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ArcCosh






Mathematica Notation

Traditional Notation









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





http://functions.wolfram.com/01.26.27.1496.01









  


  










Input Form





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










Standard Form





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MathML Form







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





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










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





2003-08-21