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http://functions.wolfram.com/01.02.19.0015.01
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Im[(I y)^a] == Sign[y] Abs[y]^a Sin[(Pi a)/2] /;
Element[a, Reals] && Element[y, Reals]
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Im", "[", SuperscriptBox[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", "y"]], ")"]], "a"], "]"]], "\[Equal]", RowBox[List[RowBox[List["Sign", "[", "y", "]"]], " ", SuperscriptBox[RowBox[List["Abs", "[", "y", "]"]], "a"], " ", RowBox[List["Sin", "[", FractionBox[RowBox[List["\[Pi]", " ", "a"]], "2"], "]"]]]]]], "/;", RowBox[List[RowBox[List["a", "\[Element]", "Reals"]], "&&", 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> <mi> Im </mi> <mo> ⁡ </mo> <mo> ( </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> y </mi> </mrow> <mo> ) </mo> </mrow> <mi> a </mi> </msup> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mi> sgn </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> y </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mi> y </mi> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> <mi> a </mi> </msup> <mo> ⁢ </mo> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> a </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> a </mi> <mo> ∈ </mo> <semantics> <mi> ℝ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalR]", Function[Reals]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <mi> y </mi> <mo> ∈ </mo> <semantics> <mi> ℝ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalR]", Function[Reals]] </annotation> </semantics> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <imaginary /> <apply> <power /> <apply> <times /> <imaginaryi /> <ci> y </ci> </apply> <ci> a </ci> </apply> </apply> <apply> <times /> <apply> <ci> Sign </ci> <ci> y </ci> </apply> <apply> <power /> <apply> <abs /> <ci> y </ci> </apply> <ci> a </ci> </apply> <apply> <sin /> <apply> <times /> <pi /> <ci> a </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <ci> a </ci> <reals /> </apply> <apply> <in /> <ci> y </ci> <reals /> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Im", "[", SuperscriptBox[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", "y_"]], ")"]], "a_"], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["Sign", "[", "y", "]"]], " ", SuperscriptBox[RowBox[List["Abs", "[", "y", "]"]], "a"], " ", RowBox[List["Sin", "[", FractionBox[RowBox[List["\[Pi]", " ", "a"]], "2"], "]"]]]], "/;", RowBox[List[RowBox[List["a", "\[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}
