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http://functions.wolfram.com/09.29.06.0005.01
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JacobiDN[z, m] \[Proportional]
((-1)^(s - 1) I)/(z - 2 r EllipticK[m] - I (2 s + 1) EllipticK[1 - m]) +
O[1] /; (z -> 2 r EllipticK[m] + I (2 s + 1) EllipticK[1 - m]) &&
Element[{r, s}, Integers]
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["JacobiDN", "[", RowBox[List["z", ",", "m"]], "]"]], "\[Proportional]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["s", "-", "1"]]], "\[ImaginaryI]"]], RowBox[List["z", "-", RowBox[List["2", "r", " ", RowBox[List["EllipticK", "[", "m", "]"]]]], "-", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "+", "1"]], ")"]], RowBox[List["EllipticK", "[", RowBox[List["1", "-", "m"]], "]"]]]]]]], "+", RowBox[List["O", "[", "1", "]"]]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List["z", "\[Rule]", RowBox[List[RowBox[List["2", "r", " ", RowBox[List["EllipticK", "[", "m", "]"]]]], "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "+", "1"]], ")"]], RowBox[List["EllipticK", "[", RowBox[List["1", "-", "m"]], "]"]]]]]]]], ")"]], "\[And]", RowBox[List[RowBox[List["{", RowBox[List["r", ",", "s"]], "}"]], "\[Element]", "Integers"]]]]]]]]
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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> <mi> dn </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> ∝ </mo> <mrow> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> s </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ⁢ </mo> <mi> ⅈ </mi> </mrow> <mrow> <mi> z </mi> <mo> - </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> s </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> K </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> r </mi> <mo> ⁢ </mo> <mrow> <mi> K </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> </mrow> </mrow> </mfrac> <mo> + </mo> <mrow> <mi> O </mi> <mo> ⁡ </mo> <mo> ( </mo> <mn> 1 </mn> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <semantics> <mo> → </mo> <annotation encoding='Mathematica'> "\[Rule]" </annotation> </semantics> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> s </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mrow> <mi> K </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> r </mi> <mo> ⁢ </mo> <mrow> <mi> K </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mo> { </mo> <mrow> <mi> r </mi> <mo> , </mo> <mi> s </mi> </mrow> <mo> } </mo> </mrow> <mo> ∈ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] </annotation> </semantics> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <ci> Proportional </ci> <apply> <ci> JacobiDN </ci> <ci> z </ci> <ci> m </ci> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <plus /> <ci> s </ci> <cn type='integer'> -1 </cn> </apply> </apply> <imaginaryi /> <apply> <power /> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> s </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <ci> EllipticK </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> r </ci> <apply> <ci> EllipticK </ci> <ci> m </ci> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> O </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <and /> <apply> <ci> Rule </ci> <ci> z </ci> <apply> <plus /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> s </ci> </apply> <cn type='integer'> 1 </cn> </apply> <imaginaryi /> <apply> <ci> EllipticK </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> r </ci> <apply> <ci> EllipticK </ci> <ci> m </ci> </apply> </apply> </apply> </apply> <apply> <in /> <list> <ci> r </ci> <ci> s </ci> </list> <integers /> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["JacobiDN", "[", RowBox[List["z_", ",", "m_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["s", "-", "1"]]], " ", "\[ImaginaryI]"]], RowBox[List["z", "-", RowBox[List["2", " ", "r", " ", RowBox[List["EllipticK", "[", "m", "]"]]]], "-", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "+", "1"]], ")"]], " ", RowBox[List["EllipticK", "[", RowBox[List["1", "-", "m"]], "]"]]]]]]], "+", RowBox[List["O", "[", "1", "]"]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List["z", "\[Rule]", RowBox[List[RowBox[List["2", " ", "r", " ", RowBox[List["EllipticK", "[", "m", "]"]]]], "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "+", "1"]], ")"]], " ", RowBox[List["EllipticK", "[", RowBox[List["1", "-", "m"]], "]"]]]]]]]], ")"]], "&&", RowBox[List[RowBox[List["{", RowBox[List["r", ",", "s"]], "}"]], "\[Element]", "Integers"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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