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http://functions.wolfram.com/06.17.07.0002.01
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HarmonicNumber[z, r] == ((-1)^(r - 1)/(r - 1)!)
Integrate[((t^z - 1)/(t - 1)) Log[t]^(r - 1), {t, 0, 1}] /;
Re[z] > -1 && Element[r, Integers] && r > 0
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["HarmonicNumber", "[", RowBox[List["z", ",", "r"]], "]"]], "\[Equal]", RowBox[List[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["r", "-", "1"]]], RowBox[List[RowBox[List["(", RowBox[List["r", "-", "1"]], ")"]], "!"]]], RowBox[List[SubsuperscriptBox["\[Integral]", "0", "1"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox["t", "z"], "-", "1"]], RowBox[List["t", "-", "1"]]], SuperscriptBox[RowBox[List["Log", "[", "t", "]"]], RowBox[List["r", "-", "1"]]], RowBox[List["\[DifferentialD]", "t"]]]]]]]]]], "/;", " ", RowBox[List[RowBox[List[RowBox[List["Re", "[", "z", "]"]], ">", RowBox[List["-", "1"]]]], "\[And]", RowBox[List["r", "\[Element]", "Integers"]], "\[And]", RowBox[List["r", ">", "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> <msubsup> <semantics> <mi> H </mi> <annotation-xml encoding='MathML-Content'> <ci> HarmonicNumber </ci> </annotation-xml> </semantics> <mi> z </mi> <mrow> <mo> ( </mo> <mi> r </mi> <mo> ) </mo> </mrow> </msubsup> <mo> ⩵ </mo> <mrow> <mfrac> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> r </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> r </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <msubsup> <mo> ∫ </mo> <mn> 0 </mn> <mn> 1 </mn> </msubsup> <mrow> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <msup> <mi> t </mi> <mi> z </mi> </msup> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msup> <mi> log </mi> <mrow> <mi> r </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> t </mi> <mo> ) </mo> </mrow> </mrow> <mrow> <mi> t </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> t </mi> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> > </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ∧ </mo> <mrow> <mi> r </mi> <mo> ∈ </mo> <msup> <mi> ℕ </mi> <mo> + </mo> </msup> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> HarmonicNumber </ci> <ci> z </ci> <ci> r </ci> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <plus /> <ci> r </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <factorial /> <apply> <plus /> <ci> r </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <int /> <bvar> <ci> t </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <cn type='integer'> 1 </cn> </uplimit> <apply> <times /> <apply> <plus /> <apply> <power /> <ci> t </ci> <ci> z </ci> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <apply> <ln /> <ci> t </ci> </apply> <apply> <plus /> <ci> r </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> t </ci> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <gt /> <apply> <real /> <ci> z </ci> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <in /> <ci> r </ci> <apply> <ci> SuperPlus </ci> <ci> ℕ </ci> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["HarmonicNumber", "[", RowBox[List["z_", ",", "r_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["r", "-", "1"]]], " ", RowBox[List[SubsuperscriptBox["\[Integral]", "0", "1"], RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["t", "z"], "-", "1"]], ")"]], " ", SuperscriptBox[RowBox[List["Log", "[", "t", "]"]], RowBox[List["r", "-", "1"]]]]], RowBox[List["t", "-", "1"]]], RowBox[List["\[DifferentialD]", "t"]]]]]]]], RowBox[List[RowBox[List["(", RowBox[List["r", "-", "1"]], ")"]], "!"]]], "/;", RowBox[List[RowBox[List[RowBox[List["Re", "[", "z", "]"]], ">", RowBox[List["-", "1"]]]], "&&", RowBox[List["r", "\[Element]", "Integers"]], "&&", RowBox[List["r", ">", "0"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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