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ArcSinh






Mathematica Notation

Traditional Notation









Elementary Functions > ArcSinh[z] > Representations through equivalent functions > With related functions > Involving coth-1 > Involving sinh-1(((z-(z2-1)1/2)/(2(z2-1)1/2))1/2) > Involving sinh-1(((z-(z2-1)1/2)/(2(z2-1)1/2))1/2) and coth-1(z)





http://functions.wolfram.com/01.25.27.1728.01









  


  










Input Form





ArcSinh[Sqrt[(z - Sqrt[z^2 - 1])/(2 Sqrt[z^2 - 1])]] == (I/2) Sqrt[-(I/z)] Sqrt[(-I) z] ArcCoth[z] + (Pi/4) ((-Sqrt[z]) Sqrt[-(1/z)] - 2 I Sqrt[z] Sqrt[1/z] + 2 I + (1/Sqrt[z]) Sqrt[-(1/z)] Sqrt[z^2])










Standard Form





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MathML Form







<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> <msqrt> <mfrac> <mrow> <mi> z </mi> <mo> - </mo> <msqrt> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msqrt> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> </mrow> </mfrac> </msqrt> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mfrac> <mi> &#960; </mi> <mrow> <mn> 4 </mn> <mtext> </mtext> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <msqrt> <mi> z </mi> </msqrt> </mrow> <mo> &#8290; </mo> <msqrt> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mi> z </mi> </mfrac> </mrow> </msqrt> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; 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Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ArcSinh", "[", SqrtBox[FractionBox[RowBox[List["z_", "-", SqrtBox[RowBox[List[SuperscriptBox["z_", "2"], "-", "1"]]]]], RowBox[List["2", " ", SqrtBox[RowBox[List[SuperscriptBox["z_", "2"], "-", "1"]]]]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", "\[ImaginaryI]", " ", SqrtBox[RowBox[List["-", FractionBox["\[ImaginaryI]", "z"]]]], " ", SqrtBox[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "z"]]], " ", RowBox[List["ArcCoth", "[", "z", "]"]]]], "+", RowBox[List[FractionBox["1", "4"], " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", SqrtBox["z"]]], " ", SqrtBox[RowBox[List["-", FractionBox["1", "z"]]]]]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", SqrtBox["z"], " ", SqrtBox[FractionBox["1", "z"]]]], "+", RowBox[List["2", " ", "\[ImaginaryI]"]], "+", FractionBox[RowBox[List[SqrtBox[RowBox[List["-", FractionBox["1", "z"]]]], " ", SqrtBox[SuperscriptBox["z", "2"]]]], SqrtBox["z"]]]], ")"]]]]]]]]]]










Date Added to functions.wolfram.com (modification date)





2003-08-21