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http://functions.wolfram.com/01.20.16.0132.01
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Cosh[a ArcTan[z]] == ((1/2) ((1 - I z)^(a I) + (1 + I z)^(a I)))/
(1 + z^2)^((a I)/2)
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Cell[BoxData[RowBox[List[RowBox[List["Cosh", "[", RowBox[List["a", " ", RowBox[List["ArcTan", "[", "z", "]"]]]], "]"]], "\[Equal]", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", RowBox[List["\[ImaginaryI]", " ", "z"]]]], ")"]], RowBox[List["a", " ", "\[ImaginaryI]"]]], "+", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", RowBox[List["\[ImaginaryI]", " ", "z"]]]], ")"]], RowBox[List["a", " ", "\[ImaginaryI]"]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", SuperscriptBox["z", "2"]]], ")"]], RowBox[List["-", FractionBox[RowBox[List["a", " ", "\[ImaginaryI]"]], "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> <mi> cosh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mrow> <msup> <mi> tan </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> ⅈ </mi> </mrow> </msup> <mo> + </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> ⅈ </mi> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> ⅈ </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </msup> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <cosh /> <apply> <times /> <ci> a </ci> <apply> <arctan /> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <apply> <power /> <apply> <plus /> <apply> <times /> <imaginaryi /> <ci> z </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <ci> a </ci> <imaginaryi /> </apply> </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> <apply> <times /> <ci> a </ci> <imaginaryi /> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <times /> <ci> a </ci> <imaginaryi /> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Cosh", "[", RowBox[List["a_", " ", RowBox[List["ArcTan", "[", "z_", "]"]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", RowBox[List["\[ImaginaryI]", " ", "z"]]]], ")"]], RowBox[List["a", " ", "\[ImaginaryI]"]]], "+", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", RowBox[List["\[ImaginaryI]", " ", "z"]]]], ")"]], RowBox[List["a", " ", "\[ImaginaryI]"]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", SuperscriptBox["z", "2"]]], ")"]], RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", RowBox[List["(", RowBox[List["a", " ", "\[ImaginaryI]"]], ")"]]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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