|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
http://functions.wolfram.com/09.29.16.0039.01
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
n JacobiDN[n z, m] == (-1)^((1 - n)/2)
Sum[JacobiDN[z + (4 (r EllipticK[m] + s \[Tau] EllipticK[m]))/n, m],
{r, 0, n - 1}, {s, 0, n - 1}] /; Element[(n + 1)/2, Integers] && n > 0
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["n", " ", RowBox[List["JacobiDN", "[", RowBox[List[RowBox[List["n", " ", "z"]], ",", "m"]], "]"]]]], "\[Equal]", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], FractionBox[RowBox[List["1", "-", "n"]], "2"]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["r", "=", "0"]], RowBox[List["n", "-", "1"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["s", "=", "0"]], RowBox[List["n", "-", "1"]]], RowBox[List["JacobiDN", "[", RowBox[List[RowBox[List["z", "+", FractionBox[RowBox[List["4", RowBox[List["(", RowBox[List[RowBox[List["r", " ", RowBox[List["EllipticK", "[", "m", "]"]]]], "+", RowBox[List["s", " ", "\[Tau]", " ", RowBox[List["EllipticK", "[", "m", "]"]]]]]], ")"]]]], "n"]]], ",", "m"]], "]"]]]]]]]]]], "/;", RowBox[List[RowBox[List[FractionBox[RowBox[List["n", "+", "1"]], "2"], "\[Element]", "Integers"]], "\[And]", RowBox[List["n", ">", "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> n </mi> <mo> ⁢ </mo> <mrow> <mi> dn </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mi> n </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mfrac> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> n </mi> </mrow> <mn> 2 </mn> </mfrac> </msup> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> r </mi> <mo> , </mo> <mrow> <mi> s </mi> <mo> = </mo> <mn> 0 </mn> </mrow> </mrow> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <mi> dn </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mi> z </mi> <mo> + </mo> <mfrac> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mi> K </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> r </mi> </mrow> <mo> + </mo> <mrow> <mrow> <mi> K </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> s </mi> <mo> ⁢ </mo> <mi> τ </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mi> n </mi> </mfrac> </mrow> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mfrac> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> <mo> ∈ </mo> <msup> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] </annotation> </semantics> <mo> + </mo> </msup> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> FormBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> n </ms> <apply> <ci> RowBox </ci> <list> <ms> dn </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> n </ms> <ms> z </ms> </list> </apply> <ms> ❘ </ms> <ms> m </ms> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> <ms> ⩵ </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SuperscriptBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> - </ms> <ms> 1 </ms> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <ms> 1 </ms> <ms> - </ms> <ms> n </ms> </list> </apply> <ms> 2 </ms> </apply> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> ErrorBox </ci> <apply> <ci> UnderoverscriptBox </ci> <ms> ∑ </ms> <apply> <ci> RowBox </ci> <list> <ms> r </ms> <ms> , </ms> <apply> <ci> RowBox </ci> <list> <ms> s </ms> <ms> = </ms> <ms> 0 </ms> </list> </apply> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> n </ms> <ms> - </ms> <ms> 1 </ms> </list> </apply> </apply> </apply> <apply> <ci> RowBox </ci> <list> <ms> dn </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> z </ms> <ms> + </ms> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <ms> 4 </ms> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> K </ms> <ms> ( </ms> <ms> m </ms> <ms> ) </ms> </list> </apply> <ms> r </ms> </list> </apply> <ms> + </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> K </ms> <ms> ( </ms> <ms> m </ms> <ms> ) </ms> </list> </apply> <ms> s </ms> <ms> τ </ms> </list> </apply> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> <ms> n </ms> </apply> </list> </apply> <ms> ❘ </ms> <ms> m </ms> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> </list> </apply> </list> </apply> <ms> /; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <ms> n </ms> <ms> + </ms> <ms> 1 </ms> </list> </apply> <ms> 2 </ms> </apply> <ms> ∈ </ms> <apply> <ci> SuperscriptBox </ci> <apply> <ci> TagBox </ci> <ms> ℤ </ms> <apply> <ci> Function </ci> <integers /> </apply> </apply> <ms> + </ms> </apply> </list> </apply> </list> </apply> <ci> TraditionalForm </ci> </apply> </annotation-xml> </semantics> </math>
|
|
|
|
|
|
|
|
|
|
| |
|
|
|
|
| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["n_", " ", RowBox[List["JacobiDN", "[", RowBox[List[RowBox[List["n_", " ", "z_"]], ",", "m_"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], FractionBox[RowBox[List["1", "-", "n"]], "2"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["r", "=", "0"]], RowBox[List["n", "-", "1"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["s", "=", "0"]], RowBox[List["n", "-", "1"]]], RowBox[List["JacobiDN", "[", RowBox[List[RowBox[List["z", "+", FractionBox[RowBox[List["4", " ", RowBox[List["(", RowBox[List[RowBox[List["r", " ", RowBox[List["EllipticK", "[", "m", "]"]]]], "+", RowBox[List["s", " ", "\[Tau]", " ", RowBox[List["EllipticK", "[", "m", "]"]]]]]], ")"]]]], "n"]]], ",", "m"]], "]"]]]]]]]], "/;", RowBox[List[RowBox[List[FractionBox[RowBox[List["n", "+", "1"]], "2"], "\[Element]", "Integers"]], "&&", RowBox[List["n", ">", "0"]]]]]]]]]] |
|
|
|
|
|
|
|
|
|
|
Date Added to functions.wolfram.com (modification date)
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|