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SixJSymbol






Mathematica Notation

Traditional Notation









Hypergeometric Functions > SixJSymbol[{j1,j2,j3},{j4,j5,j6}] > Summation > Finite summation > Involving one 6j symbol and two 9j symbols





http://functions.wolfram.com/07.40.23.0016.01









  


  










Input Form





Sum[(2 k + 1) (2 l + 1) NineJSymbol[{Subscript[j, 1], Subscript[j, 2], k}, {Subscript[j, 3], Subscript[j, 4], Subscript[j, 5]}, {Subscript[j, 6], Subscript[j, 7], Subscript[j, 8]}] NineJSymbol[{Subscript[j, 1], Subscript[j, 2], k}, {Subscript[j, 9], Subscript[j, 10], Subscript[j, 5]}, {l, Subscript[j, 11], Subscript[j, 8]}] SixJSymbol[{Subscript[j, 1], Subscript[j, 9], l}, {Subscript[j, 11], Subscript[j, 8], Subscript[j, 12]}], {k, Max[Abs[Subscript[j, 1] - Subscript[j, 2]], Abs[Subscript[j, 5] - Subscript[j, 8]]], Min[Subscript[j, 1] + Subscript[j, 2], Subscript[j, 5] + Subscript[j, 8]]}, {l, Max[Abs[Subscript[j, 1] - Subscript[j, 9]], Abs[Subscript[j, 8] - Subscript[j, 11]]], Min[Subscript[j, 1] + Subscript[j, 9], Subscript[j, 8] + Subscript[j, 11]]}] == SixJSymbol[{Subscript[j, 1], Subscript[j, 8], Subscript[j, 12]}, {Subscript[j, 7], Subscript[j, 3], Subscript[j, 6]}] SixJSymbol[{Subscript[j, 7], Subscript[j, 3], Subscript[j, 12]}, {Subscript[j, 5], Subscript[j, 2], Subscript[j, 4]}] SixJSymbol[{Subscript[j, 5], Subscript[j, 2], Subscript[j, 12]}, {Subscript[j, 11], Subscript[j, 9], Subscript[j, 10]}]










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", RowBox[List["Max", "[", RowBox[List[RowBox[List["Abs", "[", RowBox[List[SubscriptBox["j", "1"], "-", SubscriptBox["j", "2"]]], "]"]], ",", RowBox[List["Abs", "[", RowBox[List[SubscriptBox["j", "5"], "-", SubscriptBox["j", "8"]]], "]"]]]], "]"]]]], RowBox[List["Min", "[", RowBox[List[RowBox[List[SubscriptBox["j", "1"], "+", SubscriptBox["j", "2"]]], ",", RowBox[List[SubscriptBox["j", "5"], "+", SubscriptBox["j", "8"]]]]], "]"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["l", "=", RowBox[List["Max", "[", RowBox[List[RowBox[List["Abs", "[", RowBox[List[SubscriptBox["j", "1"], "-", SubscriptBox["j", "9"]]], "]"]], ",", RowBox[List["Abs", "[", RowBox[List[SubscriptBox["j", "8"], "-", SubscriptBox["j", "11"]]], "]"]]]], "]"]]]], RowBox[List["Min", "[", RowBox[List[RowBox[List[SubscriptBox["j", "1"], "+", SubscriptBox["j", "9"]]], ",", RowBox[List[SubscriptBox["j", "8"], "+", SubscriptBox["j", "11"]]]]], "]"]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", "k"]], "+", "1"]], ")"]], RowBox[List["(", RowBox[List[RowBox[List["2", "l"]], "+", "1"]], ")"]], RowBox[List["NineJSymbol", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["j", "1"], ",", SubscriptBox["j", "2"], ",", "k"]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["j", "3"], ",", SubscriptBox["j", "4"], ",", SubscriptBox["j", "5"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["j", "6"], ",", SubscriptBox["j", "7"], ",", SubscriptBox["j", "8"]]], "}"]]]], "]"]], RowBox[List["NineJSymbol", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["j", "1"], ",", SubscriptBox["j", "2"], ",", "k"]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["j", "9"], ",", SubscriptBox["j", "10"], ",", SubscriptBox["j", "5"]]], "}"]], ",", RowBox[List["{", RowBox[List["l", ",", SubscriptBox["j", "11"], ",", SubscriptBox["j", "8"]]], "}"]]]], "]"]], RowBox[List["SixJSymbol", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["j", "1"], ",", SubscriptBox["j", "9"], ",", "l"]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["j", "11"], ",", SubscriptBox["j", "8"], ",", SubscriptBox["j", "12"]]], "}"]]]], "]"]]]]]]]], "\[Equal]", RowBox[List[RowBox[List["SixJSymbol", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["j", "1"], ",", SubscriptBox["j", "8"], ",", SubscriptBox["j", "12"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["j", "7"], ",", SubscriptBox["j", "3"], ",", SubscriptBox["j", "6"]]], "}"]]]], "]"]], RowBox[List["SixJSymbol", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["j", "7"], ",", SubscriptBox["j", "3"], ",", SubscriptBox["j", "12"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["j", "5"], ",", SubscriptBox["j", "2"], ",", SubscriptBox["j", "4"]]], "}"]]]], "]"]], RowBox[List["SixJSymbol", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["j", "5"], ",", SubscriptBox["j", "2"], ",", SubscriptBox["j", "12"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["j", "11"], ",", SubscriptBox["j", "9"], ",", SubscriptBox["j", "10"]]], "}"]]]], "]"]]]]]]]]










MathML Form







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<mtd> <mi> l </mi> </mtd> <mtd> <msub> <mi> j </mi> <mn> 11 </mn> </msub> </mtd> <mtd> <msub> <mi> j </mi> <mn> 8 </mn> </msub> </mtd> </mtr> </mtable> <mo> } </mo> </mrow> <mo> &#8290; </mo> <mrow> <semantics> <mo> { </mo> <annotation encoding='Mathematica'> TagBox[StyleBox[&quot;{&quot;, Rule[SpanMaxSize, DirectedInfinity[1]]], SixJSymbol] </annotation> </semantics> <mtext> &#8287; </mtext> <mtable> <mtr> <mtd> <msub> <mi> j </mi> <mn> 1 </mn> </msub> </mtd> <mtd> <msub> <mi> j </mi> <mn> 9 </mn> </msub> </mtd> <mtd> <mi> l </mi> </mtd> </mtr> <mtr> <mtd> <msub> <mi> j </mi> <mn> 11 </mn> </msub> </mtd> <mtd> <msub> <mi> j </mi> <mn> 8 </mn> </msub> </mtd> <mtd> <msub> <mi> j </mi> <mn> 12 </mn> </msub> </mtd> </mtr> </mtable> <mtext> &#8287; </mtext> <semantics> <mo> } </mo> <annotation encoding='Mathematica'> TagBox[StyleBox[&quot;}&quot;, Rule[SpanMaxSize, DirectedInfinity[1]]], SixJSymbol] </annotation> </semantics> </mrow> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <semantics> <mo> { </mo> <annotation encoding='Mathematica'> TagBox[StyleBox[&quot;{&quot;, Rule[SpanMaxSize, DirectedInfinity[1]]], SixJSymbol] </annotation> </semantics> <mtext> &#8287; </mtext> <mtable> <mtr> <mtd> <msub> <mi> j </mi> <mn> 1 </mn> </msub> </mtd> <mtd> <msub> <mi> j </mi> <mn> 8 </mn> </msub> </mtd> <mtd> <msub> <mi> j </mi> <mn> 12 </mn> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi> j </mi> <mn> 7 </mn> </msub> </mtd> <mtd> <msub> <mi> j </mi> <mn> 3 </mn> </msub> </mtd> <mtd> <msub> <mi> j </mi> <mn> 6 </mn> </msub> </mtd> </mtr> </mtable> <mtext> &#8287; </mtext> <semantics> <mo> } </mo> <annotation encoding='Mathematica'> TagBox[StyleBox[&quot;}&quot;, Rule[SpanMaxSize, DirectedInfinity[1]]], SixJSymbol] </annotation> </semantics> </mrow> <mo> &#8290; </mo> <mrow> <semantics> <mo> { </mo> <annotation encoding='Mathematica'> TagBox[StyleBox[&quot;{&quot;, Rule[SpanMaxSize, DirectedInfinity[1]]], SixJSymbol] </annotation> </semantics> <mtext> &#8287; </mtext> <mtable> <mtr> <mtd> <msub> <mi> j </mi> <mn> 7 </mn> </msub> </mtd> <mtd> <msub> <mi> j </mi> <mn> 3 </mn> </msub> </mtd> <mtd> <msub> <mi> j </mi> <mn> 12 </mn> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi> j </mi> <mn> 5 </mn> </msub> </mtd> <mtd> <msub> <mi> j </mi> <mn> 2 </mn> </msub> </mtd> <mtd> <msub> <mi> j </mi> <mn> 4 </mn> </msub> </mtd> </mtr> </mtable> <mtext> &#8287; </mtext> <semantics> <mo> } </mo> <annotation encoding='Mathematica'> TagBox[StyleBox[&quot;}&quot;, Rule[SpanMaxSize, DirectedInfinity[1]]], SixJSymbol] </annotation> </semantics> </mrow> <mo> &#8290; </mo> <mrow> <semantics> <mo> { </mo> <annotation encoding='Mathematica'> TagBox[StyleBox[&quot;{&quot;, Rule[SpanMaxSize, DirectedInfinity[1]]], SixJSymbol] </annotation> </semantics> <mtext> &#8287; </mtext> <mtable> <mtr> <mtd> <msub> <mi> j </mi> <mn> 5 </mn> </msub> </mtd> <mtd> <msub> <mi> j </mi> <mn> 2 </mn> </msub> </mtd> <mtd> <msub> <mi> j </mi> <mn> 12 </mn> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi> j </mi> <mn> 11 </mn> </msub> </mtd> <mtd> <msub> <mi> j </mi> <mn> 9 </mn> </msub> </mtd> <mtd> <msub> <mi> j </mi> <mn> 10 </mn> </msub> </mtd> </mtr> </mtable> <mtext> &#8287; </mtext> <semantics> <mo> } </mo> <annotation encoding='Mathematica'> TagBox[StyleBox[&quot;}&quot;, Rule[SpanMaxSize, DirectedInfinity[1]]], SixJSymbol] </annotation> </semantics> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <sum /> <bvar> <ci> l </ci> </bvar> <lowlimit> <apply> <max /> <apply> <abs /> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 9 </cn> </apply> </apply> </apply> </apply> <apply> <abs /> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 8 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 11 </cn> </apply> </apply> </apply> </apply> </apply> </lowlimit> <uplimit> <apply> <min /> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 9 </cn> </apply> </apply> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 8 </cn> </apply> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 11 </cn> </apply> </apply> </apply> </uplimit> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <apply> <max /> <apply> <abs /> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <apply> <abs /> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 5 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 8 </cn> </apply> </apply> </apply> </apply> </apply> </lowlimit> <uplimit> <apply> <min /> <apply> <plus /> <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> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 5 </cn> </apply> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 8 </cn> </apply> </apply> </apply> </uplimit> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> l </ci> </apply> <cn type='integer'> 1 </cn> </apply> <list> <list> <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> <ci> k </ci> </list> <list> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 3 </cn> </apply> <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> </list> <list> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 6 </cn> </apply> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 7 </cn> </apply> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 8 </cn> </apply> </list> </list> </list> <list> <list> <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> <ci> k </ci> </list> <list> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 9 </cn> </apply> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 10 </cn> </apply> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 5 </cn> </apply> </list> <list> <ci> l </ci> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 11 </cn> </apply> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 8 </cn> </apply> </list> </list> </list> <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'> 9 </cn> </apply> <ci> l </ci> </list> <list> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 11 </cn> </apply> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 8 </cn> </apply> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 12 </cn> </apply> </list> </apply> </apply> </apply> </apply> <apply> <times /> <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'> 8 </cn> </apply> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 12 </cn> </apply> </list> <list> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 7 </cn> </apply> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 3 </cn> </apply> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 6 </cn> </apply> </list> </apply> <apply> <ci> SixJSymbol </ci> <list> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 7 </cn> </apply> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 3 </cn> </apply> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 12 </cn> </apply> </list> <list> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 5 </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'> 4 </cn> </apply> </list> </apply> <apply> <ci> SixJSymbol </ci> <list> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 5 </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'> 12 </cn> </apply> </list> <list> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 11 </cn> </apply> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 9 </cn> </apply> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 10 </cn> </apply> </list> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", RowBox[List["Max", "[", RowBox[List[RowBox[List["Abs", "[", RowBox[List[SubscriptBox["j", "1"], "-", SubscriptBox["j", "2"]]], "]"]], ",", RowBox[List["Abs", "[", RowBox[List[SubscriptBox["j", "5"], "-", SubscriptBox["j", "8"]]], "]"]]]], "]"]]]], RowBox[List["Min", "[", RowBox[List[RowBox[List[SubscriptBox["j", "1"], "+", SubscriptBox["j", "2"]]], ",", RowBox[List[SubscriptBox["j", "5"], "+", SubscriptBox["j", "8"]]]]], "]"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["l_", "=", RowBox[List["Max", "[", RowBox[List[RowBox[List["Abs", "[", RowBox[List[SubscriptBox["j", "1"], "-", SubscriptBox["j", "9"]]], "]"]], ",", RowBox[List["Abs", "[", RowBox[List[SubscriptBox["j", "8"], "-", SubscriptBox["j", "11"]]], "]"]]]], "]"]]]], RowBox[List["Min", "[", RowBox[List[RowBox[List[SubscriptBox["j", "1"], "+", SubscriptBox["j", "9"]]], ",", RowBox[List[SubscriptBox["j", "8"], "+", SubscriptBox["j", "11"]]]]], "]"]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "l_"]], "+", "1"]], ")"]], " ", RowBox[List["NineJSymbol", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["j", "1"], ",", SubscriptBox["j", "2"], ",", "k"]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["j", "3"], ",", SubscriptBox["j", "4"], ",", SubscriptBox["j", "5"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["j", "6"], ",", SubscriptBox["j", "7"], ",", SubscriptBox["j", "8"]]], "}"]]]], "]"]], " ", RowBox[List["NineJSymbol", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["j", "1"], ",", SubscriptBox["j", "2"], ",", "k"]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["j", "9"], ",", SubscriptBox["j", "10"], ",", SubscriptBox["j", "5"]]], "}"]], ",", RowBox[List["{", RowBox[List["l_", ",", SubscriptBox["j", "11"], ",", SubscriptBox["j", "8"]]], "}"]]]], "]"]], " ", RowBox[List["SixJSymbol", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["j", "1"], ",", SubscriptBox["j", "9"], ",", "l_"]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["j", "11"], ",", SubscriptBox["j", "8"], ",", SubscriptBox["j", "12"]]], "}"]]]], "]"]]]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["SixJSymbol", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["j", "1"], ",", SubscriptBox["j", "8"], ",", SubscriptBox["j", "12"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["j", "7"], ",", SubscriptBox["j", "3"], ",", SubscriptBox["j", "6"]]], "}"]]]], "]"]], " ", RowBox[List["SixJSymbol", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["j", "7"], ",", SubscriptBox["j", "3"], ",", SubscriptBox["j", "12"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["j", "5"], ",", SubscriptBox["j", "2"], ",", SubscriptBox["j", "4"]]], "}"]]]], "]"]], " ", RowBox[List["SixJSymbol", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["j", "5"], ",", SubscriptBox["j", "2"], ",", SubscriptBox["j", "12"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["j", "11"], ",", SubscriptBox["j", "9"], ",", SubscriptBox["j", "10"]]], "}"]]]], "]"]]]]]]]]










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





2001-12-21





© 1998- Wolfram Research, Inc.