|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
http://functions.wolfram.com/01.25.27.1074.01
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
ArcSinh[z Sqrt[(Sqrt[1 - z^2] - 1)/(2 z^2)]] ==
(1/2) I ArcSec[1/z] - (Pi I)/4 /; 0 < Arg[z] < Pi/2 ||
-Pi < Arg[z] < -(Pi/2)
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["ArcSinh", "[", RowBox[List["z", " ", SqrtBox[FractionBox[RowBox[List[SqrtBox[RowBox[List["1", "-", SuperscriptBox["z", "2"]]]], "-", "1"]], RowBox[List["2", SuperscriptBox["z", "2"]]]]]]], "]"]], "\[Equal]", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", "\[ImaginaryI]", " ", RowBox[List["ArcSec", "[", FractionBox["1", "z"], "]"]]]], "-", FractionBox[RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], "4"]]]]], "/;", RowBox[List[RowBox[List["0", "<", RowBox[List["Arg", "[", "z", "]"]], "<", FractionBox["\[Pi]", "2"]]], "\[Or]", RowBox[List[RowBox[List["-", "\[Pi]"]], "<", RowBox[List["Arg", "[", "z", "]"]], "<", RowBox[List["-", FractionBox["\[Pi]", "2"]]]]]]]]]]]
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<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> sinh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mi> z </mi> <mo> ⁢ </mo> <msqrt> <mfrac> <mrow> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> <mo> - </mo> <mn> 1 </mn> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </mfrac> </msqrt> </mrow> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mrow> <msup> <mi> sec </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <mn> 1 </mn> <mi> z </mi> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mfrac> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> ⅈ </mi> </mrow> <mn> 4 </mn> </mfrac> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mn> 0 </mn> <mo> < </mo> <mrow> <mi> arg </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> < </mo> <mfrac> <mi> π </mi> <mn> 2 </mn> </mfrac> </mrow> <mo> ∨ </mo> <mrow> <mrow> <mo> - </mo> <mi> π </mi> </mrow> <mo> < </mo> <mrow> <mi> arg </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> < </mo> <mrow> <mo> - </mo> <mfrac> <mi> π </mi> <mn> 2 </mn> </mfrac> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <arcsinh /> <apply> <times /> <ci> z </ci> <apply> <power /> <apply> <times /> <apply> <plus /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <imaginaryi /> <apply> <arcsec /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <pi /> <imaginaryi /> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <or /> <apply> <lt /> <cn type='integer'> 0 </cn> <apply> <arg /> <ci> z </ci> </apply> <apply> <times /> <pi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <lt /> <apply> <times /> <cn type='integer'> -1 </cn> <pi /> </apply> <apply> <arg /> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <pi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
|
|
|
|
|
|
|
|
|
|
| |
|
|
|
|
| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ArcSinh", "[", RowBox[List["z_", " ", SqrtBox[FractionBox[RowBox[List[SqrtBox[RowBox[List["1", "-", SuperscriptBox["z_", "2"]]]], "-", "1"]], RowBox[List["2", " ", SuperscriptBox["z_", "2"]]]]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[FractionBox["1", "2"], " ", "\[ImaginaryI]", " ", RowBox[List["ArcSec", "[", FractionBox["1", "z"], "]"]]]], "-", FractionBox[RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], "4"]]], "/;", RowBox[List[RowBox[List["0", "<", RowBox[List["Arg", "[", "z", "]"]], "<", FractionBox["\[Pi]", "2"]]], "||", RowBox[List[RowBox[List["-", "\[Pi]"]], "<", RowBox[List["Arg", "[", "z", "]"]], "<", RowBox[List["-", FractionBox["\[Pi]", "2"]]]]]]]]]]]]] |
|
|
|
|
|
|
|
|
|
|
Date Added to functions.wolfram.com (modification date)
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|