|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
http://functions.wolfram.com/01.30.07.0002.01
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
ArcSech[z] == -Integrate[1/(t Sqrt[(1 - t)/(1 + t)] (t + 1)), {t, 1, z}] /;
!IntervalMemberQ[Interval[{-1, 0}], z]
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["ArcSech", "[", "z", "]"]], "\[Equal]", RowBox[List["-", RowBox[List[SubsuperscriptBox["\[Integral]", "1", "z"], RowBox[List[FractionBox["1", RowBox[List["t", " ", SqrtBox[FractionBox[RowBox[List["1", "-", "t"]], RowBox[List["1", "+", "t"]]]], " ", RowBox[List["(", RowBox[List["t", "+", "1"]], ")"]]]]], RowBox[List["\[DifferentialD]", "t"]]]]]]]]]], "/;", FormBox[RowBox[List["Not", "[", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List[RowBox[List["-", "1"]], ",", "0"]], "}"]], "]"]], ",", "z"]], "]"]], "]"]], TraditionalForm]]]]]
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<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> <mo> - </mo> <mrow> <msubsup> <mo> ∫ </mo> <mn> 1 </mn> <mi> z </mi> </msubsup> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mi> t </mi> <mo> ⁢ </mo> <msqrt> <mfrac> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> t </mi> </mrow> <mrow> <mn> 1 </mn> <mo> + </mo> <mi> t </mi> </mrow> </mfrac> </msqrt> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mi> t </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> t </mi> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> z </mi> <mo> ∉ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mn> 0 </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> <mo> - </mo> <mrow> <msubsup> <mo> ∫ </mo> <mn> 1 </mn> <mi> z </mi> </msubsup> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mi> t </mi> <mo> ⁢ </mo> <msqrt> <mfrac> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> t </mi> </mrow> <mrow> <mn> 1 </mn> <mo> + </mo> <mi> t </mi> </mrow> </mfrac> </msqrt> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mi> t </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> t </mi> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> z </mi> <mo> ∉ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mn> 0 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </annotation-xml> </semantics> </math>
|
|
|
|
|
|
|
|
|
|
| |
|
|
|
|
| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ArcSech", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["-", RowBox[List[SubsuperscriptBox["\[Integral]", "1", "z"], RowBox[List[FractionBox["1", RowBox[List["t", " ", SqrtBox[FractionBox[RowBox[List["1", "-", "t"]], RowBox[List["1", "+", "t"]]]], " ", RowBox[List["(", RowBox[List["t", "+", "1"]], ")"]]]]], RowBox[List["\[DifferentialD]", "t"]]]]]]]], "/;", RowBox[List["!", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List[RowBox[List["-", "1"]], ",", "0"]], "}"]], "]"]], ",", "z"]], "]"]]]]]]]]]] |
|
|
|
|
|
|
|
|
|
|
Date Added to functions.wolfram.com (modification date)
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|