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http://functions.wolfram.com/09.33.20.0005.01
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D[JacobiNS[z, m], {z, n}] == (-1)^n z^(-1 - n) n! +
z^(-1 - n) Sum[(((-1)^(k - 1) (2^(2 k - 1) - 1) BernoulliB[2 k])/
(k (2 k - n - 1)!)) ((Pi z)/(2 EllipticK[m]))^(2 k),
{k, 1, Infinity}] + ((2^(1 - n) Pi^(n + 1))/EllipticK[m]^(n + 1))
Sum[(((2 k + 1)^n EllipticNomeQ[m]^(2 k + 1))/
(1 - EllipticNomeQ[m]^(2 k + 1)))
Sin[((2 k + 1) Pi z)/(2 EllipticK[m]) + (n Pi)/2], {k, 0, Infinity}] /;
Element[n, Integers] && n > 0
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List["z", ",", "n"]], "}"]]], RowBox[List["JacobiNS", "[", RowBox[List["z", ",", "m"]], "]"]]]], "\[Equal]", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "1"]], "-", "n"]]], " ", RowBox[List["n", "!"]]]], "+", RowBox[List[SuperscriptBox["z", RowBox[List[RowBox[List["-", "1"]], "-", "n"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["k", "-", "1"]]], " ", RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["2", " ", "k"]], "-", "1"]]], "-", "1"]], ")"]], " ", RowBox[List["BernoulliB", "[", RowBox[List["2", " ", "k"]], "]"]]]], RowBox[List["k", " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", "n", "-", "1"]], ")"]], "!"]]]]], SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["\[Pi]", " ", "z"]], RowBox[List["2", RowBox[List["EllipticK", "[", "m", "]"]]]]], ")"]], RowBox[List["2", " ", "k"]]]]]]]]], "+", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List["1", "-", "n"]]], " ", SuperscriptBox["\[Pi]", RowBox[List["n", "+", "1"]]]]], SuperscriptBox[RowBox[List["EllipticK", "[", "m", "]"]], RowBox[List["n", "+", "1"]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["2", "k"]], "+", "1"]], ")"]], "n"], SuperscriptBox[RowBox[List["EllipticNomeQ", "[", "m", "]"]], RowBox[List[RowBox[List["2", "k"]], "+", "1"]]], " "]], RowBox[List["1", "-", SuperscriptBox[RowBox[List["EllipticNomeQ", "[", "m", "]"]], RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]]]]]], RowBox[List["Sin", "[", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", "k"]], "+", "1"]], ")"]], " ", "\[Pi]", " ", "z"]], RowBox[List["2", RowBox[List["EllipticK", "[", "m", "]"]]]]], "+", FractionBox[RowBox[List["n", " ", "\[Pi]"]], "2"]]], "]"]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", ">", "0"]]]]]]]]
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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> <mfrac> <mrow> <msup> <mo> ∂ </mo> <mi> n </mi> </msup> <mrow> <mi> ns </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mo> ∂ </mo> <msup> <mi> z </mi> <mi> n </mi> </msup> </mrow> </mfrac> <mo> ⩵ </mo> <mrow> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> n </mi> </msup> <mo> ⁢ </mo> <mrow> <mi> n </mi> <mo> ! </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> z </mi> <mrow> <mrow> <mo> - </mo> <mi> n </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> </mrow> <mo> + </mo> <mtext> </mtext> <mrow> <msup> <mi> z </mi> <mrow> <mrow> <mo> - </mo> <mi> n </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> ∞ </mi> </munderover> <mrow> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> k </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mn> 2 </mn> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> k </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msub> <semantics> <mi> B </mi> <annotation encoding='Mathematica'> TagBox["B", BernoulliB] </annotation> </semantics> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> k </mi> </mrow> </msub> </mrow> <mrow> <mi> k </mi> <mo> ⁢ </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> k </mi> </mrow> <mo> - </mo> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mrow> </mfrac> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mfrac> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mi> K </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> ) </mo> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> k </mi> </mrow> </msup> </mrow> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mrow> <msup> <mn> 2 </mn> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> n </mi> </mrow> </msup> <mo> ⁢ </mo> <msup> <mi> π </mi> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msup> </mrow> <msup> <mrow> <mi> K </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msup> </mfrac> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> ∞ </mi> </munderover> <mrow> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> k </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> n </mi> </msup> <mo> ⁢ </mo> <msup> <mrow> <semantics> <mi> q </mi> <annotation-xml encoding='MathML-Content'> <ci> EllipticNomeQ </ci> </annotation-xml> </semantics> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> k </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </msup> </mrow> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mrow> <semantics> <mi> q </mi> <annotation-xml encoding='MathML-Content'> <ci> EllipticNomeQ </ci> </annotation-xml> </semantics> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> k </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </msup> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mfrac> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> n </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> + </mo> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> k </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mi> K </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> n </mi> <mo> ∈ </mo> <msup> <mi> ℕ </mi> <mo> + </mo> </msup> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <partialdiff /> <bvar> <ci> z </ci> <degree> <ci> n </ci> </degree> </bvar> <apply> <ci> JacobiNS </ci> <ci> z </ci> <ci> m </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <apply> <factorial /> <ci> n </ci> </apply> <apply> <power /> <ci> z </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <ci> z </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <plus /> <ci> k </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <ci> BernoulliB </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> </apply> <apply> <power /> <apply> <times /> <ci> k </ci> <apply> <factorial /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <pi /> <ci> z </ci> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> EllipticK </ci> <ci> m </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> </apply> </apply> <apply> <power /> <pi /> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <ci> EllipticK </ci> <ci> m </ci> </apply> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <times /> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> <cn type='integer'> 1 </cn> </apply> <ci> n </ci> </apply> <apply> <power /> <apply> <ci> EllipticNomeQ </ci> <ci> m </ci> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <apply> <ci> EllipticNomeQ </ci> <ci> m </ci> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <sin /> <apply> <plus /> <apply> <times /> <pi /> <ci> n </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> <cn type='integer'> 1 </cn> </apply> <pi /> <ci> z </ci> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> EllipticK </ci> <ci> m </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <in /> <ci> n </ci> <apply> <ci> SuperPlus </ci> <ci> ℕ </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["z_", ",", "n_"]], "}"]]]]], RowBox[List["JacobiNS", "[", RowBox[List["z_", ",", "m_"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "1"]], "-", "n"]]], " ", RowBox[List["n", "!"]]]], "+", RowBox[List[SuperscriptBox["z", RowBox[List[RowBox[List["-", "1"]], "-", "n"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["k", "-", "1"]]], " ", RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["2", " ", "k"]], "-", "1"]]], "-", "1"]], ")"]], " ", RowBox[List["BernoulliB", "[", RowBox[List["2", " ", "k"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["\[Pi]", " ", "z"]], RowBox[List["2", " ", RowBox[List["EllipticK", "[", "m", "]"]]]]], ")"]], RowBox[List["2", " ", "k"]]]]], RowBox[List["k", " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", "n", "-", "1"]], ")"]], "!"]]]]]]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List["1", "-", "n"]]], " ", SuperscriptBox["\[Pi]", RowBox[List["n", "+", "1"]]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]], ")"]], "n"], " ", SuperscriptBox[RowBox[List["EllipticNomeQ", "[", "m", "]"]], RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]]]]], ")"]], " ", RowBox[List["Sin", "[", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]], ")"]], " ", "\[Pi]", " ", "z"]], RowBox[List["2", " ", RowBox[List["EllipticK", "[", "m", "]"]]]]], "+", FractionBox[RowBox[List["n", " ", "\[Pi]"]], "2"]]], "]"]]]], RowBox[List["1", "-", SuperscriptBox[RowBox[List["EllipticNomeQ", "[", "m", "]"]], RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]]]]]]]]]], SuperscriptBox[RowBox[List["EllipticK", "[", "m", "]"]], RowBox[List["n", "+", "1"]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", ">", "0"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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){kind=small}
