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http://functions.wolfram.com/01.25.27.0979.01
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ArcSinh[z] == (Pi/4) (-(Sqrt[-z^2]/z) - I Sqrt[I/z] Sqrt[(-I) z]
Sqrt[1/(1 - I Sqrt[2] z)] Sqrt[1 - I Sqrt[2] z] +
I Sqrt[-(I/z)] Sqrt[I z] Sqrt[1/(1 + I Sqrt[2] z)]
Sqrt[1 + I Sqrt[2] z] - Sqrt[z^2 + z^4]/(z Sqrt[-1 - z^2])) -
(Sqrt[-1 - 2 z^2]/(2 z^2 Sqrt[1 + 2 z^2])) Sqrt[-(z^2/(-1 - z^2))]
Sqrt[(-z^2) (-1 - z^2)] (Pi/2 - ArcSec[1/(2 z Sqrt[-z^2 - 1])])
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Cell[BoxData[RowBox[List[RowBox[List["ArcSinh", "[", "z", "]"]], "\[Equal]", RowBox[List[RowBox[List[FractionBox["\[Pi]", "4"], RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox[SqrtBox[RowBox[List["-", SuperscriptBox["z", "2"]]]], "z"]]], "-", RowBox[List["\[ImaginaryI]", SqrtBox[FractionBox["\[ImaginaryI]", "z"]], " ", SqrtBox[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "z"]]], " ", SqrtBox[FractionBox["1", RowBox[List["1", "-", RowBox[List["\[ImaginaryI]", " ", SqrtBox["2"], " ", "z"]]]]]], " ", SqrtBox[RowBox[List["1", "-", RowBox[List["\[ImaginaryI]", " ", SqrtBox["2"], " ", "z"]]]]]]], "+", RowBox[List["\[ImaginaryI]", SqrtBox[RowBox[List["-", FractionBox["\[ImaginaryI]", "z"]]]], " ", SqrtBox[RowBox[List["\[ImaginaryI]", " ", "z"]]], " ", SqrtBox[FractionBox["1", RowBox[List["1", "+", RowBox[List["\[ImaginaryI]", " ", SqrtBox["2"], " ", "z"]]]]]], " ", SqrtBox[RowBox[List["1", "+", RowBox[List["\[ImaginaryI]", " ", SqrtBox["2"], " ", "z"]]]]]]], "-", FractionBox[SqrtBox[RowBox[List[SuperscriptBox["z", "2"], "+", SuperscriptBox["z", "4"]]]], RowBox[List["z", " ", SqrtBox[RowBox[List[RowBox[List["-", "1"]], "-", SuperscriptBox["z", "2"]]]]]]]]], ")"]]]], "-", RowBox[List[FractionBox[SqrtBox[RowBox[List[RowBox[List["-", "1"]], "-", RowBox[List["2", " ", SuperscriptBox["z", "2"]]]]]], RowBox[List["2", " ", SuperscriptBox["z", "2"], " ", SqrtBox[RowBox[List["1", "+", RowBox[List["2", " ", SuperscriptBox["z", "2"]]]]]]]]], " ", SqrtBox[RowBox[List["-", FractionBox[SuperscriptBox["z", "2"], RowBox[List[RowBox[List["-", "1"]], "-", SuperscriptBox["z", "2"]]]]]]], " ", SqrtBox[RowBox[List[RowBox[List["-", SuperscriptBox["z", "2"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", SuperscriptBox["z", "2"]]], ")"]]]]], " ", RowBox[List["(", RowBox[List[FractionBox["\[Pi]", "2"], "-", RowBox[List["ArcSec", "[", FractionBox["1", RowBox[List["2", " ", "z", " ", SqrtBox[RowBox[List[RowBox[List["-", SuperscriptBox["z", "2"]]], "-", "1"]]]]]], "]"]]]], ")"]]]]]]]]]]
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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> <mi> sinh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mfrac> <mi> π </mi> <mn> 4 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <msqrt> <mrow> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> <mi> z </mi> </mfrac> </mrow> <mo> - </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <msqrt> <mfrac> <mi> ⅈ </mi> <mi> z </mi> </mfrac> </msqrt> <mo> ⁢ </mo> <msqrt> <mrow> <mrow> <mo> - </mo> <mi> ⅈ </mi> </mrow> <mo> ⁢ </mo> <mi> z </mi> </mrow> </msqrt> <mo> ⁢ </mo> <msqrt> <mfrac> <mn> 1 </mn> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <msqrt> <mn> 2 </mn> </msqrt> <mo> ⁢ </mo> <mi> z </mi> </mrow> </mrow> </mfrac> </msqrt> <mo> ⁢ </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <msqrt> <mn> 2 </mn> </msqrt> <mo> ⁢ </mo> <mi> z </mi> </mrow> </mrow> </msqrt> </mrow> <mo> + </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <msqrt> <mrow> <mo> - </mo> <mfrac> <mi> ⅈ </mi> <mi> z </mi> </mfrac> </mrow> </msqrt> <mo> ⁢ </mo> <msqrt> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </msqrt> <mo> ⁢ </mo> <msqrt> <mfrac> <mn> 1 </mn> <mrow> <mrow> <msqrt> <mn> 2 </mn> </msqrt> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </mfrac> </msqrt> <mo> ⁢ </mo> <msqrt> <mrow> <mrow> <msqrt> <mn> 2 </mn> </msqrt> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> </mrow> <mo> - </mo> <mfrac> <msqrt> <mrow> <msup> <mi> z </mi> <mn> 4 </mn> </msup> <mo> + </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> <mrow> <mi> z </mi> <mo> ⁢ </mo> <msqrt> <mrow> <mrow> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mfrac> <mrow> <msqrt> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> <mtext> </mtext> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <msqrt> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> </mrow> </mfrac> <mo> ⁢ </mo> <msqrt> <mrow> <mo> - </mo> <mfrac> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mrow> <mrow> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </mfrac> </mrow> </msqrt> <mo> ⁢ </mo> <msqrt> <mrow> <mrow> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </msqrt> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <mi> π </mi> <mn> 2 </mn> </mfrac> <mo> - </mo> <mrow> <msup> <mi> sec </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <mn> 1 </mn> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> z </mi> <mo> ⁢ </mo> <msqrt> <mrow> <mrow> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> </mrow> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <arcsinh /> <ci> z </ci> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <pi /> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <apply> <power /> <apply> <times /> <imaginaryi /> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <ci> z </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> z </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> z </ci> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <times /> <imaginaryi /> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <imaginaryi /> <ci> z </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <plus /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <imaginaryi /> <ci> z </ci> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <imaginaryi /> <ci> z </ci> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <ci> z </ci> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <apply> <power /> <apply> <plus /> <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> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <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> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </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> <plus /> <apply> <times /> <pi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <arcsec /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ArcSinh", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox["1", "4"], " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox[SqrtBox[RowBox[List["-", SuperscriptBox["z", "2"]]]], "z"]]], "-", RowBox[List["\[ImaginaryI]", " ", SqrtBox[FractionBox["\[ImaginaryI]", "z"]], " ", SqrtBox[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "z"]]], " ", SqrtBox[FractionBox["1", RowBox[List["1", "-", RowBox[List["\[ImaginaryI]", " ", SqrtBox["2"], " ", "z"]]]]]], " ", SqrtBox[RowBox[List["1", "-", RowBox[List["\[ImaginaryI]", " ", SqrtBox["2"], " ", "z"]]]]]]], "+", RowBox[List["\[ImaginaryI]", " ", SqrtBox[RowBox[List["-", FractionBox["\[ImaginaryI]", "z"]]]], " ", SqrtBox[RowBox[List["\[ImaginaryI]", " ", "z"]]], " ", SqrtBox[FractionBox["1", RowBox[List["1", "+", RowBox[List["\[ImaginaryI]", " ", SqrtBox["2"], " ", "z"]]]]]], " ", SqrtBox[RowBox[List["1", "+", RowBox[List["\[ImaginaryI]", " ", SqrtBox["2"], " ", "z"]]]]]]], "-", FractionBox[SqrtBox[RowBox[List[SuperscriptBox["z", "2"], "+", SuperscriptBox["z", "4"]]]], RowBox[List["z", " ", SqrtBox[RowBox[List[RowBox[List["-", "1"]], "-", SuperscriptBox["z", "2"]]]]]]]]], ")"]]]], "-", FractionBox[RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "1"]], "-", RowBox[List["2", " ", SuperscriptBox["z", "2"]]]]]], " ", SqrtBox[RowBox[List["-", FractionBox[SuperscriptBox["z", "2"], RowBox[List[RowBox[List["-", "1"]], "-", SuperscriptBox["z", "2"]]]]]]], " ", SqrtBox[RowBox[List[RowBox[List["-", SuperscriptBox["z", "2"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", SuperscriptBox["z", "2"]]], ")"]]]]], " ", RowBox[List["(", RowBox[List[FractionBox["\[Pi]", "2"], "-", RowBox[List["ArcSec", "[", FractionBox["1", RowBox[List["2", " ", "z", " ", SqrtBox[RowBox[List[RowBox[List["-", SuperscriptBox["z", "2"]]], "-", "1"]]]]]], "]"]]]], ")"]]]], RowBox[List["2", " ", SuperscriptBox["z", "2"], " ", SqrtBox[RowBox[List["1", "+", RowBox[List["2", " ", SuperscriptBox["z", "2"]]]]]]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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