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http://functions.wolfram.com/06.05.19.0001.01
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Re[Gamma[x + I y]] ==
Sum[((-1)^k/k!) (k/((k + x)^2 + y^2) + x/((k + x)^2 + y^2)),
{k, 0, Infinity}] + Integrate[(t^(x - 1) Cos[y Log[t]])/E^t,
{t, 1, Infinity}]
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Cell[BoxData[RowBox[List[RowBox[List["Re", "[", RowBox[List["Gamma", "[", RowBox[List["x", "+", RowBox[List["\[ImaginaryI]", " ", "y"]]]], "]"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " "]], RowBox[List["k", "!"]]], RowBox[List["(", RowBox[List[FractionBox["k", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["k", "+", "x"]], ")"]], "2"], "+", SuperscriptBox["y", "2"]]]], "+", FractionBox["x", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["k", "+", "x"]], ")"]], "2"], "+", SuperscriptBox["y", "2"]]]]]], ")"]]]]]], "+", RowBox[List[SubsuperscriptBox["\[Integral]", "1", "\[Infinity]"], RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["-", "t"]]], " ", SuperscriptBox["t", RowBox[List["x", "-", "1"]]], " ", RowBox[List["Cos", "[", RowBox[List["y", " ", RowBox[List["Log", "[", "t", "]"]]]], "]"]]]], RowBox[List["\[DifferentialD]", "t"]]]]]]]]]]]]
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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> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> Γ </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> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <msubsup> <mo> ∫ </mo> <mn> 1 </mn> <mi> ∞ </mi> </msubsup> <mrow> <mrow> <msup> <mi> ⅇ </mi> <mrow> <mo> - </mo> <mi> t </mi> </mrow> </msup> <mo> ⁢ </mo> <msup> <mi> t </mi> <mrow> <mi> x </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> y </mi> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> t </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> t </mi> </mrow> </mrow> </mrow> <mo> + </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> ∞ </mi> </munderover> <mrow> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msup> <mtext> </mtext> </mrow> <mrow> <mi> k </mi> <mo> ! </mo> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <mi> k </mi> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mi> x </mi> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> + </mo> <msup> <mi> y </mi> <mn> 2 </mn> </msup> </mrow> </mfrac> <mo> + </mo> <mfrac> <mi> x </mi> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mi> x </mi> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> + </mo> <msup> <mi> y </mi> <mn> 2 </mn> </msup> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <real /> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> x </ci> <apply> <times /> <imaginaryi /> <ci> y </ci> </apply> </apply> </apply> </apply> <apply> <plus /> <apply> <int /> <bvar> <ci> t </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> t </ci> </apply> </apply> <apply> <power /> <ci> t </ci> <apply> <plus /> <ci> x </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <cos /> <apply> <times /> <ci> y </ci> <apply> <ln /> <ci> t </ci> </apply> </apply> </apply> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> <apply> <power /> <apply> <factorial /> <ci> k </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <ci> k </ci> <apply> <power /> <apply> <plus /> <apply> <power /> <apply> <plus /> <ci> k </ci> <ci> x </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> y </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <ci> x </ci> <apply> <power /> <apply> <plus /> <apply> <power /> <apply> <plus /> <ci> k </ci> <ci> x </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> y </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Re", "[", RowBox[List["Gamma", "[", RowBox[List["x_", "+", RowBox[List["\[ImaginaryI]", " ", "y_"]]]], "]"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List["(", RowBox[List[FractionBox["k", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["k", "+", "x"]], ")"]], "2"], "+", SuperscriptBox["y", "2"]]]], "+", FractionBox["x", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["k", "+", "x"]], ")"]], "2"], "+", SuperscriptBox["y", "2"]]]]]], ")"]]]], RowBox[List["k", "!"]]]]], "+", RowBox[List[SubsuperscriptBox["\[Integral]", "1", "\[Infinity]"], RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["-", "t"]]], " ", SuperscriptBox["t", RowBox[List["x", "-", "1"]]], " ", RowBox[List["Cos", "[", RowBox[List["y", " ", RowBox[List["Log", "[", "t", "]"]]]], "]"]]]], RowBox[List["\[DifferentialD]", "t"]]]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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