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http://functions.wolfram.com/01.03.16.0018.01
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E^((1/2) I ArcTan[z]) == (1/Sqrt[2]) (Sqrt[1 + 1/Sqrt[z^2 + 1]] +
((I z)/Sqrt[z^2]) Sqrt[1 - 1/Sqrt[z^2 + 1]])
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Cell[BoxData[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox["1", "2"], " ", "\[ImaginaryI]", " ", RowBox[List["ArcTan", "[", "z", "]"]]]]], "\[Equal]", RowBox[List[FractionBox["1", SqrtBox["2"]], RowBox[List["(", RowBox[List[SqrtBox[RowBox[List["1", "+", FractionBox["1", SqrtBox[RowBox[List[SuperscriptBox["z", "2"], "+", "1"]]]]]]], "+", RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", "z", " "]], SqrtBox[SuperscriptBox["z", "2"]]], SqrtBox[RowBox[List["1", "-", FractionBox["1", SqrtBox[RowBox[List[SuperscriptBox["z", "2"], "+", "1"]]]]]]]]]]], ")"]]]]]]]]
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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> exp </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mfrac> <mi> ⅈ </mi> <mn> 2 </mn> </mfrac> <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> <msqrt> <mn> 2 </mn> </msqrt> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mfrac> <mrow> <mtext> </mtext> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </mrow> <msqrt> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </msqrt> </mfrac> <mo> ⁢ </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <mn> 1 </mn> <msqrt> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> </mfrac> </mrow> </msqrt> </mrow> <mo> + </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> + </mo> <mfrac> <mn> 1 </mn> <msqrt> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> </mfrac> </mrow> </msqrt> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <exp /> <apply> <times /> <apply> <times /> <imaginaryi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <arctan /> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <imaginaryi /> <ci> z </ci> <apply> <power /> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox["1", "2"], " ", "\[ImaginaryI]", " ", RowBox[List["ArcTan", "[", "z_", "]"]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[SqrtBox[RowBox[List["1", "+", FractionBox["1", SqrtBox[RowBox[List[SuperscriptBox["z", "2"], "+", "1"]]]]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", "z"]], ")"]], " ", SqrtBox[RowBox[List["1", "-", FractionBox["1", SqrtBox[RowBox[List[SuperscriptBox["z", "2"], "+", "1"]]]]]]]]], SqrtBox[SuperscriptBox["z", "2"]]]]], SqrtBox["2"]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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