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variants of this functions
JacobiP






Mathematica Notation

Traditional Notation









Hypergeometric Functions > JacobiP[nu,a,b,z] > General characteristics > Poles and essential singularities > With respect to z





http://functions.wolfram.com/07.15.04.0006.01









  


  










Input Form





Singularities[JacobiP[\[Nu], a, b, z], z] == {{ComplexInfinity, -\[Alpha]}} /; (Element[\[Nu], Integers] && \[Nu] > 0 && \[Alpha] == -\[Nu]) || (Element[-a - b - \[Nu] - 1, Integers] && -a - b - \[Nu] - 1 > 0 && \[Alpha] == a + b + \[Nu] + 1) || (Element[\[Nu], Integers] && \[Nu] > 0 && Element[-a - b - \[Nu] - 1, Integers] && -a - b - \[Nu] - 1 > 0 && \[Alpha] == Min[\[Nu], -a - b - \[Nu] - 1])










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Singularities", "[", RowBox[List[RowBox[List["JacobiP", "[", RowBox[List["\[Nu]", ",", "a", ",", "b", ",", "z"]], "]"]], ",", "z"]], "]"]], "\[Equal]", RowBox[List["{", RowBox[List["{", RowBox[List["ComplexInfinity", ",", RowBox[List["-", "\[Alpha]"]]]], "}"]], "}"]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["\[Nu]", "\[Element]", "Integers"]], "\[And]", RowBox[List["\[Nu]", ">", "0"]], "\[And]", RowBox[List["\[Alpha]", "\[Equal]", RowBox[List["-", "\[Nu]"]]]]]], ")"]], "\[Or]", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List[RowBox[List["-", "a"]], "-", "b", "-", "\[Nu]", "-", "1"]], "\[Element]", "Integers"]], "\[And]", RowBox[List[RowBox[List[RowBox[List["-", "a"]], "-", "b", "-", "\[Nu]", "-", "1"]], ">", "0"]], "\[And]", RowBox[List["\[Alpha]", "\[Equal]", RowBox[List["a", "+", "b", "+", "\[Nu]", "+", "1"]]]]]], ")"]], "\[Or]", RowBox[List["(", RowBox[List[RowBox[List["\[Nu]", "\[Element]", "Integers"]], "\[And]", RowBox[List["\[Nu]", ">", "0"]], "\[And]", RowBox[List[RowBox[List[RowBox[List["-", "a"]], "-", "b", "-", "\[Nu]", "-", "1"]], "\[Element]", "Integers"]], "\[And]", RowBox[List[RowBox[List[RowBox[List["-", "a"]], "-", "b", "-", "\[Nu]", "-", "1"]], ">", "0"]], "\[And]", RowBox[List["\[Alpha]", "\[Equal]", RowBox[List["Min", "[", RowBox[List["\[Nu]", ",", RowBox[List[RowBox[List["-", "a"]], "-", "b", "-", "\[Nu]", "-", "1"]]]], "]"]]]]]], ")"]]]]]]]]










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> <msub> <mi> &#119982;&#119998;&#120003;&#8458; </mi> <mi> z </mi> </msub> <mo> ( </mo> <mrow> <msubsup> <mi> P </mi> <mi> &#957; </mi> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> , </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> </msubsup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mo> { </mo> <mrow> <mo> { </mo> <mrow> <mover> <mi> &#8734; </mi> <mo> ~ </mo> </mover> <mo> , </mo> <mrow> <mo> - </mo> <mi> &#945; </mi> </mrow> </mrow> <mo> } </mo> </mrow> <mo> } </mo> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> &#957; </mi> <mo> &#8712; </mo> <msup> <mi> &#8469; </mi> <mo> + </mo> </msup> </mrow> <mo> &#8743; </mo> <mrow> <mi> &#945; </mi> <mo> &#10869; </mo> <mrow> <mo> - </mo> <mi> &#957; </mi> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8744; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mrow> <mo> - </mo> <mi> a </mi> </mrow> <mo> - </mo> <mi> b </mi> <mo> - </mo> <mi> &#957; </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> &#8712; </mo> <msup> <mi> &#8469; </mi> <mo> + </mo> </msup> </mrow> <mo> &#8743; </mo> <mrow> <mi> &#945; </mi> <mo> &#10869; </mo> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> <mo> + </mo> <mi> &#957; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8744; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> &#957; </mi> <mo> &#8712; </mo> <msup> <mi> &#8469; </mi> <mo> + </mo> </msup> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mi> a </mi> </mrow> <mo> - </mo> <mi> b </mi> <mo> - </mo> <mi> &#957; </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> &#8712; </mo> <msup> <mi> &#8469; </mi> <mo> + </mo> </msup> </mrow> <mo> &#8743; </mo> <mrow> <mi> &#945; </mi> <mo> &#10869; </mo> <mrow> <mi> min </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> &#957; </mi> <mo> , </mo> <mrow> <mrow> <mo> - </mo> <mi> a </mi> </mrow> <mo> - </mo> <mi> b </mi> <mo> - </mo> <mi> &#957; </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <apply> <ci> Subscript </ci> <ci> &#119982;&#119998;&#120003;&#8458; </ci> <ci> z </ci> </apply> <apply> <ci> JacobiP </ci> <ci> &#957; </ci> <ci> a </ci> <ci> b </ci> <ci> z </ci> </apply> </apply> <list> <list> <apply> <ci> OverTilde </ci> <infinity /> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#945; </ci> </apply> </list> </list> </apply> <apply> <or /> <apply> <and /> <apply> <in /> <ci> &#957; </ci> <apply> <ci> SuperPlus </ci> <ci> &#8469; </ci> </apply> </apply> <apply> <eq /> <ci> &#945; </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <ci> SuperPlus </ci> <ci> &#8469; </ci> </apply> </apply> <apply> <eq /> <ci> &#945; </ci> <apply> <plus /> <ci> a </ci> <ci> b </ci> <ci> &#957; </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <ci> &#957; </ci> <apply> <ci> SuperPlus </ci> <ci> &#8469; </ci> </apply> </apply> <apply> <in /> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <ci> SuperPlus </ci> <ci> &#8469; </ci> </apply> </apply> <apply> <eq /> <ci> &#945; </ci> <apply> <min /> <ci> &#957; </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Singularities", "[", RowBox[List[RowBox[List["JacobiP", "[", RowBox[List["\[Nu]_", ",", "a_", ",", "b_", ",", "z_"]], "]"]], ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["{", RowBox[List["{", RowBox[List["ComplexInfinity", ",", RowBox[List["-", "\[Alpha]"]]]], "}"]], "}"]], "/;", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["\[Nu]", "\[Element]", "Integers"]], "&&", RowBox[List["\[Nu]", ">", "0"]], "&&", RowBox[List["\[Alpha]", "\[Equal]", RowBox[List["-", "\[Nu]"]]]]]], ")"]], "||", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List[RowBox[List["-", "a"]], "-", "b", "-", "\[Nu]", "-", "1"]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List[RowBox[List["-", "a"]], "-", "b", "-", "\[Nu]", "-", "1"]], ">", "0"]], "&&", RowBox[List["\[Alpha]", "\[Equal]", RowBox[List["a", "+", "b", "+", "\[Nu]", "+", "1"]]]]]], ")"]], "||", RowBox[List["(", RowBox[List[RowBox[List["\[Nu]", "\[Element]", "Integers"]], "&&", RowBox[List["\[Nu]", ">", "0"]], "&&", RowBox[List[RowBox[List[RowBox[List["-", "a"]], "-", "b", "-", "\[Nu]", "-", "1"]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List[RowBox[List["-", "a"]], "-", "b", "-", "\[Nu]", "-", "1"]], ">", "0"]], "&&", RowBox[List["\[Alpha]", "\[Equal]", RowBox[List["Min", "[", RowBox[List["\[Nu]", ",", RowBox[List[RowBox[List["-", "a"]], "-", "b", "-", "\[Nu]", "-", "1"]]]], "]"]]]]]], ")"]]]]]]]]]]










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





2001-10-29





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