Log[EllipticTheta[3, a + b, q]/EllipticTheta[3, a - b, q]] == 4 Sum[((-1)^k/k) (q^k/(1 - q^(2 k))) Sin[2 k a] Sin[2 k b], {k, 1, Infinity}]

Cell[BoxData[RowBox[List[RowBox[List["Log", "[", FractionBox[RowBox[List["EllipticTheta", "[", RowBox[List["3", ",", RowBox[List["a", "+", "b"]], ",", "q"]], "]"]], RowBox[List["EllipticTheta", "[", RowBox[List["3", ",", RowBox[List["a", "-", "b"]], ",", "q"]], "]"]]], "]"]], "\[Equal]", RowBox[List["4", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], RowBox[List[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], "k"], " ", FractionBox[SuperscriptBox["q", "k"], RowBox[List["1", "-", SuperscriptBox["q", RowBox[List["2", "k"]]]]]], " ", RowBox[List["Sin", "[", RowBox[List["2", "k", " ", "a"]], "]"]], " ", RowBox[List["Sin", "[", RowBox[List["2", "k", " ", "b"]], "]"]]]]]]]]]]]]

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mrow> <msub> <mi> &#977; </mi> <mn> 3 </mn> </msub> <mo> ( </mo> <mrow> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> </mrow> <mo> , </mo> <mi> q </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <msub> <mi> &#977; </mi> <mn> 3 </mn> </msub> <mo> ( </mo> <mrow> <mrow> <mi> a </mi> <mo> - </mo> <mi> b </mi> </mrow> <mo> , </mo> <mi> q </mi> </mrow> <mo> ) </mo> </mrow> </mfrac> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <mfrac> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msup> <mi> k </mi> </mfrac> <mo> &#8290; </mo> <mfrac> <msup> <mi> q </mi> <mrow> <mtext> </mtext> <mi> k </mi> </mrow> </msup> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> q </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> </msup> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> k </mi> <mo> &#8290; </mo> <mi> a </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> k </mi> <mo> &#8290; </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ln /> <apply> <times /> <apply> <ci> EllipticTheta </ci> <cn type='integer'> 3 </cn> <apply> <plus /> <ci> a </ci> <ci> b </ci> </apply> <ci> q </ci> </apply> <apply> <power /> <apply> <ci> EllipticTheta </ci> <cn type='integer'> 3 </cn> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> <ci> q </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 4 </cn> <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> <ci> k </ci> </apply> <apply> <power /> <ci> k </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <power /> <ci> q </ci> <ci> k </ci> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> q </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <sin /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> <ci> a </ci> </apply> </apply> <apply> <sin /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> <ci> b </ci> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Log", "[", FractionBox[RowBox[List["EllipticTheta", "[", RowBox[List["3", ",", RowBox[List["a_", "+", "b_"]], ",", "q_"]], "]"]], RowBox[List["EllipticTheta", "[", RowBox[List["3", ",", RowBox[List["a_", "-", "b_"]], ",", "q_"]], "]"]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List["4", " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", SuperscriptBox["q", "k"], " ", RowBox[List["Sin", "[", RowBox[List["2", " ", "k", " ", "a"]], "]"]], " ", RowBox[List["Sin", "[", RowBox[List["2", " ", "k", " ", "b"]], "]"]]]], RowBox[List["k", " ", RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["q", RowBox[List["2", " ", "k"]]]]], ")"]]]]]]]]]]]]]

2001-10-29