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variants of this functions
Gamma




Mathematica Notation

Traditional Notation









Gamma, Beta, Erf > Gamma[z] > Integral transforms > Inverse Mellin transforms





http://functions.wolfram.com/06.05.22.0004.01









  


  








Input Form





InverseMellinTransform[Gamma[s]/Gamma[a + s], s, t] == ((1 - t)^(a - 1) UnitStep[1 - t])/Gamma[a] /; Re[a] > 0 && Re[s] > 0








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["InverseMellinTransform", "[", RowBox[List[FractionBox[RowBox[List["Gamma", "[", "s", "]"]], RowBox[List["Gamma", "[", RowBox[List["a", "+", "s"]], "]"]]], ",", "s", ",", "t"]], "]"]], "\[Equal]", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "t"]], ")"]], RowBox[List["a", "-", "1"]]], RowBox[List["UnitStep", "[", RowBox[List["1", "-", "t"]], "]"]]]], RowBox[List["Gamma", "[", "a", "]"]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Re", "[", "a", "]"]], ">", FormBox["0", TraditionalForm]]], "\[And]", RowBox[List[RowBox[List["Re", "[", "s", "]"]], ">", FormBox["0", TraditionalForm]]]]]]]]]








MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <mrow> <msubsup> <mi> &#8499; </mi> <mi> s </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msubsup> <mo> [ </mo> <mfrac> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> s </mi> <mo> ) </mo> </mrow> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mi> s </mi> </mrow> <mo> ) </mo> </mrow> </mfrac> <mo> ] </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mi> t </mi> <mo> ) </mo> </mrow> </mrow> <mo> &#10869; </mo> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> t </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> a </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> &#8290; </mo> <mrow> <semantics> <mi> &#952; </mi> <annotation-xml encoding='MathML-Content'> <ci> UnitStep </ci> </annotation-xml> </semantics> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> t </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> </mfrac> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> <mo> &gt; </mo> <mn> 0 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> s </mi> <mo> ) </mo> </mrow> <mo> &gt; </mo> <mn> 0 </mn> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <times /> <apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> &#8499; </ci> <ci> s </ci> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <times /> <apply> <ci> Gamma </ci> <ci> s </ci> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> a </ci> <ci> s </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <ci> t </ci> </apply> <apply> <times /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> t </ci> </apply> </apply> <apply> <plus /> <ci> a </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> UnitStep </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> t </ci> </apply> </apply> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <ci> a </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <and /> <apply> <gt /> <apply> <real /> <ci> a </ci> </apply> <cn type='integer'> 0 </cn> </apply> <apply> <gt /> <apply> <real /> <ci> s </ci> </apply> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["InverseMellinTransform", "[", RowBox[List[FractionBox[RowBox[List["Gamma", "[", "s_", "]"]], RowBox[List["Gamma", "[", RowBox[List["a_", "+", "s_"]], "]"]]], ",", "s_", ",", "t_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "t"]], ")"]], RowBox[List["a", "-", "1"]]], " ", RowBox[List["UnitStep", "[", RowBox[List["1", "-", "t"]], "]"]]]], RowBox[List["Gamma", "[", "a", "]"]]], "/;", RowBox[List[RowBox[List[RowBox[List["Re", "[", "a", "]"]], ">", "0"]], "&&", RowBox[List[RowBox[List["Re", "[", "s", "]"]], ">", "0"]]]]]]]]]]


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





2001-10-29