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ArcCsch




Mathematica Notation

Traditional Notation









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





http://functions.wolfram.com/01.29.27.0671.01









  


  








Input Form





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








Standard Form





Cell[BoxData[RowBox[List[RowBox[List["ArcCsch", "[", "z", "]"]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[FractionBox["1", "2"], "-", RowBox[List[SqrtBox[FractionBox["z", RowBox[List["z", "+", "\[ImaginaryI]"]]]], SqrtBox[FractionBox[RowBox[List["z", "+", "\[ImaginaryI]"]], "z"]]]]]], ")"]], "\[Pi]", " ", "\[ImaginaryI]"]], "-", RowBox[List[FractionBox[RowBox[List["2", " ", SqrtBox[RowBox[List[RowBox[List["-", "1"]], "-", RowBox[List["\[ImaginaryI]", " ", "z"]]]]], " ", SqrtBox[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "z"]]]]], RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], "+", "z"]]], " ", SqrtBox[RowBox[List["\[ImaginaryI]", "+", "z"]]]]]], SqrtBox[FractionBox[RowBox[List["\[ImaginaryI]", "+", "z"]], "z"]], RowBox[List["ArcCot", "[", FractionBox[SqrtBox[RowBox[List["z", "+", "\[ImaginaryI]"]]], SqrtBox[RowBox[List["z", "-", "\[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> csch </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> - </mo> <mrow> <msqrt> <mfrac> <mi> z </mi> <mrow> <mi> z </mi> <mo> + </mo> <mi> &#8520; </mi> </mrow> </mfrac> </msqrt> <mo> &#8290; </mo> <msqrt> <mfrac> <mrow> <mi> z </mi> <mo> + </mo> <mi> &#8520; </mi> </mrow> <mi> z </mi> </mfrac> </msqrt> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mfrac> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msqrt> <mrow> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> - </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mrow> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <mrow> <mo> - </mo> <mi> &#8520; </mi> </mrow> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msqrt> </mrow> <mrow> <msqrt> <mrow> <mrow> <mo> - </mo> <mi> &#8520; </mi> </mrow> <mo> + </mo> <mi> z </mi> </mrow> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <mi> &#8520; </mi> <mo> + </mo> <mi> z </mi> </mrow> </msqrt> </mrow> </mfrac> <mo> &#8290; </mo> <msqrt> <mfrac> <mrow> <mi> &#8520; </mi> <mo> + </mo> <mi> z </mi> </mrow> <mi> z </mi> </mfrac> </msqrt> <mo> &#8290; </mo> <mrow> <msup> <mi> cot </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <msqrt> <mrow> <mi> z </mi> <mo> + </mo> <mi> &#8520; </mi> </mrow> </msqrt> <msqrt> <mrow> <mi> z </mi> <mo> - </mo> <mi> &#8520; </mi> </mrow> </msqrt> </mfrac> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <arccsch /> <ci> z </ci> </apply> <apply> <plus /> <apply> <times /> <imaginaryi /> <pi /> <apply> <plus /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <apply> <times /> <ci> z </ci> <apply> <power /> <apply> <plus /> <ci> z </ci> <imaginaryi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <ci> z </ci> <imaginaryi /> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <cn type='integer'> 2 </cn> <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> <power /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <ci> z </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <ci> z </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <imaginaryi /> <ci> z </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <imaginaryi /> <ci> z </ci> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <arccot /> <apply> <times /> <apply> <power /> <apply> <plus /> <ci> z </ci> <imaginaryi /> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





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


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





2003-08-21