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 EllipticTheta

 http://functions.wolfram.com/09.02.16.0006.01

 Input Form

 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

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[FractionBox[RowBox[List["EllipticTheta", "[", RowBox[List["2", ",", RowBox[List["\[Pi]", " ", RowBox[List["(", 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[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[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[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"]]]]]]]]

 MathML Form

 ϑ 2 ( π ( ( n - 1 ) z - 1 2 ) , π τ ) ϑ 2 ( π ( z - 1 2 ) , π τ ) ( k = 1 n - 1 ϑ 2 ( π ( n z + k τ - 1 2 ) , n π τ ) ϑ 2 ( π ( k τ - 1 2 ) , n π τ ) ) k = 1 n - 1 ϑ ( k τ , n τ ) ( n 2 - n - 2 k ) π z ϑ ( n z + k τ , n τ ) /; Im ( τ ) > 0 n Condition EllipticTheta 2 n -1 z -1 1 2 τ EllipticTheta 2 z -1 1 2 τ -1 k 1 n -1 EllipticTheta 2 n z k τ -1 1 2 n τ EllipticTheta 2 k τ -1 1 2 n τ -1 k 1 n -1 ϑ k τ n τ n 2 -1 n -1 2 k z ϑ n z k τ n τ -1 τ 0 n [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", FractionBox[RowBox[List["EllipticTheta", "[", RowBox[List["2", ",", RowBox[List["\[Pi]", " ", RowBox[List["(", 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[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[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[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)

 2002-12-18