|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
http://functions.wolfram.com/09.02.16.0005.01
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
EllipticTheta[2, Pi (1/2 + (-1 + n) z), E^(I Pi \[Tau])]/
EllipticTheta[2, Pi (1/2 + z), E^(I Pi \[Tau])] ==
Product[EllipticTheta[2, Pi (1/2 + n z + k \[Tau]), E^(I n Pi \[Tau])]/
EllipticTheta[2, Pi (1/2 + k \[Tau]), E^(I n Pi \[Tau])],
{k, 1, -1 + n}] Sum[(\[CurlyTheta][k \[Tau], n \[Tau]]
E^((n^2 - n - 2 k) I Pi z))/\[CurlyTheta][n z + k \[Tau], n \[Tau]],
{k, 1, n - 1}] /; Im[\[Tau]] > 0 && Element[n, Integers] && n >= 1
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Cell[BoxData[RowBox[List[RowBox[List[FractionBox[RowBox[List["EllipticTheta", "[", RowBox[List["2", ",", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List[FractionBox["1", "2"], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "n"]], ")"]], " ", "z"]]]], ")"]]]], ",", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", "\[Tau]"]]]]], "]"]], RowBox[List["EllipticTheta", "[", RowBox[List["2", ",", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List[FractionBox["1", "2"], "+", "z"]], ")"]]]], ",", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", "\[Tau]"]]]]], "]"]]], "\[Equal]", RowBox[List[RowBox[List["(", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], RowBox[List[RowBox[List["-", "1"]], "+", "n"]]], FractionBox[RowBox[List["EllipticTheta", "[", RowBox[List["2", ",", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List[FractionBox["1", "2"], "+", RowBox[List["n", " ", "z"]], "+", RowBox[List["k", " ", "\[Tau]"]]]], ")"]]]], ",", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "n", " ", "\[Pi]", " ", "\[Tau]"]]]]], "]"]], RowBox[List["EllipticTheta", "[", RowBox[List["2", ",", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List[FractionBox["1", "2"], "+", RowBox[List["k", " ", "\[Tau]"]]]], ")"]]]], ",", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "n", " ", "\[Pi]", " ", "\[Tau]"]]]]], "]"]]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], RowBox[List["n", "-", "1"]]], FractionBox[RowBox[List[RowBox[List["\[CurlyTheta]", "[", RowBox[List[RowBox[List["k", " ", "\[Tau]"]], ",", RowBox[List["n", " ", "\[Tau]"]]]], "]"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["n", "2"], "-", "n", "-", RowBox[List["2", " ", "k"]]]], ")"]], " ", "\[ImaginaryI]", " ", "\[Pi]", " ", "z"]]]]], RowBox[List["\[CurlyTheta]", "[", RowBox[List[RowBox[List[RowBox[List["n", " ", "z"]], "+", RowBox[List["k", " ", "\[Tau]"]]]], ",", RowBox[List["n", " ", "\[Tau]"]]]], "]"]]]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Im", "[", "\[Tau]", "]"]], ">", "0"]], "&&", RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "1"]]]]]]]]
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mfrac> <mrow> <msub> <mi> ϑ </mi> <mn> 2 </mn> </msub> <mo> ( </mo> <mrow> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> , </mo> <msup> <mi> ⅇ </mi> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mi> τ </mi> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> <mrow> <msub> <mi> ϑ </mi> <mn> 2 </mn> </msub> <mo> ( </mo> <mrow> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> , </mo> <msup> <mi> ⅇ </mi> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mi> τ </mi> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> </mfrac> <mo> ⩵ </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <munderover> <mo> ∏ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <mfrac> <mrow> <msub> <mi> ϑ </mi> <mn> 2 </mn> </msub> <mo> ( </mo> <mrow> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> n </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> + </mo> <mrow> <mi> k </mi> <mo> ⁢ </mo> <mi> τ </mi> </mrow> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> , </mo> <msup> <mi> ⅇ </mi> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> n </mi> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mi> τ </mi> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> <mrow> <msub> <mi> ϑ </mi> <mn> 2 </mn> </msub> <mo> ( </mo> <mrow> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> k </mi> <mo> ⁢ </mo> <mi> τ </mi> </mrow> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> , </mo> <msup> <mi> ⅇ </mi> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> n </mi> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mi> τ </mi> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <mfrac> <mrow> <mrow> <mi> ϑ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mi> k </mi> <mo> ⁢ </mo> <mi> τ </mi> </mrow> <mo> , </mo> <mrow> <mi> n </mi> <mo> ⁢ </mo> <mi> τ </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mrow> <mrow> <mo> ( </mo> <mrow> <msup> <mi> n </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mi> n </mi> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> k </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </msup> </mrow> <mrow> <mi> ϑ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mrow> <mi> n </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> + </mo> <mrow> <mi> k </mi> <mo> ⁢ </mo> <mi> τ </mi> </mrow> </mrow> <mo> , </mo> <mrow> <mi> n </mi> <mo> ⁢ </mo> <mi> τ </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mfrac> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <mi> Im </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> τ </mi> <mo> ) </mo> </mrow> <mo> > </mo> <mn> 0 </mn> </mrow> <mo> ∧ </mo> <mrow> <mi> n </mi> <mo> ∈ </mo> <mi> ℕ </mi> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <times /> <apply> <ci> EllipticTheta </ci> <cn type='integer'> 2 </cn> <apply> <times /> <pi /> <apply> <plus /> <apply> <times /> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> <ci> z </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <imaginaryi /> <pi /> <ci> τ </ci> </apply> </apply> </apply> <apply> <power /> <apply> <ci> EllipticTheta </ci> <cn type='integer'> 2 </cn> <apply> <times /> <pi /> <apply> <plus /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <imaginaryi /> <pi /> <ci> τ </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <product /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </uplimit> <apply> <times /> <apply> <ci> EllipticTheta </ci> <cn type='integer'> 2 </cn> <apply> <times /> <pi /> <apply> <plus /> <apply> <times /> <ci> n </ci> <ci> z </ci> </apply> <apply> <times /> <ci> k </ci> <ci> τ </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <imaginaryi /> <ci> n </ci> <pi /> <ci> τ </ci> </apply> </apply> </apply> <apply> <power /> <apply> <ci> EllipticTheta </ci> <cn type='integer'> 2 </cn> <apply> <times /> <pi /> <apply> <plus /> <apply> <times /> <ci> k </ci> <ci> τ </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <imaginaryi /> <ci> n </ci> <pi /> <ci> τ </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </uplimit> <apply> <times /> <apply> <ci> ϑ </ci> <apply> <times /> <ci> k </ci> <ci> τ </ci> </apply> <apply> <times /> <ci> n </ci> <ci> τ </ci> </apply> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <plus /> <apply> <power /> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> </apply> </apply> <imaginaryi /> <pi /> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <ci> ϑ </ci> <apply> <plus /> <apply> <times /> <ci> n </ci> <ci> z </ci> </apply> <apply> <times /> <ci> k </ci> <ci> τ </ci> </apply> </apply> <apply> <times /> <ci> n </ci> <ci> τ </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <gt /> <apply> <imaginary /> <ci> τ </ci> </apply> <cn type='integer'> 0 </cn> </apply> <apply> <in /> <ci> n </ci> <ci> ℕ </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
|
|
|
|
|
|
|
|
|
|
| |
|
|
|
|
| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", FractionBox[RowBox[List["EllipticTheta", "[", RowBox[List["2", ",", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List[FractionBox["1", "2"], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "n_"]], ")"]], " ", "z_"]]]], ")"]]]], ",", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", "\[Tau]_"]]]]], "]"]], RowBox[List["EllipticTheta", "[", RowBox[List["2", ",", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List[FractionBox["1", "2"], "+", "z_"]], ")"]]]], ",", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", "\[Tau]_"]]]]], "]"]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], RowBox[List[RowBox[List["-", "1"]], "+", "n"]]], FractionBox[RowBox[List["EllipticTheta", "[", RowBox[List["2", ",", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List[FractionBox["1", "2"], "+", RowBox[List["n", " ", "z"]], "+", RowBox[List["k", " ", "\[Tau]"]]]], ")"]]]], ",", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "n", " ", "\[Pi]", " ", "\[Tau]"]]]]], "]"]], RowBox[List["EllipticTheta", "[", RowBox[List["2", ",", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List[FractionBox["1", "2"], "+", RowBox[List["k", " ", "\[Tau]"]]]], ")"]]]], ",", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "n", " ", "\[Pi]", " ", "\[Tau]"]]]]], "]"]]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], RowBox[List["n", "-", "1"]]], FractionBox[RowBox[List[RowBox[List["\[CurlyTheta]", "[", RowBox[List[RowBox[List["k", " ", "\[Tau]"]], ",", RowBox[List["n", " ", "\[Tau]"]]]], "]"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["n", "2"], "-", "n", "-", RowBox[List["2", " ", "k"]]]], ")"]], " ", "\[ImaginaryI]", " ", "\[Pi]", " ", "z"]]]]], RowBox[List["\[CurlyTheta]", "[", RowBox[List[RowBox[List[RowBox[List["n", " ", "z"]], "+", RowBox[List["k", " ", "\[Tau]"]]]], ",", RowBox[List["n", " ", "\[Tau]"]]]], "]"]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Im", "[", "\[Tau]", "]"]], ">", "0"]], "&&", RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "1"]]]]]]]]]] |
|
|
|
|
|
|
|
|
|
|
|
Date Added to functions.wolfram.com (modification date)
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
){kind=small}
