Sqrt[1 + z] == 1 + z/ (2 (1 + z/(2 (2 + (3 z)/(2 (3 + (3 z)/(2 (2 + (5 z)/(2 (5 + \[Ellipsis]))\ )))))))) /; !IntervalMemberQ[Interval[{-Infinity, -1}], z]

Cell[BoxData[RowBox[List[RowBox[List[SqrtBox[RowBox[List["1", "+", "z"]]], "\[Equal]", RowBox[List["1", "+", FractionBox["z", RowBox[List["2", " ", RowBox[List["(", RowBox[List["1", "+", FractionBox["z", RowBox[List["2", " ", RowBox[List["(", RowBox[List["2", "+", FractionBox[RowBox[List["3", " ", "z"]], RowBox[List["2", " ", RowBox[List["(", RowBox[List["3", "+", FractionBox[RowBox[List["3", " ", "z"]], RowBox[List["2", " ", RowBox[List["(", RowBox[List["2", "+", FractionBox[RowBox[List["5", " ", "z"]], RowBox[List["2", " ", RowBox[List["(", RowBox[List["5", "+", "\[Ellipsis]"]], ")"]]]]]]], ")"]]]]]]], ")"]]]]]]], ")"]]]]]]], ")"]]]]]]]]], "/;", "\[InvisibleSpace]", RowBox[List["!", RowBox[List["(", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List[RowBox[List["-", "\[Infinity]"]], ",", RowBox[List["-", "1"]]]], "}"]], "]"]], ",", "z"]], "]"]], ")"]]]]]]]]

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <msqrt> <mrow> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> &#10869; </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mfrac> <mi> z </mi> <mstyle scriptlevel='0'> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mfrac> <mi> z </mi> <mstyle scriptlevel='0'> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> + </mo> <mfrac> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mstyle scriptlevel='0'> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 3 </mn> <mo> + </mo> <mfrac> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mstyle scriptlevel='0'> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> + </mo> <mfrac> <mrow> <mn> 5 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mstyle scriptlevel='0'> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 5 </mn> <mo> + </mo> <mo> &#8230; </mo> </mrow> <mo> ) </mo> </mrow> </mrow> </mstyle> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mstyle> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mstyle> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mstyle> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mstyle> </mfrac> </mrow> </mrow> <mo> /; </mo> <mtext> &#8203; </mtext> <mrow> <mi> z </mi> <mo> &#8713; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> &#8734; </mi> </mrow> <mo> , </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <mrow> <mrow> <msqrt> <mrow> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> &#10869; </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mfrac> <mi> z </mi> <mstyle scriptlevel='0'> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mfrac> <mi> z </mi> <mstyle scriptlevel='0'> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> + </mo> <mfrac> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mstyle scriptlevel='0'> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 3 </mn> <mo> + </mo> <mfrac> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mstyle scriptlevel='0'> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> + </mo> <mfrac> <mrow> <mn> 5 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mstyle scriptlevel='0'> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 5 </mn> <mo> + </mo> <mo> &#8230; </mo> </mrow> <mo> ) </mo> </mrow> </mrow> </mstyle> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mstyle> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mstyle> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mstyle> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mstyle> </mfrac> </mrow> </mrow> <mo> /; </mo> <mtext> &#8203; </mtext> <mrow> <mi> z </mi> <mo> &#8713; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> &#8734; </mi> </mrow> <mo> , </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", SqrtBox[RowBox[List["1", "+", "z_"]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["1", "+", FractionBox["z", RowBox[List["2", " ", RowBox[List["(", RowBox[List["1", "+", FractionBox["z", RowBox[List["2", " ", RowBox[List["(", RowBox[List["2", "+", FractionBox[RowBox[List["3", " ", "z"]], RowBox[List["2", " ", RowBox[List["(", RowBox[List["3", "+", FractionBox[RowBox[List["3", " ", "z"]], RowBox[List["2", " ", RowBox[List["(", RowBox[List["2", "+", FractionBox[RowBox[List["5", " ", "z"]], RowBox[List["2", " ", RowBox[List["(", RowBox[List["5", "+", "\[Ellipsis]"]], ")"]]]]]]], ")"]]]]]]], ")"]]]]]]], ")"]]]]]]], ")"]]]]]]], "/;", RowBox[List["!", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List[RowBox[List["-", "\[Infinity]"]], ",", RowBox[List["-", "1"]]]], "}"]], "]"]], ",", "z"]], "]"]]]]]]]]]]

2001-10-29