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http://functions.wolfram.com/01.30.27.1816.01
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ArcSech[(-I) z] == (1/2) I Pi (-1 - I Sqrt[I/z] Sqrt[I z] +
Sqrt[(z - I)/z] Sqrt[z/(z - I)]) + (1/2) Sqrt[1/(1 + I z)] Sqrt[1 + I z]
ArcCosh[(2 + z^2)/z^2]
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Cell[BoxData[RowBox[List[RowBox[List["ArcSech", "[", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", "\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", RowBox[List["\[ImaginaryI]", " ", SqrtBox[FractionBox["\[ImaginaryI]", "z"]], " ", SqrtBox[RowBox[List["\[ImaginaryI]", " ", "z"]]]]], "+", RowBox[List[SqrtBox[FractionBox[RowBox[List["z", "-", "\[ImaginaryI]"]], "z"]], " ", SqrtBox[FractionBox["z", RowBox[List["z", "-", "\[ImaginaryI]"]]]]]]]], ")"]]]], "+", RowBox[List[FractionBox["1", "2"], " ", SqrtBox[FractionBox["1", RowBox[List["1", "+", RowBox[List["\[ImaginaryI]", " ", "z"]]]]]], " ", SqrtBox[RowBox[List["1", "+", RowBox[List["\[ImaginaryI]", " ", "z"]]]]], " ", RowBox[List["ArcCosh", "[", FractionBox[RowBox[List["2", "+", 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> sech </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> ⅈ </mi> </mrow> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> - </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <msqrt> <mfrac> <mi> ⅈ </mi> <mi> z </mi> </mfrac> </msqrt> <mo> ⁢ </mo> <msqrt> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </msqrt> </mrow> <mo> + </mo> <mrow> <msqrt> <mfrac> <mrow> <mi> z </mi> <mo> - </mo> <mi> ⅈ </mi> </mrow> <mi> z </mi> </mfrac> </msqrt> <mo> ⁢ </mo> <msqrt> <mfrac> <mi> z </mi> <mrow> <mi> z </mi> <mo> - </mo> <mi> ⅈ </mi> </mrow> </mfrac> </msqrt> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <msqrt> <mfrac> <mn> 1 </mn> <mrow> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </mfrac> </msqrt> <mo> ⁢ </mo> <msqrt> <mrow> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> ⁢ </mo> <mrow> <msup> <mi> cosh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <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> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <arcsech /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <ci> z </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <imaginaryi /> <pi /> <apply> <plus /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <apply> <power /> <apply> <times /> <imaginaryi /> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <imaginaryi /> <ci> z </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <times /> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <ci> z </ci> <apply> <power /> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <plus /> <apply> <times /> <imaginaryi /> <ci> z </ci> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </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> <arccosh /> <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> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ArcSech", "[", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", "\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", RowBox[List["\[ImaginaryI]", " ", SqrtBox[FractionBox["\[ImaginaryI]", "z"]], " ", SqrtBox[RowBox[List["\[ImaginaryI]", " ", "z"]]]]], "+", RowBox[List[SqrtBox[FractionBox[RowBox[List["z", "-", "\[ImaginaryI]"]], "z"]], " ", SqrtBox[FractionBox["z", RowBox[List["z", "-", "\[ImaginaryI]"]]]]]]]], ")"]]]], "+", RowBox[List[FractionBox["1", "2"], " ", SqrtBox[FractionBox["1", RowBox[List["1", "+", RowBox[List["\[ImaginaryI]", " ", "z"]]]]]], " ", SqrtBox[RowBox[List["1", "+", RowBox[List["\[ImaginaryI]", " ", "z"]]]]], " ", RowBox[List["ArcCosh", "[", FractionBox[RowBox[List["2", "+", SuperscriptBox["z", "2"]]], SuperscriptBox["z", "2"]], "]"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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