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http://functions.wolfram.com/09.35.16.0032.01
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JacobiSD[z + (4 r + 1) EllipticK[m] + I (4 s - 1) EllipticK[1 - m], m] ==
(I/(Sqrt[m] Sqrt[1 - m])) JacobiDS[z, m] /; Element[{r, s}, Integers]
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["JacobiSD", "[", RowBox[List[RowBox[List["z", "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["4", "r"]], "+", "1"]], ")"]], RowBox[List["EllipticK", "[", "m", "]"]]]], "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["4", "s"]], "-", "1"]], ")"]], RowBox[List["EllipticK", "[", RowBox[List["1", "-", "m"]], "]"]]]]]], ",", "m"]], "]"]], "\[Equal]", RowBox[List[FractionBox["\[ImaginaryI]", RowBox[List[SqrtBox["m"], SqrtBox[RowBox[List["1", "-", "m"]]]]]], RowBox[List["JacobiDS", "[", RowBox[List["z", ",", "m"]], "]"]]]]]], "/;", 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> sd </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mi> z </mi> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 4 </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> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> r </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> K </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo>  </mo> <mrow> <mfrac> <mi> ⅈ </mi> <mrow> <msqrt> <mi> m </mi> </msqrt> <mo> ⁢ </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> m </mi> </mrow> </msqrt> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mi> ds </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </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[List[], Integers]] </annotation> </semantics> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> JacobiSD </ci> <apply> <plus /> <ci> z </ci> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </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 /> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> r </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <ci> EllipticK </ci> <ci> m </ci> </apply> </apply> </apply> <ci> m </ci> </apply> <apply> <times /> <apply> <times /> <imaginaryi /> <apply> <power /> <apply> <times /> <apply> <power /> <ci> m </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> JacobiDS </ci> <ci> z </ci> <ci> m </ci> </apply> </apply> </apply> <apply> <in /> <list> <ci> r </ci> <ci> s </ci> </list> <integers /> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["JacobiSD", "[", RowBox[List[RowBox[List["z_", "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["4", " ", "r_"]], "+", "1"]], ")"]], " ", RowBox[List["EllipticK", "[", "m_", "]"]]]], "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["4", " ", "s_"]], "-", "1"]], ")"]], " ", RowBox[List["EllipticK", "[", RowBox[List["1", "-", "m_"]], "]"]]]]]], ",", "m_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["JacobiDS", "[", RowBox[List["z", ",", "m"]], "]"]]]], RowBox[List[SqrtBox["m"], " ", SqrtBox[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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