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http://functions.wolfram.com/04.06.22.0006.01
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MellinTransform[Mod[m, t], t, z] == -((m^(z + 1) Zeta[z + 1])/(z + 1)) /;
-1 < Re[z] < 0
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["MellinTransform", "[", RowBox[List[RowBox[List["Mod", "[", RowBox[List["m", ",", "t"]], "]"]], ",", "t", ",", "z"]], "]"]], "\[Equal]", RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox["m", RowBox[List["z", "+", "1"]]], " ", RowBox[List["Zeta", "[", RowBox[List["z", "+", "1"]], "]"]]]], RowBox[List["z", "+", "1"]]]]]]], "/;", RowBox[List[RowBox[List["-", "1"]], "<", RowBox[List["Re", "[", "z", "]"]], "<", "0"]]]]]]
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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> <mrow> <mrow> <msub> <mi> ℳ </mi> <mi> t </mi> </msub> <mo> [ </mo> <semantics> <mrow> <mi> m </mi> <mo> ⁢ </mo> <mi> mod </mi> <mo> ⁢ </mo> <mi> t </mi> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <rem /> <ci> $CellContext`m </ci> <ci> $CellContext`t </ci> </apply> </annotation-xml> </semantics> <mo> ] </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mo> - </mo> <mfrac> <mrow> <msup> <mi> m </mi> <mrow> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msup> <mo> ⁢ </mo> <semantics> <mrow> <mi> ζ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Zeta]", "(", TagBox[RowBox[List["z", "+", "1"]], Rule[Editable, True]], ")"]], InterpretTemplate[Function[$CellContext`e, Zeta[$CellContext`e]]]] </annotation> </semantics> </mrow> <mrow> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mfrac> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> < </mo> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> < </mo> <mn> 0 </mn> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <times /> <apply> <apply> <ci> Subscript </ci> <ci> ℳ </ci> <ci> t </ci> </apply> <apply> <rem /> <ci> $CellContext`m </ci> <ci> $CellContext`t </ci> </apply> </apply> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <ci> m </ci> <apply> <plus /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <ci> Zeta </ci> <apply> <plus /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <lt /> <cn type='integer'> -1 </cn> <apply> <real /> <ci> z </ci> </apply> <cn type='integer'> 0 </cn> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["MellinTransform", "[", RowBox[List[RowBox[List["Mod", "[", RowBox[List["m_", ",", "t_"]], "]"]], ",", "t_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox["m", RowBox[List["z", "+", "1"]]], " ", RowBox[List["Zeta", "[", RowBox[List["z", "+", "1"]], "]"]]]], RowBox[List["z", "+", "1"]]]]], "/;", RowBox[List[RowBox[List["-", "1"]], "<", RowBox[List["Re", "[", "z", "]"]], "<", "0"]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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