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JacobiSN






Mathematica Notation

Traditional Notation









Elliptic Functions > JacobiSN[z,m] > Differentiation > Fractional integro-differentiation > With respect to z





http://functions.wolfram.com/09.36.20.0007.01









  


  










Input Form





D[JacobiSN[z, m], {z, \[Alpha]}] == ((2^(\[Alpha] - 1) Pi^(5/2) z^(1 - \[Alpha]))/(m^(1/2) EllipticK[m]^2)) Sum[(((2 k + 1) EllipticNomeQ[m]^(k + 1/2))/ (1 - EllipticNomeQ[m]^(2 k + 1))) HypergeometricPFQRegularized[{1}, {1 - \[Alpha]/2, (3 - \[Alpha])/2}, -(((2 k + 1)^2 Pi^2 z^2)/(16 EllipticK[m]^2))], {k, 0, Infinity}]










Standard Form





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MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mfrac> <mrow> <msup> <mo> &#8706; </mo> <mi> &#945; </mi> </msup> <mrow> <mi> sn </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> &#10072; </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mo> &#8706; </mo> <msup> <mi> z </mi> <mi> &#945; </mi> </msup> </mrow> </mfrac> <mo> &#10869; </mo> <mrow> <mfrac> <mrow> <msup> <mn> 2 </mn> <mrow> <mi> &#945; </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> &#8290; </mo> <msup> <mi> &#960; </mi> <mrow> <mn> 5 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> &#945; </mi> </mrow> </msup> </mrow> <mrow> <msqrt> <mi> m </mi> </msqrt> <mo> &#8290; </mo> <msup> <mrow> <mi> K </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </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> <mi> k </mi> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </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> &#8290; </mo> <mi> k </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </msup> </mrow> </mfrac> <mo> &#8290; </mo> <semantics> <mrow> <mrow> <msub> <mo> &#8202; 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Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["z_", ",", "\[Alpha]_"]], "}"]]]]], RowBox[List["JacobiSN", "[", RowBox[List["z_", ",", "m_"]], "]"]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List["\[Alpha]", "-", "1"]]], " ", SuperscriptBox["\[Pi]", RowBox[List["5", "/", "2"]]], " ", SuperscriptBox["z", RowBox[List["1", "-", "\[Alpha]"]]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]], ")"]], " ", SuperscriptBox[RowBox[List["EllipticNomeQ", "[", "m", "]"]], RowBox[List["k", "+", FractionBox["1", "2"]]]]]], ")"]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", "1", "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "-", FractionBox["\[Alpha]", "2"]]], ",", FractionBox[RowBox[List["3", "-", "\[Alpha]"]], "2"]]], "}"]], ",", RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]], ")"]], "2"], " ", SuperscriptBox["\[Pi]", "2"], " ", SuperscriptBox["z", "2"]]], RowBox[List["16", " ", SuperscriptBox[RowBox[List["EllipticK", "[", "m", "]"]], "2"]]]]]]]], "]"]]]], RowBox[List["1", "-", SuperscriptBox[RowBox[List["EllipticNomeQ", "[", "m", "]"]], RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]]]]]]]]]], RowBox[List[SqrtBox["m"], " ", SuperscriptBox[RowBox[List["EllipticK", "[", "m", "]"]], "2"]]]]]]]]










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





2001-10-29





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