriptBox["z", "2"]]], "}"]], ",", RowBox[List["{", RowBox[List["a", ",", "b"]], "}"]]]], "]"]], "\[Equal]", RowBox[List[FractionBox[SqrtBox[SubsuperscriptBox["z", "2", "2"]], RowBox[List["2", " ", SubscriptBox["z", "2"]]]], RowBox[List[SubsuperscriptBox["\[Integral]", "\[Infinity]", SubscriptBox["z", "1"]], RowBox[List[FractionBox["1", SqrtBox[RowBox[List[SuperscriptBox["t", "3"], "+", RowBox[List["a", " ", SuperscriptBox["t", "2"]]], "+", RowBox[List["b", " ", "t"]]]]]], RowBox[List["\[DifferentialD]", "t"]]]]]]]]]], "/;", " ", RowBox[List[RowBox[List[RowBox[List[SubsuperscriptBox["z", "1", "3"], "+", RowBox[List["a", " ", SubsuperscriptBox["z", "1", "2"]]], "+", RowBox[List["b", " ", SubscriptBox["z", "1"]]], "-", SubsuperscriptBox["z", "2", "2"]]], "\[Equal]", "0"]], "\[And]", RowBox[List["a", "\[Element]", "Reals"]], "\[And]", RowBox[List["b", "\[Element]", "Reals"]]]]]]]]










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> elog </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <msub> <mi> z </mi> <mn> 1 </mn> </msub> <mo> , </mo> <mrow> <msub> <mi> z </mi> <mn> 2 </mn> </msub> <mo> ; </mo> <mi> a </mi> </mrow> <mo> , </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <mrow> <msqrt> <msubsup> <mi> z </mi> <mn> 2 </mn> <mn> 2 </mn> </msubsup> </msqrt> <mtext> </mtext> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msub> <mi> z </mi> <mn> 2 </mn> </msub> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <msubsup> <mo> &#8747; </mo> <mi> &#8734; </mi> <msub> <mi> z </mi> <mn> 1 </mn> </msub> </msubsup> <mrow> <mfrac> <mn> 1 </mn> <msqrt> <mrow> <msup> <mi> t </mi> <mn> 3 </mn> </msup> <mo> + </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <msup> <mi> t </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mi> t </mi> </mrow> </mrow> </msqrt> </mfrac> <mo> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> t </mi> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <msubsup> <mi> z </mi> <mn> 1 </mn> <mn> 3 </mn> </msubsup> <mo> + </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <msubsup> <mi> z </mi> <mn> 1 </mn> <mn> 2 </mn> </msubsup> </mrow> <mo> + </mo> <mrow> <mi> b </mi> <mo> &#8290; </mo> <msub> <mi> z </mi> <mn> 1 </mn> </msub> </mrow> <mo> - </mo> <msubsup> <mi> z </mi> <mn> 2 </mn> <mn> 2 </mn> </msubsup> </mrow> <mo> &#10869; </mo> <mn> 0 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mi> a </mi> <mo> &#8712; </mo> <semantics> <mi> &#8477; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalR]&quot;, Function[Reals]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mi> b </mi> <mo> &#8712; </mo> <semantics> <mi> &#8477; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalR]&quot;, Function[Reals]] </annotation> </semantics> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> elog </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> CompoundExpression </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <ci> a </ci> </apply> <ci> b </ci> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <apply> <power /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <int /> <bvar> <ci> t </ci> </bvar> <lowlimit> <infinity /> </lowlimit> <uplimit> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> </uplimit> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> t </ci> <cn type='integer'> 3 </cn> </apply> <apply> <times /> <ci> a </ci> <apply> <power /> <ci> t </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <ci> b </ci> <ci> t </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <eq /> <apply> <plus /> <apply> <power /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 3 </cn> </apply> <apply> <times /> <ci> a </ci> <apply> <power /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <ci> b </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='integer'> 0 </cn> </apply> <apply> <in /> <ci> a </ci> <reals /> </apply> <apply> <in /> <ci> b </ci> <reals /> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["EllipticLog", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["z_", "1"], ",", SubscriptBox["z_", "2"]]], "}"]], ",", RowBox[List["{", RowBox[List["a_", ",", "b_"]], "}"]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[SqrtBox[SubsuperscriptBox["zz", "2", "2"]], " ", RowBox[List[SubsuperscriptBox["\[Integral]", "\[Infinity]", SubscriptBox["zz", "1"]], RowBox[List[FractionBox["1", SqrtBox[RowBox[List[SuperscriptBox["t", "3"], "+", RowBox[List["a", " ", SuperscriptBox["t", "2"]]], "+", RowBox[List["b", " ", "t"]]]]]], RowBox[List["\[DifferentialD]", "t"]]]]]]]], RowBox[List["2", " ", SubscriptBox["zz", "2"]]]], "/;", RowBox[List[RowBox[List[RowBox[List[SubsuperscriptBox["zz", "1", "3"], "+", RowBox[List["a", " ", SubsuperscriptBox["zz", "1", "2"]]], "+", RowBox[List["b", " ", SubscriptBox["zz", "1"]]], "-", SubsuperscriptBox["zz", "2", "2"]]], "\[Equal]", "0"]], "&&", RowBox[List["a", "\[Element]", "Reals"]], "&&", RowBox[List["b", "\[Element]", "Reals"]]]]]]]]]]










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





2001-10-29