Re[z^a] == Sum[(Pochhammer[-a, j + l]/((l - j)! j! Pochhammer[1/2, j])) (1 - Re[z])^l (Im[z]^2/(4 (Re[z] - 1)))^j, {j, 0, Infinity}, {l, 0, Infinity}] /; Element[a, Reals]

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Re", "[", SuperscriptBox["z", "a"], "]"]], "\[Equal]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "\[Infinity]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["l", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", "a"]], ",", RowBox[List["j", "+", "l"]]]], "]"]], " "]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["l", "-", "j"]], ")"]], "!"]], " ", RowBox[List["j", "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "j"]], "]"]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", RowBox[List["Re", "[", "z", "]"]]]], ")"]], "l"], " ", SuperscriptBox[RowBox[List["(", FractionBox[SuperscriptBox[RowBox[List["Im", "[", "z", "]"]], "2"], RowBox[List["4", " ", RowBox[List["(", RowBox[List[RowBox[List["Re", "[", "z", "]"]], "-", "1"]], ")"]]]]], ")"]], "j"]]]]]]]]], "/;", RowBox[List["Element", "[", RowBox[List["a", ",", "Reals"]], "]"]]]]]]

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <msup> <mi> z </mi> <mi> a </mi> </msup> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> l </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <mfrac> <mrow> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mi> a </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> j </mi> <mo> + </mo> <mi> l </mi> </mrow> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[&quot;-&quot;, &quot;a&quot;]], &quot;)&quot;]], RowBox[List[&quot;j&quot;, &quot;+&quot;, &quot;l&quot;]]], Pochhammer] </annotation> </semantics> <mtext> </mtext> </mrow> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> l </mi> <mo> - </mo> <mi> j </mi> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> j </mi> <mo> ! </mo> </mrow> <mo> &#8290; </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mi> j </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, FractionBox[&quot;1&quot;, &quot;2&quot;], &quot;)&quot;]], &quot;j&quot;], Pochhammer] </annotation> </semantics> </mrow> </mfrac> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mi> l </mi> </msup> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mfrac> <msup> <mrow> <mi> Im </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> ) </mo> </mrow> <mi> j </mi> </msup> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> a </mi> <mo> &#8712; </mo> <semantics> <mi> &#8477; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalR]&quot;, Function[Reals]] </annotation> </semantics> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <real /> <apply> <power /> <ci> z </ci> <ci> a </ci> </apply> </apply> <apply> <sum /> <bvar> <ci> l </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <times /> <apply> <ci> Pochhammer </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> <apply> <plus /> <ci> j </ci> <ci> l </ci> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <factorial /> <apply> <plus /> <ci> l </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> </apply> </apply> <apply> <factorial /> <ci> j </ci> </apply> <apply> <ci> Pochhammer </ci> <cn type='rational'> 1 <sep /> 2 </cn> <ci> j </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <real /> <ci> z </ci> </apply> </apply> </apply> <ci> l </ci> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <apply> <imaginary /> <ci> z </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <plus /> <apply> <real /> <ci> z </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <ci> j </ci> </apply> </apply> </apply> </apply> </apply> <apply> <in /> <ci> a </ci> <reals /> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Re", "[", SuperscriptBox["z_", "a_"], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "\[Infinity]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["l", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", "a"]], ",", RowBox[List["j", "+", "l"]]]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", RowBox[List["Re", "[", "z", "]"]]]], ")"]], "l"], " ", SuperscriptBox[RowBox[List["(", FractionBox[SuperscriptBox[RowBox[List["Im", "[", "z", "]"]], "2"], RowBox[List["4", " ", RowBox[List["(", RowBox[List[RowBox[List["Re", "[", "z", "]"]], "-", "1"]], ")"]]]]], ")"]], "j"]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["l", "-", "j"]], ")"]], "!"]], " ", RowBox[List["j", "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "j"]], "]"]]]]]]]]], "/;", RowBox[List["a", "\[Element]", "Reals"]]]]]]]]

2001-10-29