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http://functions.wolfram.com/01.06.19.0008.01
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Abs[Sin[x + I y]] == Abs[Sin[x] + Sin[I y]] /;
Element[x, Reals] && Element[y, Reals]
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Abs", "[", RowBox[List["Sin", "[", RowBox[List["x", "+", RowBox[List["\[ImaginaryI]", " ", "y"]]]], "]"]], "]"]], "\[Equal]", RowBox[List["Abs", "[", RowBox[List[RowBox[List["Sin", "[", "x", "]"]], "+", RowBox[List["Sin", "[", RowBox[List["\[ImaginaryI]", " ", "y"]], "]"]]]], "]"]]]], "/;", RowBox[List[RowBox[List["x", "\[Element]", "Reals"]], "\[And]", RowBox[List["y", "\[Element]", "Reals"]]]]]]]]
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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> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> x </mi> <mo> + </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> y </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> <mo> ⩵ </mo> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mrow> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> y </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> x </mi> <mo> ∈ </mo> <mi> ℝ </mi> </mrow> <mo> ∧ </mo> <mrow> <mi> y </mi> <mo> ∈ </mo> <mi> ℝ </mi> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <abs /> <apply> <sin /> <apply> <plus /> <ci> x </ci> <apply> <times /> <imaginaryi /> <ci> y </ci> </apply> </apply> </apply> </apply> <apply> <abs /> <apply> <plus /> <apply> <sin /> <ci> x </ci> </apply> <apply> <sin /> <apply> <times /> <imaginaryi /> <ci> y </ci> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <ci> x </ci> <ci> ℝ </ci> </apply> <apply> <in /> <ci> y </ci> <ci> ℝ </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Abs", "[", RowBox[List["Sin", "[", RowBox[List["x_", "+", RowBox[List["\[ImaginaryI]", " ", "y_"]]]], "]"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["Abs", "[", RowBox[List[RowBox[List["Sin", "[", "x", "]"]], "+", RowBox[List["Sin", "[", RowBox[List["\[ImaginaryI]", " ", "y"]], "]"]]]], "]"]], "/;", RowBox[List[RowBox[List["x", "\[Element]", "Reals"]], "&&", RowBox[List["y", "\[Element]", "Reals"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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){kind=small}
