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Sinh






Mathematica Notation

Traditional Notation









Elementary Functions > Sinh[z] > Representations through equivalent functions > With inverse function





http://functions.wolfram.com/01.19.27.0068.01









  


  










Input Form





ArcSinh[Sinh[z]] == (-1)^k (z - Pi I k) /; (-(Pi/2) + k Pi < Im[z] < Pi/2 + k Pi || (Im[z] == -(Pi/2) + k Pi && Re[z] <= 0) || (Im[z] == Pi/2 + k Pi && Re[z] >= 0)) && Element[k, Integers]










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["ArcSinh", "[", RowBox[List["Sinh", "[", "z", "]"]], "]"]], "\[Equal]", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List["(", RowBox[List["z", "-", RowBox[List["\[Pi]", " ", "\[ImaginaryI]", " ", "k"]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List[RowBox[List["-", FractionBox["\[Pi]", "2"]]], "+", RowBox[List["k", " ", "\[Pi]"]]]], "<", RowBox[List["Im", "[", "z", "]"]], "<", RowBox[List[FractionBox["\[Pi]", "2"], "+", RowBox[List["k", " ", "\[Pi]"]]]]]], "\[Or]", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Im", "[", "z", "]"]], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox["\[Pi]", "2"]]], "+", RowBox[List["k", " ", "\[Pi]"]]]]]], "\[And]", RowBox[List[RowBox[List["Re", "[", "z", "]"]], "\[LessEqual]", "0"]]]], ")"]], "\[Or]", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Im", "[", "z", "]"]], "\[Equal]", RowBox[List[FractionBox["\[Pi]", "2"], "+", RowBox[List["k", " ", "\[Pi]"]]]]]], "\[And]", RowBox[List[RowBox[List["Re", "[", "z", "]"]], "\[GreaterEqual]", "0"]]]], ")"]]]], ")"]], "\[And]", RowBox[List["k", "\[Element]", "Integers"]]]]]]]]










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> <msup> <mi> sinh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mi> sinh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> k </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mrow> <mi> k </mi> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mo> - </mo> <mfrac> <mi> &#960; </mi> <mn> 2 </mn> </mfrac> </mrow> <mo> &lt; </mo> <mrow> <mi> Im </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &lt; </mo> <mrow> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> + </mo> <mfrac> <mi> &#960; </mi> <mn> 2 </mn> </mfrac> </mrow> </mrow> <mo> &#8744; </mo> <mrow> <mrow> <mrow> <mi> Im </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mi> k </mi> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mo> - </mo> <mfrac> <mi> &#960; </mi> <mn> 2 </mn> </mfrac> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#8804; </mo> <mn> 0 </mn> </mrow> </mrow> <mo> &#8744; </mo> <mrow> <mrow> <mrow> <mi> Im </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> + </mo> <mfrac> <mi> &#960; </mi> <mn> 2 </mn> </mfrac> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#8805; </mo> <mn> 0 </mn> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8743; </mo> <mrow> <mi> k </mi> <mo> &#8712; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[Integers]] </annotation> </semantics> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <arcsinh /> <apply> <sinh /> <ci> z </ci> </apply> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <pi /> <imaginaryi /> <ci> k </ci> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <or /> <apply> <lt /> <apply> <plus /> <apply> <times /> <ci> k </ci> <pi /> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <pi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <imaginary /> <ci> z </ci> </apply> <apply> <plus /> <apply> <times /> <pi /> <ci> k </ci> </apply> <apply> <times /> <pi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <eq /> <apply> <imaginary /> <ci> z </ci> </apply> <apply> <plus /> <apply> <times /> <ci> k </ci> <pi /> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <pi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <leq /> <apply> <real /> <ci> z </ci> </apply> <cn type='integer'> 0 </cn> </apply> </apply> <apply> <and /> <apply> <eq /> <apply> <imaginary /> <ci> z </ci> </apply> <apply> <plus /> <apply> <times /> <pi /> <ci> k </ci> </apply> <apply> <times /> <pi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <geq /> <apply> <real /> <ci> z </ci> </apply> <cn type='integer'> 0 </cn> </apply> </apply> </apply> <apply> <in /> <ci> k </ci> <integers /> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ArcSinh", "[", RowBox[List["Sinh", "[", "z_", "]"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List["(", RowBox[List["z", "-", RowBox[List["\[Pi]", " ", "\[ImaginaryI]", " ", "k"]]]], ")"]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List[RowBox[List["-", FractionBox["\[Pi]", "2"]]], "+", RowBox[List["k", " ", "\[Pi]"]]]], "<", RowBox[List["Im", "[", "z", "]"]], "<", RowBox[List[FractionBox["\[Pi]", "2"], "+", RowBox[List["k", " ", "\[Pi]"]]]]]], "||", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Im", "[", "z", "]"]], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox["\[Pi]", "2"]]], "+", RowBox[List["k", " ", "\[Pi]"]]]]]], "&&", RowBox[List[RowBox[List["Re", "[", "z", "]"]], "\[LessEqual]", "0"]]]], ")"]], "||", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Im", "[", "z", "]"]], "\[Equal]", RowBox[List[FractionBox["\[Pi]", "2"], "+", RowBox[List["k", " ", "\[Pi]"]]]]]], "&&", RowBox[List[RowBox[List["Re", "[", "z", "]"]], "\[GreaterEqual]", "0"]]]], ")"]]]], ")"]], "&&", RowBox[List["k", "\[Element]", "Integers"]]]]]]]]]]










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





2003-08-21