html, body, form { margin: 0; padding: 0; width: 100%; } #calculate { position: relative; width: 177px; height: 110px; background: transparent url(/images/alphabox/embed_functions_inside.gif) no-repeat scroll 0 0; } #i { position: relative; left: 18px; top: 44px; width: 133px; border: 0 none; outline: 0; font-size: 11px; } #eq { width: 9px; height: 10px; background: transparent; position: absolute; top: 47px; right: 18px; cursor: pointer; }

 ThreeJSymbol

 http://functions.wolfram.com/07.39.17.0015.01

 Input Form

 ThreeJSymbol[{Subscript[j, 1], Subscript[m, 1]}, {Subscript[j, 2], Subscript[m, 2]}, {Subscript[j, 3], Subscript[m, 3]}] == ((Sqrt[(Subscript[j, 1] + Subscript[j, 2] - Subscript[j, 3])!] Sqrt[(-Subscript[j, 1] + Subscript[j, 2] + Subscript[j, 3])!] Sqrt[(Subscript[j, 1] + Subscript[j, 2] + Subscript[j, 3] + 1)!] Sqrt[(-2 n + Subscript[j, 2] + Subscript[m, 2])!] Sqrt[(Subscript[j, 3] - Subscript[m, 3])!])/ (Sqrt[(Subscript[j, 1] - Subscript[j, 2] + Subscript[j, 3])!] Sqrt[(Subscript[j, 2] + Subscript[m, 2])!] Sqrt[(Subscript[j, 3] + Subscript[m, 3])!])) Sum[(-1)^(-Subscript[j, 3] + k + n) (((2 k + 1) (Subscript[j, 3] + k - n)! (2 n)! Sqrt[(k - -Subscript[m, 3] + n)!] Sqrt[(Subscript[j, 1] - Subscript[j, 2] + k + n)!])/ ((Subscript[j, 3] - k + n)! (Subscript[j, 3] + k + n + 1)! (-Subscript[j, 3] + k + n)! Sqrt[(k - Subscript[m, 3] - n)!] Sqrt[(-Subscript[j, 1] + Subscript[j, 2] + k - n)!] Sqrt[(Subscript[j, 1] + Subscript[j, 2] - k - n)!] Sqrt[(Subscript[j, 1] + Subscript[j, 2] + k - n + 1)!])) ThreeJSymbol[{Subscript[j, 1], Subscript[m, 1]}, {Subscript[j, 2] - n, Subscript[m, 2] - n}, {k, Subscript[m, 3] + n}], {k, Subscript[j, 3] - n, Subscript[j, 3] + n}] /; Element[2 n, Integers] && 0 <= n <= (Subscript[j, 2] + Subscript[m, 2])/2 && \[ScriptCapitalP]\[ScriptH]\[ScriptY]\[ScriptS]\[ScriptI]\[ScriptC]\ \[ScriptA]\[ScriptL]\[ScriptCapitalQ][{Subscript[j, 1], Subscript[m, 1]}, {Subscript[j, 2], Subscript[m, 2]}, {Subscript[j, 3], Subscript[m, 3]}]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["ThreeJSymbol", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["j", "1"], ",", SubscriptBox["m", "1"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["j", "2"], ",", SubscriptBox["m", "2"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["j", "3"], ",", SubscriptBox["m", "3"]]], "}"]]]], "]"]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[SqrtBox[RowBox[List[RowBox[List["(", RowBox[List[SubscriptBox["j", "1"], "+", SubscriptBox["j", "2"], "-", SubscriptBox["j", "3"]]], ")"]], "!"]]], " ", SqrtBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", SubscriptBox["j", "1"]]], "+", SubscriptBox["j", "2"], "+", SubscriptBox["j", "3"]]], ")"]], "!"]]], " ", SqrtBox[RowBox[List[RowBox[List["(", RowBox[List[SubscriptBox["j", "1"], "+", SubscriptBox["j", "2"], "+", SubscriptBox["j", "3"], "+", "1"]], ")"]], "!"]]], " ", SqrtBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "n"]], "+", SubscriptBox["j", "2"], "+", SubscriptBox["m", "2"]]], ")"]], "!"]]], " ", SqrtBox[RowBox[List[RowBox[List["(", RowBox[List[SubscriptBox["j", "3"], "-", SubscriptBox["m", "3"]]], ")"]], "!"]]]]], ")"]], "/", RowBox[List["(", RowBox[List[SqrtBox[RowBox[List[RowBox[List["(", RowBox[List[SubscriptBox["j", "1"], "-", SubscriptBox["j", "2"], "+", SubscriptBox["j", "3"]]], ")"]], "!"]]], " ", SqrtBox[RowBox[List[RowBox[List["(", RowBox[List[SubscriptBox["j", "2"], "+", SubscriptBox["m", "2"]]], ")"]], "!"]]], " ", SqrtBox[RowBox[List[RowBox[List["(", RowBox[List[SubscriptBox["j", "3"], "+", SubscriptBox["m", "3"]]], ")"]], "!"]]]]], ")"]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", RowBox[List[SubscriptBox["j", "3"], "-", "n"]]]], RowBox[List[SubscriptBox["j", "3"], "+", "n"]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["-", SubscriptBox["j", "3"]]], "+", "k", "+", "n"]]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", "k"]], "+", "1"]], ")"]], RowBox[List[RowBox[List["(", RowBox[List[SubscriptBox["j", "3"], "+", "k", "-", "n"]], ")"]], "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["2", " ", "n"]], ")"]], "!"]], " ", SqrtBox[RowBox[List[RowBox[List["(", RowBox[List["k", "-", RowBox[List["-", SubscriptBox["m", "3"]]], "+", "n"]], ")"]], "!"]]], " ", SqrtBox[RowBox[List[RowBox[List["(", RowBox[List[SubscriptBox["j", "1"], "-", SubscriptBox["j", "2"], "+", "k", "+", "n"]], ")"]], "!"]]]]], ")"]], "/", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[SubscriptBox["j", "3"], "-", "k", "+", "n"]], ")"]], "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List[SubscriptBox["j", "3"], "+", "k", "+", "n", "+", "1"]], ")"]], "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", SubscriptBox["j", "3"]]], "+", "k", "+", "n"]], ")"]], "!"]], " ", SqrtBox[RowBox[List[RowBox[List["(", RowBox[List["k", "-", SubscriptBox["m", "3"], "-", "n"]], ")"]], "!"]]], " ", SqrtBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", SubscriptBox["j", "1"]]], "+", SubscriptBox["j", "2"], "+", "k", "-", "n"]], ")"]], "!"]]], " ", SqrtBox[RowBox[List[RowBox[List["(", RowBox[List[SubscriptBox["j", "1"], "+", SubscriptBox["j", "2"], "-", "k", "-", "n"]], ")"]], "!"]]], " ", SqrtBox[RowBox[List[RowBox[List["(", RowBox[List[SubscriptBox["j", "1"], "+", SubscriptBox["j", "2"], "+", "k", "-", "n", "+", "1"]], ")"]], "!"]]]]], ")"]]]], RowBox[List["ThreeJSymbol", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["j", "1"], ",", SubscriptBox["m", "1"]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[SubscriptBox["j", "2"], "-", "n"]], ",", RowBox[List[SubscriptBox["m", "2"], "-", "n"]]]], "}"]], ",", RowBox[List["{", RowBox[List["k", ",", RowBox[List[SubscriptBox["m", "3"], "+", "n"]]]], "}"]]]], "]"]]]]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["2", "n"]], "\[Element]", "Integers"]], "\[And]", RowBox[List["0", "\[LessEqual]", "n", "\[LessEqual]", FractionBox[RowBox[List[SubscriptBox["j", "2"], "+", SubscriptBox["m", "2"]]], "2"]]], "\[And]", RowBox[List["\[ScriptCapitalP]\[ScriptH]\[ScriptY]\[ScriptS]\[ScriptI]\[ScriptC]\[ScriptA]\[ScriptL]\[ScriptCapitalQ]", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["j", "1"], ",", SubscriptBox["m", "1"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["j", "2"], ",", SubscriptBox["m", "2"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["j", "3"], ",", SubscriptBox["m", "3"]]], "}"]]]], "]"]]]]]]]]

 MathML Form

 ( TagBox[StyleBox["(", Rule[SpanMaxSize, DirectedInfinity[1]]], ThreeJSymbol] j 1 j 2 j 3 m 1 m 2 m 3 ) TagBox[StyleBox[")", Rule[SpanMaxSize, DirectedInfinity[1]]], ThreeJSymbol] ( ( j 1 + j 2 - j 3 ) ! ( - j 1 + j 2 + j 3 ) ! ( j 1 + j 2 + j 3 + 1 ) ! ( - 2 n + j 2 + m 2 ) ! ( j 3 - m 3 ) ! ) / ( ( j 1 - j 2 + j 3 ) ! ( j 2 + m 2 ) ! ( j 3 + m 3 ) ! ) k = j 3 - n j 3 + n ( - 1 ) - j 3 + k + n ( ( 2 k + 1 ) ( j 3 + k - n ) ! ( 2 n ) ! ( k - - m 3 + n ) ! ( j 1 - j 2 + k + n ) ! ) / ( ( j 3 - k + n ) ! ( j 3 + k + n + 1 ) ! ( - j 3 + k + n ) ! ( k - m 3 - n ) ! ( - j 1 + j 2 + k - n ) ! ( j 1 + j 2 - k - n ) ! ( j 1 + j 2 + k - n + 1 ) ! ) ( TagBox[StyleBox["(", Rule[SpanMaxSize, DirectedInfinity[1]]], ThreeJSymbol] j 1 j 2 - n k m 1 m 2 - n n + m 3 ) TagBox[StyleBox[")", Rule[SpanMaxSize, DirectedInfinity[1]]], ThreeJSymbol] /; 2 n n j 2 + m 2 2 𝒫𝒽𝓎𝓈𝒾𝒸𝒶ℓ𝒬 ( { j 1 , m 1 } , { j 2 , m 2 } , { j 3 , m 3 } ) Condition ThreeJSymbol Subscript j 1 Subscript m 1 Subscript j 2 Subscript m 2 Subscript j 3 Subscript m 3 Subscript j 1 Subscript j 2 -1 Subscript j 3 1 2 -1 Subscript j 1 Subscript j 2 Subscript j 3 1 2 Subscript j 1 Subscript j 2 Subscript j 3 1 1 2 -2 n Subscript j 2 Subscript m 2 1 2 Subscript j 3 -1 Subscript m 3 1 2 Subscript j 1 -1 Subscript j 2 Subscript j 3 1 2 Subscript j 2 Subscript m 2 1 2 Subscript j 3 Subscript m 3 1 2 -1 k Subscript j 3 -1 n Subscript j 3 n -1 -1 Subscript j 3 k n 2 k 1 Subscript j 3 k -1 n 2 n k -1 -1 Subscript m 3 n 1 2 Subscript j 1 -1 Subscript j 2 k n 1 2 Subscript j 3 -1 k n Subscript j 3 k n 1 -1 Subscript j 3 k n k -1 Subscript m 3 -1 n 1 2 -1 Subscript j 1 Subscript j 2 k -1 n 1 2 Subscript j 1 Subscript j 2 -1 k -1 n 1 2 Subscript j 1 Subscript j 2 k -1 n 1 1 2 -1 ThreeJSymbol Subscript j 1 Subscript m 1 Subscript j 2 -1 n Subscript m 2 -1 n k n Subscript m 3 2 n n Subscript j 2 Subscript m 2 2 -1 𝒫𝒽𝓎𝓈𝒾𝒸𝒶ℓ𝒬 Subscript j 1 Subscript m 1 Subscript j 2 Subscript m 2 Subscript j 3 Subscript m 3 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ThreeJSymbol", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["j", "1"], ",", SubscriptBox["m_", "1"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["j", "2"], ",", SubscriptBox["m_", "2"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["j", "3"], ",", SubscriptBox["m_", "3"]]], "}"]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SqrtBox[RowBox[List[RowBox[List["(", RowBox[List[SubscriptBox["j", "1"], "+", SubscriptBox["j", "2"], "-", SubscriptBox["j", "3"]]], ")"]], "!"]]], " ", SqrtBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", SubscriptBox["j", "1"]]], "+", SubscriptBox["j", "2"], "+", SubscriptBox["j", "3"]]], ")"]], "!"]]], " ", SqrtBox[RowBox[List[RowBox[List["(", RowBox[List[SubscriptBox["j", "1"], "+", SubscriptBox["j", "2"], "+", SubscriptBox["j", "3"], "+", "1"]], ")"]], "!"]]], " ", SqrtBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "n"]], "+", SubscriptBox["j", "2"], "+", SubscriptBox["mm", "2"]]], ")"]], "!"]]], " ", SqrtBox[RowBox[List[RowBox[List["(", RowBox[List[SubscriptBox["j", "3"], "-", SubscriptBox["mm", "3"]]], ")"]], "!"]]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", RowBox[List[SubscriptBox["j", "3"], "-", "n"]]]], RowBox[List[SubscriptBox["j", "3"], "+", "n"]]], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["-", SubscriptBox["j", "3"]]], "+", "k", "+", "n"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]], ")"]], " ", RowBox[List[RowBox[List["(", RowBox[List[SubscriptBox["j", "3"], "+", "k", "-", "n"]], ")"]], "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["2", " ", "n"]], ")"]], "!"]], " ", SqrtBox[RowBox[List[RowBox[List["(", RowBox[List["k", "-", RowBox[List["-", SubscriptBox["mm", "3"]]], "+", "n"]], ")"]], "!"]]], " ", SqrtBox[RowBox[List[RowBox[List["(", RowBox[List[SubscriptBox["j", "1"], "-", SubscriptBox["j", "2"], "+", "k", "+", "n"]], ")"]], "!"]]]]], ")"]], " ", RowBox[List["ThreeJSymbol", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["j", "1"], ",", SubscriptBox["mm", "1"]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[SubscriptBox["j", "2"], "-", "n"]], ",", RowBox[List[SubscriptBox["mm", "2"], "-", "n"]]]], "}"]], ",", RowBox[List["{", RowBox[List["k", ",", RowBox[List[SubscriptBox["mm", "3"], "+", "n"]]]], "}"]]]], "]"]]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[SubscriptBox["j", "3"], "-", "k", "+", "n"]], ")"]], "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List[SubscriptBox["j", "3"], "+", "k", "+", "n", "+", "1"]], ")"]], "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", SubscriptBox["j", "3"]]], "+", "k", "+", "n"]], ")"]], "!"]], " ", SqrtBox[RowBox[List[RowBox[List["(", RowBox[List["k", "-", SubscriptBox["mm", "3"], "-", "n"]], ")"]], "!"]]], " ", SqrtBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", SubscriptBox["j", "1"]]], "+", SubscriptBox["j", "2"], "+", "k", "-", "n"]], ")"]], "!"]]], " ", SqrtBox[RowBox[List[RowBox[List["(", RowBox[List[SubscriptBox["j", "1"], "+", SubscriptBox["j", "2"], "-", "k", "-", "n"]], ")"]], "!"]]], " ", SqrtBox[RowBox[List[RowBox[List["(", RowBox[List[SubscriptBox["j", "1"], "+", SubscriptBox["j", "2"], "+", "k", "-", "n", "+", "1"]], ")"]], "!"]]]]]]]]]], RowBox[List[SqrtBox[RowBox[List[RowBox[List["(", RowBox[List[SubscriptBox["j", "1"], "-", SubscriptBox["j", "2"], "+", SubscriptBox["j", "3"]]], ")"]], "!"]]], " ", SqrtBox[RowBox[List[RowBox[List["(", RowBox[List[SubscriptBox["j", "2"], "+", SubscriptBox["mm", "2"]]], ")"]], "!"]]], " ", SqrtBox[RowBox[List[RowBox[List["(", RowBox[List[SubscriptBox["j", "3"], "+", SubscriptBox["mm", "3"]]], ")"]], "!"]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["2", " ", "n"]], "\[Element]", "Integers"]], "&&", RowBox[List["0", "\[LessEqual]", "n", "\[LessEqual]", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[SubscriptBox["j", "2"], "+", SubscriptBox["mm", "2"]]], ")"]]]]]], "&&", RowBox[List["\[ScriptCapitalP]\[ScriptH]\[ScriptY]\[ScriptS]\[ScriptI]\[ScriptC]\[ScriptA]\[ScriptL]\[ScriptCapitalQ]", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["j", "1"], ",", SubscriptBox["mm", "1"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["j", "2"], ",", SubscriptBox["mm", "2"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["j", "3"], ",", SubscriptBox["mm", "3"]]], "}"]]]], "]"]]]]]]]]]]

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

 2001-12-21