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http://functions.wolfram.com/01.12.27.0654.01
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ArcSin[Sqrt[Sqrt[1 + z^2] - z]/(Sqrt[2] (1 + z^2)^(1/4))] ==
(1 - Sqrt[z^2]/z + Sqrt[(z + I)/z] Sqrt[z/(z + I)] -
Sqrt[(z - I)/z] Sqrt[z/(z - I)]) (Pi/4) + (1/2) Sqrt[(-I) z + 1]
Sqrt[1/((-I) z + 1)] ArcTan[1/z]
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Cell[BoxData[RowBox[List[RowBox[List["ArcSin", "[", FractionBox[SqrtBox[RowBox[List[SqrtBox[RowBox[List["1", "+", SuperscriptBox["z", "2"]]]], "-", "z"]]], RowBox[List[SqrtBox["2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", SuperscriptBox["z", "2"]]], ")"]], RowBox[List["1", "/", "4"]]]]]], "]"]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["1", "-", FractionBox[SqrtBox[SuperscriptBox["z", "2"]], "z"], "+", RowBox[List[SqrtBox[FractionBox[RowBox[List["z", "+", "\[ImaginaryI]"]], "z"]], SqrtBox[FractionBox["z", RowBox[List["z", "+", "\[ImaginaryI]"]]]]]], "-", RowBox[List[SqrtBox[FractionBox[RowBox[List["z", "-", "\[ImaginaryI]"]], "z"]], SqrtBox[FractionBox["z", RowBox[List["z", "-", "\[ImaginaryI]"]]]]]]]], ")"]], FractionBox["\[Pi]", "4"]]], "+", RowBox[List[FractionBox["1", "2"], SqrtBox[RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "z"]], "+", "1"]]], SqrtBox[FractionBox["1", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "z"]], "+", "1"]]]], RowBox[List["ArcTan", "[", FractionBox["1", "z"], "]"]]]]]]]]]]
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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> sin </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <msqrt> <mrow> <msqrt> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> - </mo> <mi> z </mi> </mrow> </msqrt> <mrow> <msqrt> <mn> 2 </mn> </msqrt> <mo> ⁢ </mo> <mroot> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> <mn> 4 </mn> </mroot> </mrow> </mfrac> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mfrac> <mi> π </mi> <mn> 4 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <msqrt> <mfrac> <mi> z </mi> <mrow> <mi> z </mi> <mo> - </mo> <mi> ⅈ </mi> </mrow> </mfrac> </msqrt> </mrow> <mo> ⁢ </mo> <msqrt> <mfrac> <mrow> <mi> z </mi> <mo> - </mo> <mi> ⅈ </mi> </mrow> <mi> z </mi> </mfrac> </msqrt> </mrow> <mo> + </mo> <mrow> <msqrt> <mfrac> <mi> z </mi> <mrow> <mi> z </mi> <mo> + </mo> <mi> ⅈ </mi> </mrow> </mfrac> </msqrt> <mo> ⁢ </mo> <msqrt> <mfrac> <mrow> <mi> z </mi> <mo> + </mo> <mi> ⅈ </mi> </mrow> <mi> z </mi> </mfrac> </msqrt> </mrow> <mo> - </mo> <mfrac> <msqrt> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </msqrt> <mi> z </mi> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <msqrt> <mfrac> <mn> 1 </mn> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </mrow> </mfrac> </msqrt> <mo> ⁢ </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </mrow> </msqrt> <mo> ⁢ </mo> <mrow> <msup> <mi> tan </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <mn> 1 </mn> <mi> z </mi> </mfrac> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <arcsin /> <apply> <times /> <apply> <power /> <apply> <plus /> <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> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <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 /> 4 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <pi /> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <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> <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> <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> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 1 </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='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </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> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <arctan /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> z </ci> <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["ArcSin", "[", FractionBox[SqrtBox[RowBox[List[SqrtBox[RowBox[List["1", "+", SuperscriptBox["z_", "2"]]]], "-", "z_"]]], RowBox[List[SqrtBox["2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", SuperscriptBox["z_", "2"]]], ")"]], RowBox[List["1", "/", "4"]]]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["1", "-", FractionBox[SqrtBox[SuperscriptBox["z", "2"]], "z"], "+", RowBox[List[SqrtBox[FractionBox[RowBox[List["z", "+", "\[ImaginaryI]"]], "z"]], " ", SqrtBox[FractionBox["z", RowBox[List["z", "+", "\[ImaginaryI]"]]]]]], "-", RowBox[List[SqrtBox[FractionBox[RowBox[List["z", "-", "\[ImaginaryI]"]], "z"]], " ", SqrtBox[FractionBox["z", RowBox[List["z", "-", "\[ImaginaryI]"]]]]]]]], ")"]], " ", "\[Pi]"]], "+", RowBox[List[FractionBox["1", "2"], " ", SqrtBox[RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "z"]], "+", "1"]]], " ", SqrtBox[FractionBox["1", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "z"]], "+", "1"]]]], " ", RowBox[List["ArcTan", "[", FractionBox["1", "z"], "]"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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