|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
http://functions.wolfram.com/09.01.06.0031.01
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
EllipticTheta[1, z, q] \[Proportional] EllipticTheta[1, z, Subscript[q, 0]] +
Derivative[0, 0, 1][EllipticTheta][1, z, Subscript[q, 0]]
(q - Subscript[q, 0]) + (1/2) Derivative[0, 0, 2][EllipticTheta][1, z,
Subscript[q, 0]] (q - Subscript[q, 0])^2 +
(1/6) Derivative[0, 0, 3][EllipticTheta][1, z, Subscript[q, 0]]
(q - Subscript[q, 0])^3 + O[(q - Subscript[q, 0])^4]
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Cell[BoxData[RowBox[List[RowBox[List["EllipticTheta", "[", RowBox[List["1", ",", "z", ",", "q"]], "]"]], "\[Proportional]", RowBox[List[RowBox[List["EllipticTheta", "[", RowBox[List["1", ",", "z", ",", SubscriptBox["q", "0"]]], "]"]], "+", " ", RowBox[List[RowBox[List[SuperscriptBox["EllipticTheta", TagBox[RowBox[List["(", RowBox[List["0", ",", "0", ",", "1"]], ")"]], Derivative], Rule[MultilineFunction, None]], "[", RowBox[List["1", ",", "z", ",", SubscriptBox["q", "0"]]], "]"]], RowBox[List["(", RowBox[List["q", "-", SubscriptBox["q", "0"]]], ")"]]]], "+", RowBox[List[FractionBox["1", "2"], " ", RowBox[List[SuperscriptBox["EllipticTheta", TagBox[RowBox[List["(", RowBox[List["0", ",", "0", ",", "2"]], ")"]], Derivative], Rule[MultilineFunction, None]], "[", RowBox[List["1", ",", "z", ",", SubscriptBox["q", "0"]]], "]"]], SuperscriptBox[RowBox[List["(", RowBox[List["q", "-", SubscriptBox["q", "0"]]], ")"]], "2"]]], " ", "+", RowBox[List[FractionBox["1", "6"], " ", RowBox[List[SuperscriptBox["EllipticTheta", TagBox[RowBox[List["(", RowBox[List["0", ",", "0", ",", "3"]], ")"]], Derivative], Rule[MultilineFunction, None]], "[", RowBox[List["1", ",", "z", ",", SubscriptBox["q", "0"]]], "]"]], SuperscriptBox[RowBox[List["(", RowBox[List["q", "-", SubscriptBox["q", "0"]]], ")"]], "3"]]], "+", RowBox[List["O", "[", SuperscriptBox[RowBox[List["(", RowBox[List["q", "-", SubscriptBox["q", "0"]]], ")"]], "4"], "]"]]]]]]]]
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <msub> <mi> ϑ </mi> <mn> 1 </mn> </msub> <mo> ( </mo> <mrow> <mi> z </mi> <mo> , </mo> <mi> q </mi> </mrow> <mo> ) </mo> </mrow> <mo> ∝ </mo> <mrow> <mrow> <msub> <mi> ϑ </mi> <mn> 1 </mn> </msub> <mo> ( </mo> <mrow> <mi> z </mi> <mo> , </mo> <msub> <mi> q </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mtext> </mtext> <mrow> <mrow> <msubsup> <mi> ϑ </mi> <mn> 1 </mn> <mrow> <mo> ( </mo> <mrow> <mn> 0 </mn> <mo> , </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </msubsup> <mo> ( </mo> <mrow> <mi> z </mi> <mo> , </mo> <msub> <mi> q </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> q </mi> <mo> - </mo> <msub> <mi> q </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mrow> <msubsup> <mi> ϑ </mi> <mn> 1 </mn> <mrow> <mo> ( </mo> <mrow> <mn> 0 </mn> <mo> , </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> </msubsup> <mo> ( </mo> <mrow> <mi> z </mi> <mo> , </mo> <msub> <mi> q </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mtext> </mtext> <msup> <mrow> <mo> ( </mo> <mrow> <mi> q </mi> <mo> - </mo> <msub> <mi> q </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mfrac> <mrow> <mtext> </mtext> <mrow> <msubsup> <mi> ϑ </mi> <mn> 1 </mn> <mrow> <mo> ( </mo> <mrow> <mn> 0 </mn> <mo> , </mo> <mn> 3 </mn> </mrow> <mo> ) </mo> </mrow> </msubsup> <mo> ( </mo> <mrow> <mi> z </mi> <mo> , </mo> <msub> <mi> q </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mn> 6 </mn> </mfrac> <mo> ⁢ </mo> <mtext> </mtext> <msup> <mrow> <mo> ( </mo> <mrow> <mi> q </mi> <mo> - </mo> <msub> <mi> q </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mi> O </mi> <mo> ⁡ </mo> <mo> ( </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> q </mi> <mo> - </mo> <msub> <mi> q </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mn> 4 </mn> </msup> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <mrow> <mrow> <msub> <mi> ϑ </mi> <mn> 1 </mn> </msub> <mo> ( </mo> <mrow> <mi> z </mi> <mo> , </mo> <mi> q </mi> </mrow> <mo> ) </mo> </mrow> <mo> ∝ </mo> <mrow> <mrow> <msub> <mi> ϑ </mi> <mn> 1 </mn> </msub> <mo> ( </mo> <mrow> <mi> z </mi> <mo> , </mo> <msub> <mi> q </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mtext> </mtext> <mrow> <mrow> <msubsup> <mi> ϑ </mi> <mn> 1 </mn> <mrow> <mo> ( </mo> <mrow> <mn> 0 </mn> <mo> , </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </msubsup> <mo> ( </mo> <mrow> <mi> z </mi> <mo> , </mo> <msub> <mi> q </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> q </mi> <mo> - </mo> <msub> <mi> q </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mrow> <msubsup> <mi> ϑ </mi> <mn> 1 </mn> <mrow> <mo> ( </mo> <mrow> <mn> 0 </mn> <mo> , </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> </msubsup> <mo> ( </mo> <mrow> <mi> z </mi> <mo> , </mo> <msub> <mi> q </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mtext> </mtext> <msup> <mrow> <mo> ( </mo> <mrow> <mi> q </mi> <mo> - </mo> <msub> <mi> q </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mfrac> <mrow> <mtext> </mtext> <mrow> <msubsup> <mi> ϑ </mi> <mn> 1 </mn> <mrow> <mo> ( </mo> <mrow> <mn> 0 </mn> <mo> , </mo> <mn> 3 </mn> </mrow> <mo> ) </mo> </mrow> </msubsup> <mo> ( </mo> <mrow> <mi> z </mi> <mo> , </mo> <msub> <mi> q </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mn> 6 </mn> </mfrac> <mo> ⁢ </mo> <mtext> </mtext> <msup> <mrow> <mo> ( </mo> <mrow> <mi> q </mi> <mo> - </mo> <msub> <mi> q </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mi> O </mi> <mo> ⁡ </mo> <mo> ( </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> q </mi> <mo> - </mo> <msub> <mi> q </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mn> 4 </mn> </msup> <mo> ) </mo> </mrow> </mrow> </mrow> </annotation-xml> </semantics> </math>
|
|
|
|
|
|
|
|
|
|
| |
|
|
|
|
| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["EllipticTheta", "[", RowBox[List["1", ",", "z_", ",", "q_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["EllipticTheta", "[", RowBox[List["1", ",", "z", ",", SubscriptBox["qq", "0"]]], "]"]], "+", RowBox[List[RowBox[List[SuperscriptBox["EllipticTheta", TagBox[RowBox[List["(", RowBox[List["0", ",", "0", ",", "1"]], ")"]], Derivative], Rule[MultilineFunction, None]], "[", RowBox[List["1", ",", "z", ",", SubscriptBox["qq", "0"]]], "]"]], " ", RowBox[List["(", RowBox[List["q", "-", SubscriptBox["qq", "0"]]], ")"]]]], "+", RowBox[List[FractionBox["1", "2"], " ", RowBox[List[SuperscriptBox["EllipticTheta", TagBox[RowBox[List["(", RowBox[List["0", ",", "0", ",", "2"]], ")"]], Derivative], Rule[MultilineFunction, None]], "[", RowBox[List["1", ",", "z", ",", SubscriptBox["qq", "0"]]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["q", "-", SubscriptBox["qq", "0"]]], ")"]], "2"]]], "+", RowBox[List[FractionBox["1", "6"], " ", RowBox[List[SuperscriptBox["EllipticTheta", TagBox[RowBox[List["(", RowBox[List["0", ",", "0", ",", "3"]], ")"]], Derivative], Rule[MultilineFunction, None]], "[", RowBox[List["1", ",", "z", ",", SubscriptBox["qq", "0"]]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["q", "-", SubscriptBox["qq", "0"]]], ")"]], "3"]]], "+", SuperscriptBox[RowBox[List["O", "[", RowBox[List["q", "-", SubscriptBox["qq", "0"]]], "]"]], "4"]]]]]]] |
|
|
|
|
|
|
|
|
|
|
Date Added to functions.wolfram.com (modification date)
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|