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http://functions.wolfram.com/09.05.20.0003.01
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D[EllipticThetaPrime[1, z, q], q] == EllipticThetaPrime[1, z, q]/(4 q) +
(2 Sum[(-1)^k q^(k (1 + k)) k (1 + k) (2 k + 1) Cos[(2 k + 1) z],
{k, 1, Infinity}])/q^(3/4) /; Abs[q] < 1
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", "q"], RowBox[List["EllipticThetaPrime", "[", RowBox[List["1", ",", "z", ",", "q"]], "]"]]]], "\[Equal]", RowBox[List[FractionBox[RowBox[List["EllipticThetaPrime", "[", RowBox[List["1", ",", "z", ",", "q"]], "]"]], RowBox[List["4", " ", "q"]]], "+", RowBox[List["2", SuperscriptBox["q", RowBox[List["-", FractionBox["3", "4"]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], SuperscriptBox["q", RowBox[List["k", " ", RowBox[List["(", RowBox[List["1", "+", "k"]], ")"]]]]], "k", " ", RowBox[List["(", RowBox[List["1", "+", "k"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]], ")"]], " ", RowBox[List["Cos", "[", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]], ")"]], " ", "z"]], "]"]]]]]]]]]]]], "/;", " ", RowBox[List[RowBox[List["Abs", "[", "q", "]"]], "<", "1"]]]]]]
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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> <mo> ∂ </mo> <mrow> <msubsup> <mi> ϑ </mi> <mn> 1 </mn> <mo> ′ </mo> </msubsup> <mo> ( </mo> <mrow> <mi> z </mi> <mo> , </mo> <mi> q </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mo> ∂ </mo> <mi> q </mi> </mrow> </mfrac> <mo> ⩵ </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msup> <mi> q </mi> <mrow> <mo> - </mo> <mfrac> <mn> 3 </mn> <mn> 4 </mn> </mfrac> </mrow> </msup> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> ∞ </mi> </munderover> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msup> <mo> ⁢ </mo> <msup> <mi> q </mi> <mrow> <mi> k </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </msup> <mo> ⁢ </mo> <mi> k </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <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> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <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> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> + </mo> <mfrac> <mrow> <msubsup> <mi> ϑ </mi> <mn> 1 </mn> <mo> ′ </mo> </msubsup> <mo> ( </mo> <mrow> <mi> z </mi> <mo> , </mo> <mi> q </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> q </mi> </mrow> </mfrac> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mi> q </mi> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> <mo> < </mo> <mn> 1 </mn> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <partialdiff /> <bvar> <ci> q </ci> </bvar> <apply> <apply> <partialdiff /> <list> <cn type='integer'> 1 </cn> </list> <apply> <ci> Subscript </ci> <ci> ϑ </ci> <cn type='integer'> 1 </cn> </apply> </apply> <ci> z </ci> <ci> q </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> q </ci> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 3 <sep /> 4 </cn> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> <apply> <power /> <ci> q </ci> <apply> <times /> <ci> k </ci> <apply> <plus /> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <ci> k </ci> <apply> <plus /> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <cos /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> <cn type='integer'> 1 </cn> </apply> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <apply> <partialdiff /> <list> <cn type='integer'> 1 </cn> </list> <apply> <ci> Subscript </ci> <ci> ϑ </ci> <cn type='integer'> 1 </cn> </apply> </apply> <ci> z </ci> <ci> q </ci> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> q </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <lt /> <apply> <abs /> <ci> q </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["q_"]]], RowBox[List["EllipticThetaPrime", "[", RowBox[List["1", ",", "z_", ",", "q_"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List["EllipticThetaPrime", "[", RowBox[List["1", ",", "z", ",", "q"]], "]"]], RowBox[List["4", " ", "q"]]], "+", RowBox[List["2", " ", SuperscriptBox["q", RowBox[List["-", FractionBox["3", "4"]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", SuperscriptBox["q", RowBox[List["k", " ", RowBox[List["(", RowBox[List["1", "+", "k"]], ")"]]]]], " ", "k", " ", RowBox[List["(", RowBox[List["1", "+", "k"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]], ")"]], " ", RowBox[List["Cos", "[", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]], ")"]], " ", "z"]], "]"]]]]]]]]]], "/;", RowBox[List[RowBox[List["Abs", "[", "q", "]"]], "<", "1"]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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