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http://functions.wolfram.com/01.29.27.1503.01
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ArcCsch[z] == ((Pi I)/4) Sqrt[z^2/(2 + z^2)] Sqrt[(2 + z^2)/z^2]
(Sqrt[1 + (I Sqrt[2])/z] Sqrt[-((I z)/(Sqrt[2] - I z))] -
Sqrt[1 - (I Sqrt[2])/z] Sqrt[(I z)/(Sqrt[2] + I z)] + (I z)/Sqrt[-z^2]) +
((z Sqrt[-2 - z^2])/(2 Sqrt[(-z^2) (2 + z^2)]))
ArcCosh[-((2 Sqrt[-1 - z^2])/z^2)]
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Cell[BoxData[RowBox[List[RowBox[List["ArcCsch", "[", "z", "]"]], "\[Equal]", RowBox[List[RowBox[List[FractionBox[RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], "4"], SqrtBox[FractionBox[SuperscriptBox["z", "2"], RowBox[List["2", "+", SuperscriptBox["z", "2"]]]]], " ", SqrtBox[FractionBox[RowBox[List["2", "+", SuperscriptBox["z", "2"]]], SuperscriptBox["z", "2"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SqrtBox[RowBox[List["1", "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", SqrtBox["2"]]], "z"]]]], " ", SqrtBox[RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "z"]], RowBox[List[SqrtBox["2"], "-", RowBox[List["\[ImaginaryI]", " ", "z"]]]]]]]]]], "-", RowBox[List[SqrtBox[RowBox[List["1", "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SqrtBox["2"]]], "z"]]]], " ", SqrtBox[FractionBox[RowBox[List["\[ImaginaryI]", " ", "z"]], RowBox[List[SqrtBox["2"], "+", RowBox[List["\[ImaginaryI]", " ", "z"]]]]]]]], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "z"]], SqrtBox[RowBox[List["-", SuperscriptBox["z", "2"]]]]]]], ")"]]]], "+", RowBox[List[FractionBox[RowBox[List["z", " ", SqrtBox[RowBox[List[RowBox[List["-", "2"]], "-", SuperscriptBox["z", "2"]]]]]], RowBox[List["2", " ", SqrtBox[RowBox[List[RowBox[List["-", SuperscriptBox["z", "2"]]], " ", RowBox[List["(", RowBox[List["2", "+", SuperscriptBox["z", "2"]]], ")"]]]]]]]], RowBox[List["ArcCosh", "[", RowBox[List["-", FractionBox[RowBox[List["2", " ", SqrtBox[RowBox[List[RowBox[List["-", "1"]], "-", SuperscriptBox["z", "2"]]]]]], SuperscriptBox["z", "2"]]]], "]"]]]]]]]]]]
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<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> ⩵ </mo> <mrow> <mrow> <mfrac> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> ⅈ </mi> </mrow> <mn> 4 </mn> </mfrac> <mo> ⁢ </mo> <msqrt> <mfrac> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 2 </mn> </mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mfrac> </msqrt> <mo> ⁢ </mo> <msqrt> <mfrac> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 2 </mn> </mrow> </mfrac> </msqrt> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <msqrt> <mrow> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> </mfrac> <mo> - </mo> <mrow> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <msqrt> <mn> 2 </mn> </msqrt> </mrow> <mi> z </mi> </mfrac> </mrow> </msqrt> <mo> ⁢ </mo> <msqrt> <mfrac> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mrow> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> + </mo> <msqrt> <mn> 2 </mn> </msqrt> </mrow> </mfrac> </msqrt> </mrow> <mo> + </mo> <mrow> <msqrt> <mrow> <mn> 1 </mn> <mo> + </mo> <mfrac> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <msqrt> <mn> 2 </mn> </msqrt> </mrow> <mi> z </mi> </mfrac> </mrow> </msqrt> <mo> ⁢ </mo> <msqrt> <mrow> <mo> - </mo> <mfrac> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mrow> <msqrt> <mn> 2 </mn> </msqrt> <mo> - </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </mrow> </mfrac> </mrow> </msqrt> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mrow> <mi> z </mi> <mo> ⁢ </mo> <msqrt> <mrow> <mrow> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mn> 2 </mn> </mrow> </msqrt> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msqrt> <mrow> <mrow> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </msqrt> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <msup> <mi> cosh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mo> - </mo> <mfrac> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msqrt> <mrow> <mrow> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> </mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <arccsch /> <ci> z </ci> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <pi /> <imaginaryi /> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <imaginaryi /> <ci> z </ci> <apply> <power /> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <imaginaryi /> <ci> z </ci> <apply> <power /> <apply> <plus /> <apply> <times /> <imaginaryi /> <ci> z </ci> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <imaginaryi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> z </ci> <apply> <power /> <apply> <plus /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> z </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <ci> z </ci> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -2 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <arccosh /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ArcCsch", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], ")"]], " ", SqrtBox[FractionBox[SuperscriptBox["z", "2"], RowBox[List["2", "+", SuperscriptBox["z", "2"]]]]], " ", SqrtBox[FractionBox[RowBox[List["2", "+", SuperscriptBox["z", "2"]]], SuperscriptBox["z", "2"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SqrtBox[RowBox[List["1", "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", SqrtBox["2"]]], "z"]]]], " ", SqrtBox[RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "z"]], RowBox[List[SqrtBox["2"], "-", RowBox[List["\[ImaginaryI]", " ", "z"]]]]]]]]]], "-", RowBox[List[SqrtBox[RowBox[List["1", "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SqrtBox["2"]]], "z"]]]], " ", SqrtBox[FractionBox[RowBox[List["\[ImaginaryI]", " ", "z"]], RowBox[List[SqrtBox["2"], "+", RowBox[List["\[ImaginaryI]", " ", "z"]]]]]]]], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "z"]], SqrtBox[RowBox[List["-", SuperscriptBox["z", "2"]]]]]]], ")"]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["z", " ", SqrtBox[RowBox[List[RowBox[List["-", "2"]], "-", SuperscriptBox["z", "2"]]]]]], ")"]], " ", RowBox[List["ArcCosh", "[", RowBox[List["-", FractionBox[RowBox[List["2", " ", SqrtBox[RowBox[List[RowBox[List["-", "1"]], "-", SuperscriptBox["z", "2"]]]]]], SuperscriptBox["z", "2"]]]], "]"]]]], RowBox[List["2", " ", SqrtBox[RowBox[List[RowBox[List["-", SuperscriptBox["z", "2"]]], " ", RowBox[List["(", RowBox[List["2", "+", SuperscriptBox["z", "2"]]], ")"]]]]]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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