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http://functions.wolfram.com/01.06.16.0165.01
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Sin[a (b z^n)^(1/n)] ==
(I/2) Sum[(((-I) a (b z^n)^(1/n))^i/i!) HypergeometricPFQ[{1},
{(i + 1)/n, (i + 2)/n, \[Ellipsis], (i + n)/n},
((-I)^n a^n b z^n)/n^n] - ((I a (b z^n)^(1/n))^i/i!)
HypergeometricPFQ[{1}, {(i + 1)/n, (i + 2)/n, \[Ellipsis], (i + n)/n},
(I^n a^n b z^n)/n^n], {i, 0, n - 1}] /; Element[n, Integers] && n > 0
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Sin", "[", RowBox[List["a", " ", SuperscriptBox[RowBox[List["(", RowBox[List["b", " ", SuperscriptBox["z", "n"]]], ")"]], RowBox[List["1", "/", "n"]]]]], "]"]], "\[Equal]", RowBox[List[FractionBox["\[ImaginaryI]", "2"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "0"]], RowBox[List["n", "-", "1"]]], RowBox[List["(", RowBox[List[RowBox[List[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a", " ", SuperscriptBox[RowBox[List["(", RowBox[List["b", " ", SuperscriptBox["z", "n"]]], ")"]], RowBox[List["1", "/", "n"]]]]], ")"]], "i"], RowBox[List["i", "!"]]], RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", "1", "}"]], ",", RowBox[List["{", RowBox[List[FractionBox[RowBox[List["i", "+", "1"]], "n"], ",", FractionBox[RowBox[List["i", "+", "2"]], "n"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List["i", "+", "n"]], "n"]]], "}"]], ",", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "\[ImaginaryI]"]], ")"]], "n"], SuperscriptBox["a", "n"], " ", "b", " ", SuperscriptBox["z", "n"]]], SuperscriptBox["n", "n"]]]], " ", "]"]]]], "-", RowBox[List[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", "a", " ", SuperscriptBox[RowBox[List["(", RowBox[List["b", " ", SuperscriptBox["z", "n"]]], ")"]], RowBox[List["1", "/", "n"]]]]], ")"]], "i"], RowBox[List["i", "!"]]], RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", "1", "}"]], ",", RowBox[List["{", RowBox[List[FractionBox[RowBox[List["i", "+", "1"]], "n"], ",", FractionBox[RowBox[List["i", "+", "2"]], "n"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List["i", "+", "n"]], "n"]]], "}"]], ",", FractionBox[RowBox[List[SuperscriptBox["\[ImaginaryI]", "n"], SuperscriptBox["a", "n"], " ", "b", " ", SuperscriptBox["z", "n"]]], SuperscriptBox["n", "n"]]]], " ", "]"]]]]]], ")"]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", ">", "0"]]]]]]]]
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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> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <msup> <mi> z </mi> <mi> n </mi> </msup> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 1 </mn> <mo> / </mo> <mi> n </mi> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> <mo>  </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> i </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <mo> ( </mo> <mrow> <mrow> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> ⅈ </mi> </mrow> <mo> ⁢ </mo> <mi> a </mi> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <msup> <mi> z </mi> <mi> n </mi> </msup> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 1 </mn> <mo> / </mo> <mi> n </mi> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> <mi> i </mi> </msup> <mtext> </mtext> </mrow> <mrow> <mi> i </mi> <mo> ! </mo> </mrow> </mfrac> <mo> ⁢ </mo> <semantics> <mrow> <mrow> <msub> <mo>   </mo> <mn> 1 </mn> </msub> <msub> <mi> F </mi> <mi> n </mi> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> ; </mo> <mrow> <mfrac> <mrow> <mi> i </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </mfrac> <mo> , </mo> <mfrac> <mrow> <mi> i </mi> <mo> + </mo> <mn> 2 </mn> </mrow> <mi> n </mi> </mfrac> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mfrac> <mrow> <mi> i </mi> <mo> + </mo> <mi> n </mi> </mrow> <mi> n </mi> </mfrac> </mrow> <mo> ; </mo> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mi> ⅈ </mi> </mrow> <mo> ) </mo> </mrow> <mi> n </mi> </msup> <mo> ⁢ </mo> <msup> <mi> a </mi> <mi> n </mi> </msup> <mo> ⁢ </mo> <mi> b </mi> <mo> ⁢ </mo> <msup> <mi> z </mi> <mi> n </mi> </msup> </mrow> <msup> <mi> n </mi> <mi> n </mi> </msup> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "1"], SubscriptBox["F", "n"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[TagBox["1", HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[FractionBox[RowBox[List["i", "+", "1"]], "n"], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[FractionBox[RowBox[List["i", "+", "2"]], "n"], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[FractionBox[RowBox[List["i", "+", "n"]], "n"], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "\[ImaginaryI]"]], ")"]], "n"], " ", SuperscriptBox["a", "n"], " ", "b", " ", SuperscriptBox["z", "n"]]], SuperscriptBox["n", "n"]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False], Rule[Selectable, False]], HypergeometricPFQ] </annotation> </semantics> </mrow> <mo> - </mo> <mrow> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> a </mi> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <msup> <mi> z </mi> <mi> n </mi> </msup> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 1 </mn> <mo> / </mo> <mi> n </mi> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> <mi> i </mi> </msup> <mtext> </mtext> </mrow> <mrow> <mi> i </mi> <mo> ! </mo> </mrow> </mfrac> <mo> ⁢ </mo> <semantics> <mrow> <mrow> <msub> <mo>   </mo> <mn> 1 </mn> </msub> <msub> <mi> F </mi> <mi> n </mi> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> ; </mo> <mrow> <mfrac> <mrow> <mi> i </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </mfrac> <mo> , </mo> <mfrac> <mrow> <mi> i </mi> <mo> + </mo> <mn> 2 </mn> </mrow> <mi> n </mi> </mfrac> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mfrac> <mrow> <mi> i </mi> <mo> + </mo> <mi> n </mi> </mrow> <mi> n </mi> </mfrac> </mrow> <mo> ; </mo> <mfrac> <mrow> <msup> <mi> ⅈ </mi> <mi> n </mi> </msup> <mo> ⁢ </mo> <msup> <mi> a </mi> <mi> n </mi> </msup> <mo> ⁢ </mo> <mi> b </mi> <mo> ⁢ </mo> <msup> <mi> z </mi> <mi> n </mi> </msup> </mrow> <msup> <mi> n </mi> <mi> n </mi> </msup> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "1"], SubscriptBox["F", "n"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[TagBox["1", HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[FractionBox[RowBox[List["i", "+", "1"]], "n"], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[FractionBox[RowBox[List["i", "+", "2"]], "n"], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[FractionBox[RowBox[List["i", "+", "n"]], "n"], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[FractionBox[RowBox[List[SuperscriptBox["\[ImaginaryI]", "n"], " ", SuperscriptBox["a", "n"], " ", "b", " ", SuperscriptBox["z", "n"]]], SuperscriptBox["n", "n"]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False], Rule[Selectable, False]], HypergeometricPFQ] </annotation> </semantics> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> n </mi> <mo> ∈ </mo> <msup> <mi> ℕ </mi> <mo> + </mo> </msup> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <sin /> <apply> <times /> <ci> a </ci> <apply> <power /> <apply> <times /> <ci> b </ci> <apply> <power /> <ci> z </ci> <ci> n </ci> </apply> </apply> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <imaginaryi /> <apply> <sum /> <bvar> <ci> i </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </uplimit> <apply> <plus /> <apply> <times /> <apply> <times /> <apply> <power /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <ci> a </ci> <apply> <power /> <apply> <times /> <ci> b </ci> <apply> <power /> <ci> z </ci> <ci> n </ci> </apply> </apply> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <ci> i </ci> </apply> <apply> <power /> <apply> <factorial /> <ci> i </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> HypergeometricPFQ </ci> <list> <cn type='integer'> 1 </cn> </list> <list> <apply> <times /> <apply> <plus /> <ci> i </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <plus /> <ci> i </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> <ci> … </ci> <apply> <times /> <apply> <plus /> <ci> i </ci> <ci> n </ci> </apply> <apply> <power /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> </list> <apply> <times /> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <ci> n </ci> </apply> <apply> <power /> <ci> a </ci> <ci> n </ci> </apply> <ci> b </ci> <apply> <power /> <ci> z </ci> <ci> n </ci> </apply> <apply> <power /> <apply> <power /> <ci> n </ci> <ci> n </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <apply> <power /> <apply> <times /> <imaginaryi /> <ci> a </ci> <apply> <power /> <apply> <times /> <ci> b </ci> <apply> <power /> <ci> z </ci> <ci> n </ci> </apply> </apply> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <ci> i </ci> </apply> <apply> <power /> <apply> <factorial /> <ci> i </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> HypergeometricPFQ </ci> <list> <cn type='integer'> 1 </cn> </list> <list> <apply> <times /> <apply> <plus /> <ci> i </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <plus /> <ci> i </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> <ci> … </ci> <apply> <times /> <apply> <plus /> <ci> i </ci> <ci> n </ci> </apply> <apply> <power /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> </list> <apply> <times /> <apply> <power /> <imaginaryi /> <ci> n </ci> </apply> <apply> <power /> <ci> a </ci> <ci> n </ci> </apply> <ci> b </ci> <apply> <power /> <ci> z </ci> <ci> n </ci> </apply> <apply> <power /> <apply> <power /> <ci> n </ci> <ci> n </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <in /> <ci> n </ci> <apply> <ci> SuperPlus </ci> <ci> ℕ </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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Date Added to functions.wolfram.com (modification date)
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){kind=small}
