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http://functions.wolfram.com/06.05.19.0006.01
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Abs[Gamma[x + I y]/Gamma[x]]^2 == Product[1/(1 + y^2/(x + k)^2),
{k, 0, Infinity}]
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Cell[BoxData[RowBox[List[SuperscriptBox[RowBox[List["Abs", "[", FractionBox[RowBox[List["Gamma", "[", RowBox[List["x", "+", RowBox[List["\[ImaginaryI]", " ", "y"]]]], "]"]], RowBox[List["Gamma", "[", "x", "]"]]], "]"]], "2"], "\[Equal]", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox["1", RowBox[List["1", "+", FractionBox[SuperscriptBox["y", "2"], SuperscriptBox[RowBox[List["(", RowBox[List["x", "+", "k"]], ")"]], "2"]]]]]]]]]]]
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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> <msup> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mfrac> <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> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> </mfrac> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> <mn> 2 </mn> </msup> <mo> ⩵ </mo> <mrow> <munderover> <mo> ∏ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> ∞ </mi> </munderover> <mfrac> <mn> 1 </mn> <mrow> <mfrac> <msup> <mi> y </mi> <mn> 2 </mn> </msup> <msup> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mi> x </mi> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> </mfrac> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <power /> <apply> <abs /> <apply> <times /> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> x </ci> <apply> <times /> <imaginaryi /> <ci> y </ci> </apply> </apply> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <ci> x </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <product /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> y </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <ci> k </ci> <ci> x </ci> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", SuperscriptBox[RowBox[List["Abs", "[", FractionBox[RowBox[List["Gamma", "[", RowBox[List["x_", "+", RowBox[List["\[ImaginaryI]", " ", "y_"]]]], "]"]], RowBox[List["Gamma", "[", "x_", "]"]]], "]"]], "2"], "]"]], "\[RuleDelayed]", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox["1", RowBox[List["1", "+", FractionBox[SuperscriptBox["y", "2"], SuperscriptBox[RowBox[List["(", RowBox[List["x", "+", "k"]], ")"]], "2"]]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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