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ArcCosh




Mathematica Notation

Traditional Notation









Elementary Functions > ArcCosh[z] > Transformations > Related transformations > Sums involving the direct function > Involving sinh-1(z)





http://functions.wolfram.com/01.26.16.0016.01









  


  








Input Form





ArcCosh[x] + ArcSinh[y] == Pi I - ArcSinh[x y + Sqrt[1 + y^2] Sqrt[x^2 - 1]] /; -1 < x < 0








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List["ArcCosh", "[", "x", "]"]], "+", RowBox[List["ArcSinh", "[", "y", "]"]]]], "\[Equal]", RowBox[List[RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], "-", RowBox[List["ArcSinh", "[", RowBox[List[RowBox[List["x", " ", "y"]], "+", RowBox[List[SqrtBox[RowBox[List["1", "+", SuperscriptBox["y", "2"]]]], " ", SqrtBox[RowBox[List[SuperscriptBox["x", "2"], "-", "1"]]]]]]], "]"]]]]]], "/;", RowBox[List[RowBox[List["-", "1"]], "<", "x", "<", "0"]]]]]]








MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <mrow> <msup> <mi> cosh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <msup> <mi> sinh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> y </mi> <mo> ) </mo> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#8520; </mi> </mrow> <mo> - </mo> <mrow> <msup> <mi> sinh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mrow> <mi> x </mi> <mo> &#8290; </mo> <mi> y </mi> </mrow> <mo> + </mo> <mrow> <msqrt> <mrow> <msup> <mi> x </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <msup> <mi> y </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> &lt; </mo> <mi> x </mi> <mo> &lt; </mo> <mn> 0 </mn> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <plus /> <apply> <arccosh /> <ci> x </ci> </apply> <apply> <arcsinh /> <ci> y </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <pi /> <imaginaryi /> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <arcsinh /> <apply> <plus /> <apply> <times /> <ci> x </ci> <ci> y </ci> </apply> <apply> <times /> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> x </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> y </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <lt /> <cn type='integer'> -1 </cn> <ci> x </ci> <cn type='integer'> 0 </cn> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[RowBox[List["ArcCosh", "[", "x_", "]"]], "+", RowBox[List["ArcSinh", "[", "y_", "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], "-", RowBox[List["ArcSinh", "[", RowBox[List[RowBox[List["x", " ", "y"]], "+", RowBox[List[SqrtBox[RowBox[List["1", "+", SuperscriptBox["y", "2"]]]], " ", SqrtBox[RowBox[List[SuperscriptBox["x", "2"], "-", "1"]]]]]]], "]"]]]], "/;", RowBox[List[RowBox[List["-", "1"]], "<", "x", "<", "0"]]]]]]]]


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





2001-10-29