![](/common/images/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
http://functions.wolfram.com/01.33.03.0004.01
|
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
|
|
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
Root[Function[z, Subscript[a, 0] + Subscript[a, 1] z + Subscript[a, 2] z^2 +
Subscript[a, 3] z^3], k] == -(Subscript[a, 2]/(3 Subscript[a, 3])) +
(E^((2 I Pi (k - 1))/3) p)/(3 2^(1/3) Subscript[a, 3]) -
(2^(1/3) r)/(E^((2 I Pi (k - 1))/3) (3 p Subscript[a, 3])) /;
p == (q + Sqrt[q^2 + 4 r^3])^(1/3) &&
q == 9 Subscript[a, 1] Subscript[a, 2] Subscript[a, 3] -
2 Subscript[a, 2]^3 - 27 Subscript[a, 0] Subscript[a, 3]^2 &&
r == 3 Subscript[a, 1] Subscript[a, 3] - Subscript[a, 2]^2 && 1 <= k <= 3
|
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Root", "[", RowBox[List[RowBox[List["Function", "[", RowBox[List["z", ",", RowBox[List[SubscriptBox["a", "0"], " ", "+", RowBox[List[SubscriptBox["a", "1"], "z"]], "+", RowBox[List[SubscriptBox["a", "2"], SuperscriptBox["z", "2"]]], "+", RowBox[List[SubscriptBox["a", "3"], SuperscriptBox["z", "3"]]]]]]], "]"]], ",", "k"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox[SubscriptBox["a", "2"], RowBox[List["3", " ", SubscriptBox["a", "3"]]]]]], "+", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["2", "\[ImaginaryI]", " ", "\[Pi]", RowBox[List["(", RowBox[List["k", "-", "1"]], ")"]]]], "3"]], "p"]], RowBox[List["3", " ", SuperscriptBox["2", RowBox[List["1", "/", "3"]]], " ", SubscriptBox["a", "3"]]]], "-", FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List["1", "/", "3"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[RowBox[List["2", "\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["(", RowBox[List["k", "-", "1"]], ")"]]]], "3"]]]], "r"]], RowBox[List["3", " ", "p", " ", SubscriptBox["a", "3"]]]]]]]], "/;", RowBox[List[RowBox[List["p", "\[Equal]", SuperscriptBox[RowBox[List["(", RowBox[List["q", "+", SqrtBox[RowBox[List[SuperscriptBox["q", "2"], "+", RowBox[List["4", " ", SuperscriptBox["r", "3"]]]]]]]], ")"]], RowBox[List["1", "/", "3"]]]]], "\[And]", RowBox[List["q", "\[Equal]", RowBox[List[RowBox[List["9", " ", SubscriptBox["a", "1"], " ", SubscriptBox["a", "2"], " ", SubscriptBox["a", "3"]]], "-", RowBox[List["2", " ", SubsuperscriptBox["a", "2", "3"]]], "-", RowBox[List["27", " ", SubscriptBox["a", "0"], " ", SubsuperscriptBox["a", "3", "2"]]]]]]], "\[And]", RowBox[List["r", "\[Equal]", RowBox[List[RowBox[List["3", " ", SubscriptBox["a", "1"], " ", SubscriptBox["a", "3"]]], "-", SubsuperscriptBox["a", "2", "2"]]]]], "\[And]", RowBox[List["1", "\[LessEqual]", "k", "\[LessEqual]", "3"]]]]]]]]
|
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <msubsup> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> ; </mo> <mrow> <msub> <mi> a </mi> <mn> 0 </mn> </msub> <mo> + </mo> <mtext> </mtext> <mrow> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> + </mo> <mrow> <msub> <mi> a </mi> <mn> 2 </mn> </msub> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <msub> <mi> a </mi> <mn> 3 </mn> </msub> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 3 </mn> </msup> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msubsup> <mo> ⩵ </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <msub> <mi> a </mi> <mn> 2 </mn> </msub> <mrow> <mn> 3 </mn> <mo> ⁢ </mo> <msub> <mi> a </mi> <mn> 3 </mn> </msub> </mrow> </mfrac> </mrow> <mo> + </mo> <mfrac> <mrow> <msup> <mi> ⅇ </mi> <mfrac> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mn> 3 </mn> </mfrac> </msup> <mo> ⁢ </mo> <mi> p </mi> </mrow> <mrow> <mn> 3 </mn> <mo> ⁢ </mo> <msup> <mn> 2 </mn> <mrow> <mn> 1 </mn> <mo> / </mo> <mn> 3 </mn> </mrow> </msup> <mo> ⁢ </mo> <msub> <mi> a </mi> <mn> 3 </mn> </msub> </mrow> </mfrac> <mo> - </mo> <mfrac> <mrow> <msup> <mn> 2 </mn> <mrow> <mn> 1 </mn> <mo> / </mo> <mn> 3 </mn> </mrow> </msup> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mrow> <mo> - </mo> <mfrac> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mn> 3 </mn> </mfrac> </mrow> </msup> <mo> ⁢ </mo> <mi> r </mi> </mrow> <mrow> <mn> 3 </mn> <mo> ⁢ </mo> <mi> p </mi> <mo> ⁢ </mo> <msub> <mi> a </mi> <mn> 3 </mn> </msub> </mrow> </mfrac> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> p </mi> <mo> ⩵ </mo> <mroot> <mrow> <mi> q </mi> <mo> + </mo> <msqrt> <mrow> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <msup> <mi> r </mi> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <msup> <mi> q </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> </mrow> <mn> 3 </mn> </mroot> </mrow> <mo> ∧ </mo> <mrow> <mi> q </mi> <mo> ⩵ </mo> <mrow> <mrow> <mn> 9 </mn> <mo> ⁢ </mo> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> ⁢ </mo> <msub> <mi> a </mi> <mn> 3 </mn> </msub> <mo> ⁢ </mo> <msub> <mi> a </mi> <mn> 2 </mn> </msub> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msubsup> <mi> a </mi> <mn> 2 </mn> <mn> 3 </mn> </msubsup> </mrow> <mo> - </mo> <mrow> <mn> 27 </mn> <mo> ⁢ </mo> <msub> <mi> a </mi> <mn> 0 </mn> </msub> <mo> ⁢ </mo> <msubsup> <mi> a </mi> <mn> 3 </mn> <mn> 2 </mn> </msubsup> </mrow> </mrow> </mrow> <mo> ∧ </mo> <mrow> <mi> r </mi> <mo> ⩵ </mo> <mrow> <mrow> <mn> 3 </mn> <mo> ⁢ </mo> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> ⁢ </mo> <msub> <mi> a </mi> <mn> 3 </mn> </msub> </mrow> <mo> - </mo> <msubsup> <mi> a </mi> <mn> 2 </mn> <mn> 2 </mn> </msubsup> </mrow> </mrow> <mo> ∧ </mo> <mrow> <mn> 1 </mn> <mo> ≤ </mo> <mi> k </mi> <mo> ≤ </mo> <mn> 3 </mn> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <power /> <apply> <ci> Subscript </ci> <apply> <ci> CompoundExpression </ci> <ci> z </ci> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> <apply> <times /> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 1 </cn> </apply> <ci> z </ci> </apply> <apply> <times /> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 3 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> </apply> <ci> k </ci> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 3 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <pi /> <apply> <plus /> <ci> k </ci> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <ci> p </ci> <apply> <power /> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 3 </cn> </apply> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 3 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 3 </cn> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <pi /> <apply> <plus /> <ci> k </ci> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <ci> r </ci> <apply> <power /> <apply> <times /> <cn type='integer'> 3 </cn> <ci> p </ci> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 3 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <eq /> <ci> p </ci> <apply> <power /> <apply> <plus /> <ci> q </ci> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <ci> r </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <power /> <ci> q </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 3 </cn> </apply> </apply> <apply> <eq /> <ci> q </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 9 </cn> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 3 </cn> </apply> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 27 </cn> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 3 </cn> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <eq /> <ci> r </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <apply> <leq /> <cn type='integer'> 1 </cn> <ci> k </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
|
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/clear.gif)
| ![](/images/home/spacer.gif)
| ![](/images/home/spacer.gif)
| ![](/images/home/spacer.gif)
| ![](/images/home/spacer.gif)
| | ![](/images/home/spacer.gif)
| ![](/images/home/spacer.gif)
| ![](/images/home/spacer.gif)
| ![](/images/home/spacer.gif)
| ![](/images/home/spacer.gif)
| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Root", "[", RowBox[List[RowBox[List["Function", "[", RowBox[List["z_", ",", RowBox[List[SubscriptBox["a_", "0"], "+", RowBox[List[SubscriptBox["a_", "1"], " ", "z_"]], "+", RowBox[List[SubscriptBox["a_", "2"], " ", SuperscriptBox["z_", "2"]]], "+", RowBox[List[SubscriptBox["a_", "3"], " ", SuperscriptBox["z_", "3"]]]]]]], "]"]], ",", "k_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["-", FractionBox[SubscriptBox["aa", "2"], RowBox[List["3", " ", SubscriptBox["aa", "3"]]]]]], "+", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox["2", "3"], " ", "\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["(", RowBox[List["k", "-", "1"]], ")"]]]]], " ", "p"]], RowBox[List["3", " ", SuperscriptBox["2", RowBox[List["1", "/", "3"]]], " ", SubscriptBox["aa", "3"]]]], "-", FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List["1", "/", "3"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox["1", "3"]]], " ", RowBox[List["(", RowBox[List["2", " ", "\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["(", RowBox[List["k", "-", "1"]], ")"]]]], ")"]]]]], " ", "r"]], RowBox[List["3", " ", "p", " ", SubscriptBox["aa", "3"]]]]]], "/;", RowBox[List[RowBox[List["p", "\[Equal]", SuperscriptBox[RowBox[List["(", RowBox[List["q", "+", SqrtBox[RowBox[List[SuperscriptBox["q", "2"], "+", RowBox[List["4", " ", SuperscriptBox["r", "3"]]]]]]]], ")"]], RowBox[List["1", "/", "3"]]]]], "&&", RowBox[List["q", "\[Equal]", RowBox[List[RowBox[List["9", " ", SubscriptBox["a", "1"], " ", SubscriptBox["aa", "2"], " ", SubscriptBox["aa", "3"]]], "-", RowBox[List["2", " ", SubsuperscriptBox["aa", "2", "3"]]], "-", RowBox[List["27", " ", SubscriptBox["a", "0"], " ", SubsuperscriptBox["aa", "3", "2"]]]]]]], "&&", RowBox[List["r", "\[Equal]", RowBox[List[RowBox[List["3", " ", SubscriptBox["a", "1"], " ", SubscriptBox["aa", "3"]]], "-", SubsuperscriptBox["aa", "2", "2"]]]]], "&&", RowBox[List["1", "\[LessEqual]", "k", "\[LessEqual]", "3"]]]]]]]]]] |
| ![](/images/home/spacer.gif)
| ![](/images/home/spacer.gif)
| ![](/images/home/spacer.gif)
| ![](/images/home/spacer.gif)
|
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
Date Added to functions.wolfram.com (modification date)
|
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
|
![](/common/images/spacer.gif) |
|
|