Sin[z^(1/6)]/E^(Sqrt[3] z^(1/6)) == (Sqrt[3]/(4 Pi^(3/2))) MeijerG[{{}, {}}, {{1/6, 1/3, 1/2, 2/3, 5/6}, {0}}, z/729]

Cell[BoxData[RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", SqrtBox["3"]]], " ", SuperscriptBox["z", RowBox[List["1", "/", "6"]]]]]], " ", RowBox[List["Sin", "[", SuperscriptBox["z", RowBox[List["1", "/", "6"]]], "]"]]]], "\[Equal]", RowBox[List[FractionBox[SqrtBox["3"], RowBox[List["4", " ", SuperscriptBox["\[Pi]", RowBox[List["3", "/", "2"]]]]]], RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["1", "6"], ",", FractionBox["1", "3"], ",", FractionBox["1", "2"], ",", FractionBox["2", "3"], ",", FractionBox["5", "6"]]], "}"]], ",", RowBox[List["{", "0", "}"]]]], "}"]], ",", FractionBox["z", "729"]]], "]"]]]]]]]]

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <msup> <mi> &#8519; </mi> <mrow> <mrow> <mo> - </mo> <msqrt> <mn> 3 </mn> </msqrt> </mrow> <mo> &#8290; </mo> <mroot> <mi> z </mi> <mn> 6 </mn> </mroot> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mroot> <mi> z </mi> <mn> 6 </mn> </mroot> <mo> ) </mo> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <msqrt> <mn> 3 </mn> </msqrt> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <msup> <mi> &#960; </mi> <mrow> <mn> 3 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> </mrow> </mfrac> <mo> &#8290; </mo> <semantics> <mrow> <msubsup> <mi> G </mi> <mrow> <mn> 0 </mn> <mo> , </mo> <mn> 6 </mn> </mrow> <mrow> <mn> 5 </mn> <mo> , </mo> <mn> 0 </mn> </mrow> </msubsup> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <mi> z </mi> <mn> 729 </mn> </mfrac> <mo> &#10072; </mo> <mtable> <mtr> <mtd> <mrow> <mfrac> <mn> 1 </mn> <mn> 6 </mn> </mfrac> <mo> , </mo> <mfrac> <mn> 1 </mn> <mn> 3 </mn> </mfrac> <mo> , </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> , </mo> <mfrac> <mn> 2 </mn> <mn> 3 </mn> </mfrac> <mo> , </mo> <mfrac> <mn> 5 </mn> <mn> 6 </mn> </mfrac> <mo> , </mo> <mn> 0 </mn> </mrow> </mtd> </mtr> </mtable> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[SubsuperscriptBox[TagBox[&quot;G&quot;, MeijerG], RowBox[List[&quot;0&quot;, &quot;,&quot;, &quot;6&quot;]], RowBox[List[&quot;5&quot;, &quot;,&quot;, &quot;0&quot;]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[FractionBox[&quot;z&quot;, &quot;729&quot;], MeijerG, Rule[Editable, True]], &quot;\[VerticalSeparator]&quot;, GridBox[List[List[RowBox[List[TagBox[FractionBox[&quot;1&quot;, &quot;6&quot;], MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[FractionBox[&quot;1&quot;, &quot;3&quot;], MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[FractionBox[&quot;1&quot;, &quot;2&quot;], MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[FractionBox[&quot;2&quot;, &quot;3&quot;], MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[FractionBox[&quot;5&quot;, &quot;6&quot;], MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[&quot;0&quot;, MeijerG, Rule[Editable, True]]]]]]]]], &quot;)&quot;]]]], MeijerG, Rule[Editable, False]] </annotation> </semantics> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 6 </cn> </apply> </apply> </apply> <apply> <sin /> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 6 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <pi /> <cn type='rational'> 3 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> MeijerG </ci> <list> <list /> <list /> </list> <list> <list> <cn type='rational'> 1 <sep /> 6 </cn> <cn type='rational'> 1 <sep /> 3 </cn> <cn type='rational'> 1 <sep /> 2 </cn> <cn type='rational'> 2 <sep /> 3 </cn> <cn type='rational'> 5 <sep /> 6 </cn> </list> <list> <cn type='integer'> 0 </cn> </list> </list> <apply> <times /> <ci> z </ci> <apply> <power /> <cn type='integer'> 729 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", SqrtBox["3"]]], " ", SuperscriptBox["z_", RowBox[List["1", "/", "6"]]]]]], " ", RowBox[List["Sin", "[", SuperscriptBox["z_", RowBox[List["1", "/", "6"]]], "]"]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[SqrtBox["3"], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["1", "6"], ",", FractionBox["1", "3"], ",", FractionBox["1", "2"], ",", FractionBox["2", "3"], ",", FractionBox["5", "6"]]], "}"]], ",", RowBox[List["{", "0", "}"]]]], "}"]], ",", FractionBox["z", "729"]]], "]"]]]], RowBox[List["4", " ", SuperscriptBox["\[Pi]", RowBox[List["3", "/", "2"]]]]]]]]]]

2003-08-21