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http://functions.wolfram.com/07.36.04.0006.01
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Singularities[AppellF1[a, Subscript[b, 1], Subscript[b, 2], c,
Subscript[z, 1], Subscript[z, 2]], Subscript[z, k]] ==
{{ComplexInfinity, -\[Alpha]}} /;
(Element[-a, Integers] && -a > 0 && \[Alpha] == a) ||
(Element[-Subscript[b, k], Integers] && -Subscript[b, k] > 0 &&
\[Alpha] == Subscript[b, k]) || ((Element[-a, Integers] && -a > 0 &&
Element[-Subscript[b, k], Integers] && -Subscript[b, k] > 0 &&
\[Alpha] == Min[-a, -Subscript[b, k]]) && 1 <= k <= 2)
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Singularities", "[", RowBox[List[RowBox[List["AppellF1", "[", RowBox[List["a", ",", SubscriptBox["b", "1"], ",", SubscriptBox["b", "2"], ",", "c", ",", SubscriptBox["z", "1"], ",", SubscriptBox["z", "2"]]], "]"]], ",", SubscriptBox["z", "k"]]], "]"]], "\[Equal]", RowBox[List["{", RowBox[List["{", RowBox[List["ComplexInfinity", ",", RowBox[List["-", "\[Alpha]"]]]], "}"]], "}"]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "a"]], "\[Element]", "Integers"]], "\[And]", RowBox[List[RowBox[List["-", "a"]], ">", "0"]], "\[And]", RowBox[List["\[Alpha]", "\[Equal]", "a"]]]], ")"]], "\[Or]", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", SubscriptBox["b", "k"]]], "\[Element]", "Integers"]], "\[And]", RowBox[List[RowBox[List["-", SubscriptBox["b", "k"]]], ">", "0"]], "\[And]", RowBox[List["\[Alpha]", "\[Equal]", SubscriptBox["b", "k"]]]]], ")"]], "\[Or]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "a"]], "\[Element]", "Integers"]], "\[And]", RowBox[List[RowBox[List["-", "a"]], ">", "0"]], "\[And]", RowBox[List[RowBox[List["-", SubscriptBox["b", "k"]]], "\[Element]", "Integers"]], "\[And]", RowBox[List[RowBox[List["-", SubscriptBox["b", "k"]]], ">", "0"]], "\[And]", RowBox[List["\[Alpha]", "\[Equal]", RowBox[List["Min", "[", RowBox[List[RowBox[List["-", "a"]], ",", RowBox[List["-", SubscriptBox["b", "k"]]]]], "]"]]]]]], ")"]], "\[And]", RowBox[List["1", "\[LessEqual]", "k", "\[LessEqual]", "2"]]]]]]]]]]
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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <msub> <mi> 𝒮𝒾𝓃ℊ </mi> <msub> <mi> z </mi> <mi> k </mi> </msub> </msub> <mo> ( </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> <mi> a </mi> <mo> ; </mo> <msub> <mi> b </mi> <mn> 1 </mn> </msub> </mrow> <mo> , </mo> <mrow> <msub> <mi> b </mi> <mn> 2 </mn> </msub> <mo> ; </mo> <mi> c </mi> <mo> ; </mo> <msub> <mi> z </mi> <mn> 1 </mn> </msub> </mrow> <mo> , </mo> <msub> <mi> z </mi> <mn> 2 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mo> { </mo> <mrow> <mo> { </mo> <mrow> <mover> <mi> ∞ </mi> <mo> ~ </mo> </mover> <mo> , </mo> <mrow> <mo> - </mo> <mi> α </mi> </mrow> </mrow> <mo> } </mo> </mrow> <mo> } </mo> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mi> a </mi> </mrow> <mo> ∈ </mo> <msup> <mi> ℕ </mi> <mo> + </mo> </msup> </mrow> <mo> ∧ </mo> <mrow> <mi> α </mi> <mo> ⩵ </mo> <mi> a </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ∨ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <msub> <mi> b </mi> <mi> k </mi> </msub> </mrow> <mo> ∈ </mo> <msup> <mi> ℕ </mi> <mo> + </mo> </msup> </mrow> <mo> ∧ </mo> <mrow> <mi> α </mi> <mo> ⩵ </mo> <msub> <mi> b </mi> <mi> k </mi> </msub> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ∨ </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mi> a </mi> </mrow> <mo> ∈ </mo> <msup> <mi> ℕ </mi> <mo> + </mo> </msup> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mo> - </mo> <msub> <mi> b </mi> <mi> k </mi> </msub> </mrow> <mo> ∈ </mo> <msup> <mi> ℕ </mi> <mo> + </mo> </msup> </mrow> <mo> ∧ </mo> <mrow> <mi> α </mi> <mo> ⩵ </mo> <mrow> <mi> min </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> a </mi> </mrow> <mo> , </mo> <mrow> <mo> - </mo> <msub> <mi> b </mi> <mi> k </mi> </msub> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ∧ </mo> <mrow> <mn> 1 </mn> <mo> ≤ </mo> <mi> k </mi> <mo> ≤ </mo> <mn> 2 </mn> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <apply> <ci> Subscript </ci> <ci> 𝒮𝒾𝓃ℊ </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <ci> k </ci> </apply> </apply> <apply> <ci> AppellF1 </ci> <ci> a </ci> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <ci> c </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <list> <list> <apply> <ci> OverTilde </ci> <infinity /> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> α </ci> </apply> </list> </list> </apply> <apply> <or /> <apply> <and /> <apply> <in /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> <apply> <ci> SuperPlus </ci> <ci> ℕ </ci> </apply> </apply> <apply> <eq /> <ci> α </ci> <ci> a </ci> </apply> </apply> <apply> <and /> <apply> <in /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> b </ci> <ci> k </ci> </apply> </apply> <apply> <ci> SuperPlus </ci> <ci> ℕ </ci> </apply> </apply> <apply> <eq /> <ci> α </ci> <apply> <ci> Subscript </ci> <ci> b </ci> <ci> k </ci> </apply> </apply> </apply> <apply> <and /> <apply> <and /> <apply> <in /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> <apply> <ci> SuperPlus </ci> <ci> ℕ </ci> </apply> </apply> <apply> <in /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> b </ci> <ci> k </ci> </apply> </apply> <apply> <ci> SuperPlus </ci> <ci> ℕ </ci> </apply> </apply> <apply> <eq /> <ci> α </ci> <apply> <min /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> b </ci> <ci> k </ci> </apply> </apply> </apply> </apply> </apply> <apply> <leq /> <cn type='integer'> 1 </cn> <ci> k </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Singularities", "[", RowBox[List[RowBox[List["AppellF1", "[", RowBox[List["a_", ",", SubscriptBox["b_", "1"], ",", SubscriptBox["b_", "2"], ",", "c_", ",", SubscriptBox["z_", "1"], ",", SubscriptBox["z_", "2"]]], "]"]], ",", SubscriptBox["z_", "k"]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["{", RowBox[List["{", RowBox[List["ComplexInfinity", ",", RowBox[List["-", "\[Alpha]"]]]], "}"]], "}"]], "/;", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "a"]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List["-", "a"]], ">", "0"]], "&&", RowBox[List["\[Alpha]", "\[Equal]", "a"]]]], ")"]], "||", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", SubscriptBox["b", "k"]]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List["-", SubscriptBox["b", "k"]]], ">", "0"]], "&&", RowBox[List["\[Alpha]", "\[Equal]", SubscriptBox["b", "k"]]]]], ")"]], "||", RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "a"]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List["-", "a"]], ">", "0"]], "&&", RowBox[List[RowBox[List["-", SubscriptBox["b", "k"]]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List["-", SubscriptBox["b", "k"]]], ">", "0"]], "&&", RowBox[List["\[Alpha]", "\[Equal]", RowBox[List["Min", "[", RowBox[List[RowBox[List["-", "a"]], ",", RowBox[List["-", SubscriptBox["b", "k"]]]]], "]"]]]]]], ")"]], "&&", RowBox[List["1", "\[LessEqual]", "k", "\[LessEqual]", "2"]]]], ")"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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