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variants of this functions
Gamma




Mathematica Notation

Traditional Notation









Gamma, Beta, Erf > Gamma[z] > Inequalities





http://functions.wolfram.com/06.05.29.0009.01









  


  








Input Form





Inequality[(x + y/2)^(1 - y), Less, Gamma[x + 1]/Gamma[x + y], Greater, Exp[(1 - y) PolyGamma[x + (y + 1)/2]]] /; Element[x, Reals] && x > 0 && Element[y, Reals] && 0 < y < 1








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["x", "+", FractionBox["y", "2"]]], ")"]], RowBox[List["1", "-", "y"]]], "<", FractionBox[RowBox[List["Gamma", "[", RowBox[List["x", "+", "1"]], "]"]], RowBox[List["Gamma", "[", RowBox[List["x", "+", "y"]], "]"]]], ">", RowBox[List["Exp", "[", RowBox[List[RowBox[List["(", RowBox[List["1", "-", "y"]], ")"]], RowBox[List["PolyGamma", "[", RowBox[List["x", "+", FractionBox[RowBox[List["y", "+", "1"]], "2"]]], "]"]]]], "]"]]]], "/;", RowBox[List[RowBox[List["x", "\[Element]", "Reals"]], "\[And]", RowBox[List["x", ">", "0"]], "\[And]", RowBox[List["y", "\[Element]", "Reals"]], "\[And]", RowBox[List["0", "<", "y", "<", "1"]]]]]]]]








MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mi> x </mi> <mo> + </mo> <mfrac> <mi> y </mi> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> y </mi> </mrow> </msup> <mo> &lt; </mo> <mfrac> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> x </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> x </mi> <mo> + </mo> <mi> y </mi> </mrow> <mo> ) </mo> </mrow> </mfrac> <mo> &gt; </mo> <msup> <mi> &#8519; </mi> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> y </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mfrac> <mrow> <mi> y </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> <mo> + </mo> <mi> x </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </msup> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> x </mi> <mo> &#8712; </mo> <semantics> <mi> &#8477; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalR]&quot;, Function[Reals]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mi> x </mi> <mo> &gt; </mo> <mn> 0 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mi> y </mi> <mo> &#8712; </mo> <semantics> <mi> &#8477; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalR]&quot;, Function[Reals]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mn> 0 </mn> <mo> &lt; </mo> <mi> y </mi> <mo> &lt; </mo> <mn> 1 </mn> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <ci> Inequality </ci> <apply> <power /> <apply> <plus /> <ci> x </ci> <apply> <times /> <ci> y </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> y </ci> </apply> </apply> </apply> <lt /> <apply> <times /> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> x </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> x </ci> <ci> y </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <gt /> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> y </ci> </apply> </apply> <apply> <ci> PolyGamma </ci> <apply> <plus /> <apply> <times /> <apply> <plus /> <ci> y </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <ci> x </ci> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <ci> x </ci> <reals /> </apply> <apply> <gt /> <ci> x </ci> <cn type='integer'> 0 </cn> </apply> <apply> <in /> <ci> y </ci> <reals /> </apply> <apply> <lt /> <cn type='integer'> 0 </cn> <ci> y </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["x", "+", FractionBox["y", "2"]]], ")"]], RowBox[List["1", "-", "y"]]], "<", FractionBox[RowBox[List["Gamma", "[", RowBox[List["x", "+", "1"]], "]"]], RowBox[List["Gamma", "[", RowBox[List["x", "+", "y"]], "]"]]], ">", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List["1", "-", "y"]], ")"]], " ", RowBox[List["PolyGamma", "[", RowBox[List["x", "+", FractionBox[RowBox[List["y", "+", "1"]], "2"]]], "]"]]]]]]], "/;", RowBox[List[RowBox[List["x", "\[Element]", "Reals"]], "&&", RowBox[List["x", ">", "0"]], "&&", RowBox[List["y", "\[Element]", "Reals"]], "&&", RowBox[List["0", "<", "y", "<", "1"]]]]]]]]


Date Added to functions.wolfram.com (modification date)





2002-12-18