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http://functions.wolfram.com/01.02.10.0006.01
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(1 + z)^a == 1/(1 + ContinueFraction[{z (Floor[k/2] + (-1)^k a),
((3 - (-1)^k)/2) k + (-1)^k}, {k, 1, Infinity}]) /;
!IntervalMemberQ[Interval[{-Infinity, -1}], z]
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Cell[BoxData[RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "z"]], ")"]], "a"], "\[Equal]", RowBox[List["1", "/", RowBox[List["(", RowBox[List["1", "+", RowBox[List["ContinueFraction", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["z", " ", RowBox[List["(", RowBox[List[RowBox[List["Floor", "[", FractionBox["k", "2"], "]"]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", "a"]]]], ")"]]]], ",", RowBox[List[RowBox[List[FractionBox[RowBox[List["3", "-", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"]]], "2"], "k"]], "+", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"]]]]], "}"]], ",", RowBox[List["{", RowBox[List["k", ",", "1", ",", InterpretationBox["\[Infinity]", DirectedInfinity[1]]]], "}"]]]], "]"]]]], ")"]]]]]], "/;", "\[InvisibleSpace]", RowBox[List["!", RowBox[List["(", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List[RowBox[List["-", "\[Infinity]"]], ",", RowBox[List["-", "1"]]]], "}"]], "]"]], ",", "z"]], "]"]], ")"]]]]]]]]
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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mi> a </mi> </msup> <mo> ⩵ </mo> <mfrac> <mn> 1 </mn> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <msubsup> <mrow> <msub> <mi> Κ </mi> <mi> k </mi> </msub> <mo> ( </mo> <mrow> <mrow> <mi> z </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msup> <mo> ⁢ </mo> <mi> a </mi> </mrow> <mo> + </mo> <mrow> <mo> ⌊ </mo> <mfrac> <mi> k </mi> <mn> 2 </mn> </mfrac> <mo> ⌋ </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> , </mo> <mrow> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 3 </mn> <mo> - </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msup> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> k </mi> </mrow> <mo> + </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 1 </mn> <mi> ∞ </mi> </msubsup> </mrow> <mo> ) </mo> </mrow> </mfrac> </mrow> <mo> /; </mo> <mtext> ​ </mtext> <mrow> <mi> z </mi> <mo> ∉ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> ∞ </mi> </mrow> <mo> , </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <mrow> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mi> a </mi> </msup> <mo> ⩵ </mo> <mfrac> <mn> 1 </mn> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <msubsup> <mrow> <msub> <mi> Κ </mi> <mi> k </mi> </msub> <mo> ( </mo> <mrow> <mrow> <mi> z </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msup> <mo> ⁢ </mo> <mi> a </mi> </mrow> <mo> + </mo> <mrow> <mo> ⌊ </mo> <mfrac> <mi> k </mi> <mn> 2 </mn> </mfrac> <mo> ⌋ </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> , </mo> <mrow> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 3 </mn> <mo> - </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msup> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> k </mi> </mrow> <mo> + </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 1 </mn> <mi> ∞ </mi> </msubsup> </mrow> <mo> ) </mo> </mrow> </mfrac> </mrow> <mo> /; </mo> <mtext> ​ </mtext> <mrow> <mi> z </mi> <mo> ∉ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> ∞ </mi> </mrow> <mo> , </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "z_"]], ")"]], "a_"], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox["1", RowBox[List["1", "+", RowBox[List["ContinueFraction", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["z", " ", RowBox[List["(", RowBox[List[RowBox[List["Floor", "[", FractionBox["k", "2"], "]"]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", "a"]]]], ")"]]]], ",", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["3", "-", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"]]], ")"]], " ", "k"]], "+", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"]]]]], "}"]], ",", RowBox[List["{", RowBox[List["k", ",", "1", ",", "\[Infinity]"]], "}"]]]], "]"]]]]], "/;", RowBox[List["!", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List[RowBox[List["-", "\[Infinity]"]], ",", RowBox[List["-", "1"]]]], "}"]], "]"]], ",", "z"]], "]"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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