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http://functions.wolfram.com/01.30.10.0001.01
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ArcSech[z] == (Sqrt[1/z - 1]/Sqrt[1 - 1/z])
(Pi/2 - Sqrt[1 - z^(-2)]/
(z (1 - (1 2)/(z^2 (3 - (1 2)/(z^2 (5 - (3 4)/(z^2 (7 -
(3 4)/(z^2 (9 - (5 6)/(z^2 (11 - \[Ellipsis]))))))))))))) /;
!IntervalMemberQ[Interval[{-1, 1}], z]
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["ArcSech", "[", "z", "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List[SqrtBox[RowBox[List[FractionBox["1", "z"], "-", "1"]]], " "]], SqrtBox[RowBox[List["1", "-", FractionBox["1", "z"]]]]], RowBox[List["(", RowBox[List[FractionBox["\[Pi]", "2"], "-", FractionBox[RowBox[List[SuperscriptBox["z", RowBox[List["-", "1"]]], SqrtBox[RowBox[List["1", "-", SuperscriptBox["z", RowBox[List["-", "2"]]]]]]]], RowBox[List["1", "-", FractionBox[RowBox[List["1", "\[Times]", "2", SuperscriptBox["z", RowBox[List["-", "2"]]]]], RowBox[List["3", "-", FractionBox[RowBox[List["1", "\[Times]", "2", SuperscriptBox["z", RowBox[List["-", "2"]]]]], RowBox[List["5", "-", FractionBox[RowBox[List["3", " ", "\[Times]", "4", SuperscriptBox["z", RowBox[List["-", "2"]]]]], RowBox[List["7", "-", FractionBox[RowBox[List["3", " ", "\[Times]", "4", SuperscriptBox["z", RowBox[List["-", "2"]]]]], RowBox[List["9", "-", FractionBox[RowBox[List["5", " ", "\[Times]", "6", SuperscriptBox["z", RowBox[List["-", "2"]]]]], RowBox[List["11", "-", "\[Ellipsis]"]]]]]]]]]]]]]]]]]]]], ")"]]]]]], "/;", "\[InvisibleSpace]", RowBox[List["Not", "[", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List[RowBox[List["-", "1"]], ",", "1"]], "}"]], "]"]], ",", "z"]], "]"]], "]"]], "\[InvisibleSpace]"]]]]
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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> <mrow> <msup> <mi> sech </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mfrac> <mrow> <msqrt> <mrow> <mfrac> <mn> 1 </mn> <mi> z </mi> </mfrac> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> <mtext> </mtext> </mrow> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <mn> 1 </mn> <mi> z </mi> </mfrac> </mrow> </msqrt> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <mi> π </mi> <mn> 2 </mn> </mfrac> <mo> - </mo> <mfrac> <mrow> <msup> <mi> z </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ⁢ </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> z </mi> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> </msup> </mrow> </msqrt> </mrow> <mstyle scriptlevel='0'> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <mrow> <mn> 1 </mn> <mo> × </mo> <mn> 2 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> </msup> </mrow> <mstyle scriptlevel='0'> <mrow> <mn> 3 </mn> <mo> - </mo> <mfrac> <mrow> <mn> 1 </mn> <mo> × </mo> <mn> 2 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> </msup> </mrow> <mstyle scriptlevel='0'> <mrow> <mn> 5 </mn> <mo> - </mo> <mfrac> <mrow> <mn> 3 </mn> <mtext> </mtext> <mo> × </mo> <mn> 4 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> </msup> </mrow> <mstyle scriptlevel='0'> <mrow> <mn> 7 </mn> <mo> - </mo> <mfrac> <mrow> <mn> 3 </mn> <mtext> </mtext> <mo> × </mo> <mn> 4 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> </msup> </mrow> <mstyle scriptlevel='0'> <mrow> <mn> 9 </mn> <mo> - </mo> <mfrac> <mrow> <mn> 5 </mn> <mtext> </mtext> <mo> × </mo> <mn> 6 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> </msup> </mrow> <mstyle scriptlevel='0'> <mrow> <mn> 11 </mn> <mo> - </mo> <mo> … </mo> </mrow> </mstyle> </mfrac> </mrow> </mstyle> </mfrac> </mrow> </mstyle> </mfrac> </mrow> </mstyle> </mfrac> </mrow> </mstyle> </mfrac> </mrow> </mstyle> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mtext> ​​ </mtext> <mrow> <mi> z </mi> <mo> ∉ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <mrow> <mrow> <mrow> <msup> <mi> sech </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mfrac> <mrow> <msqrt> <mrow> <mfrac> <mn> 1 </mn> <mi> z </mi> </mfrac> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> <mtext> </mtext> </mrow> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <mn> 1 </mn> <mi> z </mi> </mfrac> </mrow> </msqrt> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <mi> π </mi> <mn> 2 </mn> </mfrac> <mo> - </mo> <mfrac> <mrow> <msup> <mi> z </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ⁢ </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> z </mi> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> </msup> </mrow> </msqrt> </mrow> <mstyle scriptlevel='0'> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <mrow> <mn> 1 </mn> <mo> × </mo> <mn> 2 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> </msup> </mrow> <mstyle scriptlevel='0'> <mrow> <mn> 3 </mn> <mo> - </mo> <mfrac> <mrow> <mn> 1 </mn> <mo> × </mo> <mn> 2 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> </msup> </mrow> <mstyle scriptlevel='0'> <mrow> <mn> 5 </mn> <mo> - </mo> <mfrac> <mrow> <mn> 3 </mn> <mtext> </mtext> <mo> × </mo> <mn> 4 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> </msup> </mrow> <mstyle scriptlevel='0'> <mrow> <mn> 7 </mn> <mo> - </mo> <mfrac> <mrow> <mn> 3 </mn> <mtext> </mtext> <mo> × </mo> <mn> 4 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> </msup> </mrow> <mstyle scriptlevel='0'> <mrow> <mn> 9 </mn> <mo> - </mo> <mfrac> <mrow> <mn> 5 </mn> <mtext> </mtext> <mo> × </mo> <mn> 6 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> </msup> </mrow> <mstyle scriptlevel='0'> <mrow> <mn> 11 </mn> <mo> - </mo> <mo> … </mo> </mrow> </mstyle> </mfrac> </mrow> </mstyle> </mfrac> </mrow> </mstyle> </mfrac> </mrow> </mstyle> </mfrac> </mrow> </mstyle> </mfrac> </mrow> </mstyle> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mtext> ​​ </mtext> <mrow> <mi> z </mi> <mo> ∉ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ArcSech", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[SqrtBox[RowBox[List[FractionBox["1", "z"], "-", "1"]]], " ", RowBox[List["(", RowBox[List[FractionBox["\[Pi]", "2"], "-", FractionBox[SqrtBox[RowBox[List["1", "-", FractionBox["1", SuperscriptBox["z", "2"]]]]], RowBox[List["z", " ", RowBox[List["(", RowBox[List["1", "-", FractionBox["2", RowBox[List[SuperscriptBox["z", "2"], " ", RowBox[List["(", RowBox[List["3", "-", FractionBox["2", RowBox[List[SuperscriptBox["z", "2"], " ", RowBox[List["(", RowBox[List["5", "-", FractionBox[RowBox[List["3", " ", "4"]], RowBox[List[SuperscriptBox["z", "2"], " ", RowBox[List["(", RowBox[List["7", "-", FractionBox[RowBox[List["3", " ", "4"]], RowBox[List[SuperscriptBox["z", "2"], " ", RowBox[List["(", RowBox[List["9", "-", FractionBox[RowBox[List["5", " ", "6"]], RowBox[List[SuperscriptBox["z", "2"], " ", RowBox[List["(", RowBox[List["11", "-", "\[Ellipsis]"]], ")"]]]]]]], ")"]]]]]]], ")"]]]]]]], ")"]]]]]]], ")"]]]]]]], ")"]]]]]]], ")"]]]], SqrtBox[RowBox[List["1", "-", FractionBox["1", "z"]]]]], "/;", RowBox[List["!", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List[RowBox[List["-", "1"]], ",", "1"]], "}"]], "]"]], ",", "z"]], "]"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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