EllipticLog[{Subscript[z, 1], Subscript[z, 2]}, {a, b}] == (-(Sqrt[Subscript[z, 2]^2]/(Sqrt[Subscript[z, 1]] Subscript[z, 2]))) AppellF1[1/2, 1/2, 1/2, 3/2, (-a - Sqrt[a^2 - 4 b])/(2 Subscript[z, 1]), (-a + Sqrt[a^2 - 4 b])/(2 Subscript[z, 1])] /; Subscript[z, 1]^3 + a Subscript[z, 1]^2 + b Subscript[z, 1] - Subscript[z, 2]^2 == 0

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[RowBox[List["-", FractionBox[SqrtBox[SubsuperscriptBox["z", "2", "2"]], RowBox[List[SqrtBox[SubscriptBox["z", "1"]], " ", SubscriptBox["z", "2"]]]]]], RowBox[List["AppellF1", "[", RowBox[List[FractionBox["1", "2"], ",", FractionBox["1", "2"], ",", FractionBox["1", "2"], ",", FractionBox["3", "2"], ",", FractionBox[RowBox[List[RowBox[List["-", "a"]], "-", SqrtBox[RowBox[List[SuperscriptBox["a", "2"], "-", RowBox[List["4", " ", "b"]]]]]]], RowBox[List["2", " ", SubscriptBox["z", "1"]]]], ",", FractionBox[RowBox[List[RowBox[List["-", "a"]], "+", SqrtBox[RowBox[List[SuperscriptBox["a", "2"], "-", RowBox[List["4", " ", "b"]]]]]]], RowBox[List["2", " ", SubscriptBox["z", "1"]]]]]], "]"]]]]]], "/;", 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"]]]]]]

<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> <mrow> <mo> - </mo> <mfrac> <msqrt> <msubsup> <mi> z </mi> <mn> 2 </mn> <mn> 2 </mn> </msubsup> </msqrt> <mrow> <msqrt> <msub> <mi> z </mi> <mn> 1 </mn> </msub> </msqrt> <mo> &#8290; </mo> <msub> <mi> z </mi> <mn> 2 </mn> </msub> </mrow> </mfrac> </mrow> <mo> &#8290; </mo> <mrow> <semantics> <msub> <mi> F </mi> <mn> 1 </mn> </msub> <annotation-xml encoding='MathML-Content'> <ci> AppellF1 </ci> </annotation-xml> </semantics> <mo> ( </mo> <mrow> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ; </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> , </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ; </mo> <mfrac> <mn> 3 </mn> <mn> 2 </mn> </mfrac> <mo> ; </mo> <mfrac> <mrow> <mrow> <mo> - </mo> <mi> a </mi> </mrow> <mo> - </mo> <msqrt> <mrow> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> b </mi> </mrow> </mrow> </msqrt> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msub> <mi> z </mi> <mn> 1 </mn> </msub> </mrow> </mfrac> </mrow> <mo> , </mo> <mfrac> <mrow> <msqrt> <mrow> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> b </mi> </mrow> </mrow> </msqrt> <mo> - </mo> <mi> a </mi> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msub> <mi> z </mi> <mn> 1 </mn> </msub> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <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> </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 /> <cn type='integer'> -1 </cn> <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 /> <apply> <power /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <ci> AppellF1 </ci> <cn type='rational'> 1 <sep /> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> <cn type='rational'> 3 <sep /> 2 </cn> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 4 </cn> <ci> b </ci> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 4 </cn> <ci> b </ci> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <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> </annotation-xml> </semantics> </math>

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[RowBox[List["-", FractionBox[RowBox[List[SqrtBox[SubsuperscriptBox["zz", "2", "2"]], " ", RowBox[List["AppellF1", "[", RowBox[List[FractionBox["1", "2"], ",", FractionBox["1", "2"], ",", FractionBox["1", "2"], ",", FractionBox["3", "2"], ",", FractionBox[RowBox[List[RowBox[List["-", "a"]], "-", SqrtBox[RowBox[List[SuperscriptBox["a", "2"], "-", RowBox[List["4", " ", "b"]]]]]]], RowBox[List["2", " ", SubscriptBox["zz", "1"]]]], ",", FractionBox[RowBox[List[RowBox[List["-", "a"]], "+", SqrtBox[RowBox[List[SuperscriptBox["a", "2"], "-", RowBox[List["4", " ", "b"]]]]]]], RowBox[List["2", " ", SubscriptBox["zz", "1"]]]]]], "]"]]]], RowBox[List[SqrtBox[SubscriptBox["zz", "1"]], " ", SubscriptBox["zz", "2"]]]]]], "/;", 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"]]]]]]]]

2001-10-29