Racah 6j symbol: Zeros (formula 07.40.30.0001)

SixJSymbol

$Mathematica Notation$

Hypergeometric Functions

SixJSymbol[{j₁,j₂,j₃},{j₄,j₅,j₆}]

Zeros

http://functions.wolfram.com/07.40.30.0001.01

Input Form

SixJSymbol[{Subscript[j, 1], Subscript[j, 2], Subscript[j, 3]}, {Subscript[j, 4], Subscript[j, 5], Subscript[j, 6]}] == 0 /; Subscript[j, 1] == (d e f + a d g + a b c)/2 - 1/2 && Subscript[j, 2] == (a b c + g h i + b e h)/2 - 1/2 && Subscript[j, 3] == (d e f + a d g + g h i + b e h)/2 - 1 && Subscript[j, 4] == (g i h + a d g)/2 && Subscript[j, 5] == (d e f + b e h)/2 && Subscript[j, 6] == (a b c + a d g + b e h)/2 - 1/2 && a b c + d e f + g h i + a d g + b e h - c f i == 0 && Element[a, Integers] && a > 0 && Element[b, Integers] && b > 0 && Element[c, Integers] && c > 0 && Element[d, Integers] && d > 0 && Element[e, Integers] && e > 0 && Element[f, Integers] && f > 0 && Element[g, Integers] && g > 0 && Element[h, Integers] && h > 0 && Element[i, Integers] && i > 0

Standard Form

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["SixJSymbol", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["j", "1"], ",", SubscriptBox["j", "2"], ",", SubscriptBox["j", "3"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["j", "4"], ",", SubscriptBox["j", "5"], ",", SubscriptBox["j", "6"]]], "}"]]]], "]"]], "\[Equal]", "0"]], "/;", "\[IndentingNewLine]", RowBox[List[RowBox[List[SubscriptBox["j", "1"], "\[Equal]", RowBox[List[FractionBox[RowBox[List["(", RowBox[List[RowBox[List["d", " ", "e", " ", "f"]], "+", RowBox[List["a", " ", "d", " ", "g"]], "+", " ", RowBox[List["a", " ", "b", " ", "c"]]]], ")"]], "2"], "-", FractionBox["1", "2"]]]]], "\[And]", RowBox[List[SubscriptBox["j", "2"], "\[Equal]", RowBox[List[FractionBox[RowBox[List["(", RowBox[List[RowBox[List["a", " ", "b", " ", "c"]], "+", RowBox[List["g", " ", "h", " ", "i"]], "+", " ", RowBox[List["b", " ", "e", " ", "h"]]]], ")"]], "2"], "-", FractionBox["1", "2"]]]]], "\[And]", RowBox[List[SubscriptBox["j", "3"], "\[Equal]", RowBox[List[FractionBox[RowBox[List["(", RowBox[List[RowBox[List["d", " ", "e", " ", "f"]], "+", RowBox[List["a", " ", "d", " ", "g"]], "+", RowBox[List["g", " ", "h", " ", "i"]], "+", RowBox[List["b", " ", "e", " ", "h"]]]], ")"]], "2"], "-", "1"]]]], "\[And]", RowBox[List[SubscriptBox["j", "4"], "\[Equal]", FractionBox[RowBox[List["(", RowBox[List[RowBox[List["g", " ", "i", " ", "h"]], "+", RowBox[List["a", " ", "d", " ", "g"]]]], ")"]], "2"]]], "\[And]", RowBox[List[SubscriptBox["j", "5"], "\[Equal]", FractionBox[RowBox[List["(", RowBox[List[RowBox[List["d", " ", "e", " ", "f"]], "+", RowBox[List["b", " ", "e", " ", "h"]]]], ")"]], "2"]]], "\[And]", RowBox[List[SubscriptBox["j", "6"], "\[Equal]", RowBox[List[FractionBox[RowBox[List["(", RowBox[List[RowBox[List["a", " ", "b", " ", "c"]], "+", RowBox[List["a", " ", "d", " ", "g"]], "+", RowBox[List["b", " ", "e", " ", "h"]]]], ")"]], "2"], "-", FractionBox["1", "2"]]]]], "\[And]", RowBox[List[RowBox[List[RowBox[List["a", " ", "b", " ", "c"]], "+", RowBox[List["d", " ", "e", " ", "f"]], "+", RowBox[List["g", " ", "h", " ", "i"]], "+", RowBox[List["a", " ", "d", " ", "g"]], "+", RowBox[List["b", " ", "e", " ", "h"]], "-", RowBox[List["c", " ", "f", " ", "i"]]]], "\[Equal]", "0"]], "\[And]", RowBox[List["a", "\[Element]", "Integers"]], "\[And]", RowBox[List["a", ">", "0"]], "\[And]", RowBox[List["b", "\[Element]", "Integers"]], "\[And]", RowBox[List["b", ">", "0"]], "\[And]", RowBox[List["c", "\[Element]", "Integers"]], "\[And]", RowBox[List["c", ">", "0"]], "\[And]", RowBox[List["d", "\[Element]", "Integers"]], "\[And]", RowBox[List["d", ">", "0"]], "\[And]", RowBox[List["e", "\[Element]", "Integers"]], "\[And]", RowBox[List["e", ">", "0"]], "\[And]", RowBox[List["f", "\[Element]", "Integers"]], "\[And]", RowBox[List["f", ">", "0"]], "\[And]", RowBox[List["g", "\[Element]", "Integers"]], "\[And]", RowBox[List["g", ">", "0"]], "\[And]", RowBox[List["h", "\[Element]", "Integers"]], "\[And]", RowBox[List["h", ">", "0"]], "\[And]", RowBox[List["i", "\[Element]", "Integers"]], "\[And]", RowBox[List["i", ">", "0"]]]]]]]]

MathML Form

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <semantics> <mo> { </mo> <annotation encoding='Mathematica'> TagBox[StyleBox["{", Rule[SpanMaxSize, DirectedInfinity[1]]], SixJSymbol] </annotation> </semantics> <mtext>   </mtext> <mtable> <mtr> <mtd> <msub> <mi> j </mi> <mn> 1 </mn> </msub> </mtd> <mtd> <msub> <mi> j </mi> <mn> 2 </mn> </msub> </mtd> <mtd> <msub> <mi> j </mi> <mn> 3 </mn> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi> j </mi> <mn> 4 </mn> </msub> </mtd> <mtd> <msub> <mi> j </mi> <mn> 5 </mn> </msub> </mtd> <mtd> <msub> <mi> j </mi> <mn> 6 </mn> </msub> </mtd> </mtr> </mtable> <mtext>   </mtext> <semantics> <mo> } </mo> <annotation encoding='Mathematica'> TagBox[StyleBox["}", Rule[SpanMaxSize, DirectedInfinity[1]]], SixJSymbol] </annotation> </semantics> </mrow> <mo> ⩵ </mo> <mn> 0 </mn> </mrow> <mo> /; </mo> <mrow> <mrow> <msub> <mi> j </mi> <mn> 1 </mn> </msub> <mo> ⩵ </mo> <mrow> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> b </mi> <mo> ⁢ </mo> <mi> c </mi> </mrow> <mo> + </mo> <mrow> <mi> d </mi> <mo> ⁢ </mo> <mi> e </mi> <mo> ⁢ </mo> <mi> f </mi> </mrow> <mo> + </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> d </mi> <mo> ⁢ </mo> <mi> g </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </mrow> <mo> ∧ </mo> <mrow> <msub> <mi> j </mi> <mn> 2 </mn> </msub> <mo> ⩵ </mo> <mrow> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> b </mi> <mo> ⁢ </mo> <mi> c </mi> </mrow> <mo> + </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mi> e </mi> <mo> ⁢ </mo> <mi> h </mi> </mrow> <mo> + </mo> <mrow> <mi> g </mi> <mo> ⁢ </mo> <mi> h </mi> <mo> ⁢ </mo> <mi> i </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </mrow> <mo> ∧ </mo> <mrow> <msub> <mi> j </mi> <mn> 3 </mn> </msub> <mo> ⩵ </mo> <mrow> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> d </mi> <mo> ⁢ </mo> <mi> e </mi> <mo> ⁢ </mo> <mi> f </mi> </mrow> <mo> + </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> d </mi> <mo> ⁢ </mo> <mi> g </mi> </mrow> <mo> + </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mi> e </mi> <mo> ⁢ </mo> <mi> h </mi> </mrow> <mo> + </mo> <mrow> <mi> g </mi> <mo> ⁢ </mo> <mi> h </mi> <mo> ⁢ </mo> <mi> i </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ∧ </mo> <mrow> <msub> <mi> j </mi> <mn> 4 </mn> </msub> <mo> ⩵ </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> d </mi> <mo> ⁢ </mo> <mi> g </mi> </mrow> <mo> + </mo> <mrow> <mi> h </mi> <mo> ⁢ </mo> <mi> i </mi> <mo> ⁢ </mo> <mi> g </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ∧ </mo> <mrow> <msub> <mi> j </mi> <mn> 5 </mn> </msub> <mo> ⩵ </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> d </mi> <mo> ⁢ </mo> <mi> e </mi> <mo> ⁢ </mo> <mi> f </mi> </mrow> <mo> + </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mi> e </mi> <mo> ⁢ </mo> <mi> h </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ∧ </mo> <mrow> <msub> <mi> j </mi> <mn> 6 </mn> </msub> <mo> ⩵ </mo> <mrow> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> b </mi> <mo> ⁢ </mo> <mi> c </mi> </mrow> <mo> + </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> d </mi> <mo> ⁢ </mo> <mi> g </mi> </mrow> <mo> + </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mi> e </mi> <mo> ⁢ </mo> <mi> h </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> b </mi> <mo> ⁢ </mo> <mi> c </mi> </mrow> <mo> - </mo> <mrow> <mi> f </mi> <mo> ⁢ </mo> <mi> i </mi> <mo> ⁢ </mo> <mi> c </mi> </mrow> <mo> + </mo> <mrow> <mi> d </mi> <mo> ⁢ </mo> <mi> e </mi> <mo> ⁢ </mo> <mi> f </mi> </mrow> <mo> + </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> d </mi> <mo> ⁢ </mo> <mi> g </mi> </mrow> <mo> + </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mi> e </mi> <mo> ⁢ </mo> <mi> h </mi> </mrow> <mo> + </mo> <mrow> <mi> g </mi> <mo> ⁢ </mo> <mi> h </mi> <mo> ⁢ </mo> <mi> i </mi> </mrow> </mrow> <mo> ⩵ </mo> <mn> 0 </mn> </mrow> <mo> ∧ </mo> <mrow> <mi> a </mi> <mo> ∈ </mo> <semantics> <msup> <mi> ℕ </mi> <mo> + </mo> </msup> <annotation encoding='Mathematica'> TagBox[SuperscriptBox["\[DoubleStruckCapitalN]", "+"], Function[Integers]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <mi> b </mi> <mo> ∈ </mo> <semantics> <msup> <mi> ℕ </mi> <mo> + </mo> </msup> <annotation encoding='Mathematica'> TagBox[TagBox[SuperscriptBox["\[DoubleStruckCapitalN]", "+"], Function[Integers]], Function[Integers]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <mi> c </mi> <mo> ∈ </mo> <semantics> <msup> <mi> ℕ </mi> <mo> + </mo> </msup> <annotation encoding='Mathematica'> TagBox[TagBox[SuperscriptBox["\[DoubleStruckCapitalN]", "+"], Function[Integers]], Function[Integers]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <mi> d </mi> <mo> ∈ </mo> <semantics> <msup> <mi> ℕ </mi> <mo> + </mo> </msup> <annotation encoding='Mathematica'> TagBox[TagBox[SuperscriptBox["\[DoubleStruckCapitalN]", "+"], Function[Integers]], Function[Integers]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <mi> e </mi> <mo> ∈ </mo> <semantics> <msup> <mi> ℕ </mi> <mo> + </mo> </msup> <annotation encoding='Mathematica'> TagBox[TagBox[SuperscriptBox["\[DoubleStruckCapitalN]", "+"], Function[Integers]], Function[Integers]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <mi> f </mi> <mo> ∈ </mo> <semantics> <msup> <mi> ℕ </mi> <mo> + </mo> </msup> <annotation encoding='Mathematica'> TagBox[TagBox[SuperscriptBox["\[DoubleStruckCapitalN]", "+"], Function[Integers]], Function[Integers]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <mi> g </mi> <mo> ∈ </mo> <semantics> <msup> <mi> ℕ </mi> <mo> + </mo> </msup> <annotation encoding='Mathematica'> TagBox[SuperscriptBox["\[DoubleStruckCapitalN]", "+"], Function[Integers]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <mi> h </mi> <mo> ∈ </mo> <semantics> <msup> <mi> ℕ </mi> <mo> + </mo> </msup> <annotation encoding='Mathematica'> TagBox[SuperscriptBox["\[DoubleStruckCapitalN]", "+"], Function[Integers]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <mi> i </mi> <mo> ∈ </mo> <semantics> <msup> <mi> ℕ </mi> <mo> + </mo> </msup> <annotation encoding='Mathematica'> TagBox[TagBox[SuperscriptBox["\[DoubleStruckCapitalN]", "+"], Function[Integers]], Function[Integers]] </annotation> </semantics> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> SixJSymbol </ci> <list> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 3 </cn> </apply> </list> <list> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 4 </cn> </apply> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 5 </cn> </apply> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 6 </cn> </apply> </list> </apply> <cn type='integer'> 0 </cn> </apply> <apply> <and /> <apply> <eq /> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <apply> <times /> <ci> a </ci> <ci> b </ci> <ci> c </ci> </apply> <apply> <times /> <ci> d </ci> <ci> e </ci> <ci> f </ci> </apply> <apply> <times /> <ci> a </ci> <ci> d </ci> <ci> g </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <eq /> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <apply> <times /> <ci> a </ci> <ci> b </ci> <ci> c </ci> </apply> <apply> <times /> <ci> b </ci> <ci> e </ci> <ci> h </ci> </apply> <apply> <times /> <ci> g </ci> <ci> h </ci> <ci> i </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <eq /> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 3 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <apply> <times /> <ci> d </ci> <ci> e </ci> <ci> f </ci> </apply> <apply> <times /> <ci> a </ci> <ci> d </ci> <ci> g </ci> </apply> <apply> <times /> <ci> b </ci> <ci> e </ci> <ci> h </ci> </apply> <apply> <times /> <ci> g </ci> <ci> h </ci> <ci> i </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <eq /> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 4 </cn> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <apply> <times /> <ci> a </ci> <ci> d </ci> <ci> g </ci> </apply> <apply> <times /> <ci> h </ci> <ci> i </ci> <ci> g </ci> </apply> </apply> </apply> </apply> <apply> <eq /> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 5 </cn> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <apply> <times /> <ci> d </ci> <ci> e </ci> <ci> f </ci> </apply> <apply> <times /> <ci> b </ci> <ci> e </ci> <ci> h </ci> </apply> </apply> </apply> </apply> <apply> <eq /> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 6 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <apply> <times /> <ci> a </ci> <ci> b </ci> <ci> c </ci> </apply> <apply> <times /> <ci> a </ci> <ci> d </ci> <ci> g </ci> </apply> <apply> <times /> <ci> b </ci> <ci> e </ci> <ci> h </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <eq /> <apply> <plus /> <apply> <times /> <ci> a </ci> <ci> b </ci> <ci> c </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> f </ci> <ci> i </ci> <ci> c </ci> </apply> </apply> <apply> <times /> <ci> d </ci> <ci> e </ci> <ci> f </ci> </apply> <apply> <times /> <ci> a </ci> <ci> d </ci> <ci> g </ci> </apply> <apply> <times /> <ci> b </ci> <ci> e </ci> <ci> h </ci> </apply> <apply> <times /> <ci> g </ci> <ci> h </ci> <ci> i </ci> </apply> </apply> <cn type='integer'> 0 </cn> </apply> <apply> <in /> <ci> a </ci> <integers /> </apply> <apply> <in /> <ci> b </ci> <integers /> </apply> <apply> <in /> <ci> c </ci> <integers /> </apply> <apply> <in /> <ci> d </ci> <integers /> </apply> <apply> <in /> <ci> e </ci> <integers /> </apply> <apply> <in /> <ci> f </ci> <integers /> </apply> <apply> <in /> <ci> g </ci> <integers /> </apply> <apply> <in /> <ci> h </ci> <integers /> </apply> <apply> <in /> <ci> i </ci> <integers /> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

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Date Added to functions.wolfram.com (modification date)

2002-12-18