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ArcSech




Mathematica Notation

Traditional Notation









Elementary Functions > ArcSech[z] > Transformations > Products, sums, and powers of the direct function > Sums of the direct function





http://functions.wolfram.com/01.30.16.0010.01









  


  








Input Form





ArcSech[x] + ArcSech[y] == (-Sign[(x + y)/(x y)]) ArcCosh[1/(x y) - Sqrt[1/x^2 - 1] Sqrt[1/y^2 - 1]] + Pi I (1 - Sign[(x + y)/(x y)]) /; (Abs[x] > 1 && -1 < y < 0) || (Abs[y] > 1 && -1 < x < 0)








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List["ArcSech", "[", "x", "]"]], "+", RowBox[List["ArcSech", "[", "y", "]"]]]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["-", RowBox[List["Sign", "[", FractionBox[RowBox[List["x", "+", "y"]], RowBox[List["x", " ", "y"]]], "]"]]]], " ", RowBox[List["ArcCosh", "[", RowBox[List[FractionBox["1", RowBox[List["x", " ", "y"]]], "-", RowBox[List[SqrtBox[RowBox[List[FractionBox["1", SuperscriptBox["x", "2"]], "-", "1"]]], " ", SqrtBox[RowBox[List[FractionBox["1", SuperscriptBox["y", "2"]], "-", "1"]]]]]]], "]"]]]], "+", RowBox[List["\[Pi]", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["Sign", "[", FractionBox[RowBox[List["x", "+", "y"]], RowBox[List["x", " ", "y"]]], "]"]]]], ")"]]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List[RowBox[List["Abs", "[", "x", "]"]], ">", "1"]], "&&", RowBox[List[RowBox[List["-", "1"]], "<", "y", "<", "0"]]]], "||", RowBox[List[RowBox[List[RowBox[List["Abs", "[", "y", "]"]], ">", "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> sech </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <msup> <mi> sech </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> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mi> sgn </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mrow> <mi> x </mi> <mo> + </mo> <mi> y </mi> </mrow> <mrow> <mi> x </mi> <mo> &#8290; </mo> <mi> y </mi> </mrow> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mrow> <mi> sgn </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mrow> <mi> x </mi> <mo> + </mo> <mi> y </mi> </mrow> <mrow> <mi> x </mi> <mo> &#8290; </mo> <mi> y </mi> </mrow> </mfrac> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msup> <mi> cosh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mi> x </mi> <mo> &#8290; </mo> <mi> y </mi> </mrow> </mfrac> <mo> - </mo> <mrow> <msqrt> <mrow> <mfrac> <mn> 1 </mn> <msup> <mi> x </mi> <mn> 2 </mn> </msup> </mfrac> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <mfrac> <mn> 1 </mn> <msup> <mi> y </mi> <mn> 2 </mn> </msup> </mfrac> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mi> x </mi> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <mo> &gt; </mo> <mn> 1 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> &lt; </mo> <mi> y </mi> <mo> &lt; </mo> <mn> 0 </mn> </mrow> </mrow> <mo> &#8744; </mo> <mrow> <mrow> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mi> y </mi> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <mo> &gt; </mo> <mn> 1 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> &lt; </mo> <mi> x </mi> <mo> &lt; </mo> <mn> 0 </mn> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <plus /> <apply> <arcsech /> <ci> x </ci> </apply> <apply> <arcsech /> <ci> y </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <pi /> <imaginaryi /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Sign </ci> <apply> <times /> <apply> <plus /> <ci> x </ci> <ci> y </ci> </apply> <apply> <power /> <apply> <times /> <ci> x </ci> <ci> y </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <ci> Sign </ci> <apply> <times /> <apply> <plus /> <ci> x </ci> <ci> y </ci> </apply> <apply> <power /> <apply> <times /> <ci> x </ci> <ci> y </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <arccosh /> <apply> <plus /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <ci> x </ci> <ci> y </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <ci> x </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <ci> y </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <or /> <apply> <and /> <apply> <gt /> <apply> <abs /> <ci> x </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <lt /> <cn type='integer'> -1 </cn> <ci> y </ci> <cn type='integer'> 0 </cn> </apply> </apply> <apply> <and /> <apply> <gt /> <apply> <abs /> <ci> y </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <lt /> <cn type='integer'> -1 </cn> <ci> x </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[RowBox[List["ArcSech", "[", "x_", "]"]], "+", RowBox[List["ArcSech", "[", "y_", "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[RowBox[List["-", RowBox[List["Sign", "[", FractionBox[RowBox[List["x", "+", "y"]], RowBox[List["x", " ", "y"]]], "]"]]]], " ", RowBox[List["ArcCosh", "[", RowBox[List[FractionBox["1", RowBox[List["x", " ", "y"]]], "-", RowBox[List[SqrtBox[RowBox[List[FractionBox["1", SuperscriptBox["x", "2"]], "-", "1"]]], " ", SqrtBox[RowBox[List[FractionBox["1", SuperscriptBox["y", "2"]], "-", "1"]]]]]]], "]"]]]], "+", RowBox[List["\[Pi]", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["Sign", "[", FractionBox[RowBox[List["x", "+", "y"]], RowBox[List["x", " ", "y"]]], "]"]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Abs", "[", "x", "]"]], ">", "1"]], "&&", RowBox[List[RowBox[List["-", "1"]], "<", "y", "<", "0"]]]], ")"]], "||", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Abs", "[", "y", "]"]], ">", "1"]], "&&", RowBox[List[RowBox[List["-", "1"]], "<", "x", "<", "0"]]]], ")"]]]]]]]]]]


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





2001-10-29