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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/06.05.22.0003.01









  


  








Input Form





InverseMellinTransform[Gamma[s]/Gamma[a - s], s, t] == t^((1 - a)/2) BesselJ[a - 1, 2 Sqrt[t]] /; 0 < Re[s] < (2 Re[a] + 1)/4








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]", RowBox[List[SuperscriptBox["t", FractionBox[RowBox[List["1", "-", "a"]], "2"]], " ", RowBox[List["BesselJ", "[", RowBox[List[RowBox[List["a", "-", "1"]], ",", RowBox[List["2", " ", SqrtBox["t"]]]]], "]"]]]]]], "/;", RowBox[List["0", "<", RowBox[List["Re", "[", "s", "]"]], "<", FractionBox[RowBox[List[RowBox[List["2", RowBox[List["Re", "[", "a", "]"]]]], "+", "1"]], "4"]]]]]]]








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> <mrow> <msup> <mi> t </mi> <mfrac> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> a </mi> </mrow> <mn> 2 </mn> </mfrac> </msup> <mo> &#8290; </mo> <mrow> <msub> <mi> J </mi> <mrow> <mi> a </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msub> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msqrt> <mi> t </mi> </msqrt> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mn> 0 </mn> <mo> &lt; </mo> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> s </mi> <mo> ) </mo> </mrow> <mo> &lt; </mo> <mfrac> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mn> 4 </mn> </mfrac> </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> <apply> <times /> <cn type='integer'> -1 </cn> <ci> s </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <ci> t </ci> </apply> <apply> <times /> <apply> <power /> <ci> t </ci> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <ci> BesselJ </ci> <apply> <plus /> <ci> a </ci> <cn type='integer'> -1 </cn> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> t </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <lt /> <cn type='integer'> 0 </cn> <apply> <real /> <ci> s </ci> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <real /> <ci> a </ci> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </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[RowBox[List[SuperscriptBox["t", FractionBox[RowBox[List["1", "-", "a"]], "2"]], " ", RowBox[List["BesselJ", "[", RowBox[List[RowBox[List["a", "-", "1"]], ",", RowBox[List["2", " ", SqrtBox["t"]]]]], "]"]]]], "/;", RowBox[List["0", "<", RowBox[List["Re", "[", "s", "]"]], "<", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", RowBox[List["Re", "[", "a", "]"]]]], "+", "1"]], ")"]]]]]]]]]]]]


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





2001-10-29