Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site Email Comments

View Related Information In
The Documentation Center
MathWorld

Download All Formulas For This Function
Mathematica Notebook
PDF File

 

Developed with Mathematica -- Download a Free Trial Version
 











ArcSinh




Mathematica Notation

Traditional Notation









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





http://functions.wolfram.com/01.25.27.0629.01









  


  








Input Form





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








Standard Form





Cell[BoxData[RowBox[List[RowBox[List["ArcSinh", "[", "z", "]"]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["-", FractionBox[RowBox[List["2", " ", "\[ImaginaryI]", " ", SqrtBox[RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["\[ImaginaryI]", " ", "z"]]]]], " ", SqrtBox[RowBox[List["1", "+", RowBox[List["\[ImaginaryI]", " ", "z"]]]]]]], RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "1"]], "-", RowBox[List["\[ImaginaryI]", " ", "z"]]]]], " ", SqrtBox[RowBox[List["1", "-", RowBox[List["\[ImaginaryI]", " ", "z"]]]]]]]]]], RowBox[List["ArcCot", "[", FractionBox[SqrtBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "z"]], "-", "1"]]], SqrtBox[RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "z"]], "-", "1"]]]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[SqrtBox[RowBox[List["1", "-", RowBox[List["z", " ", "\[ImaginaryI]"]]]]], " ", SqrtBox[FractionBox["1", RowBox[List["1", "-", RowBox[List["z", " ", "\[ImaginaryI]"]]]]]]]], "-", FractionBox["1", "2"]]], ")"]], "\[Pi]", " ", "\[ImaginaryI]"]]]]]]]]








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> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mfrac> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <msqrt> <mrow> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> </mrow> <mrow> <msqrt> <mrow> <mrow> <mrow> <mo> - </mo> <mi> &#8520; </mi> </mrow> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mrow> </msqrt> </mrow> </mfrac> </mrow> <mo> &#8290; </mo> <mrow> <msup> <mi> cot </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <msqrt> <mrow> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> <msqrt> <mrow> <mrow> <mrow> <mo> - </mo> <mi> &#8520; </mi> </mrow> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mi> z </mi> <mo> &#8290; </mo> <mi> &#8520; </mi> </mrow> </mrow> </msqrt> <mo> &#8290; </mo> <msqrt> <mfrac> <mn> 1 </mn> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mi> z </mi> <mo> &#8290; </mo> <mi> &#8520; </mi> </mrow> </mrow> </mfrac> </msqrt> </mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#8520; </mi> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <arcsinh /> <ci> z </ci> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <apply> <power /> <apply> <plus /> <apply> <times /> <imaginaryi /> <ci> z </ci> </apply> <cn type='integer'> -1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <imaginaryi /> <ci> z </ci> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <ci> z </ci> </apply> <cn type='integer'> -1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> z </ci> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <arccot /> <apply> <times /> <apply> <power /> <apply> <plus /> <apply> <times /> <imaginaryi /> <ci> z </ci> </apply> <cn type='integer'> -1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <ci> z </ci> </apply> <cn type='integer'> -1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> z </ci> <imaginaryi /> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> z </ci> <imaginaryi /> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <pi /> <imaginaryi /> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





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


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





2003-08-21