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http://functions.wolfram.com/06.16.07.0002.01
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HarmonicNumber[z] == Integrate[1/(E^t t) - (t + 1)^(-z - 1)/t,
{t, 0, Infinity}] + EulerGamma /; Re[z] > -1
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["HarmonicNumber", "[", "z", "]"]], "\[Equal]", " ", RowBox[List[RowBox[List[SubsuperscriptBox["\[Integral]", "0", "\[Infinity]"], RowBox[List[RowBox[List["(", RowBox[List[FractionBox[SuperscriptBox["\[ExponentialE]", RowBox[List["-", "t"]]], "t"], "-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["t", "+", "1"]], ")"]], RowBox[List[RowBox[List["-", "z"]], "-", "1"]]], "t"]]], ")"]], RowBox[List["\[DifferentialD]", "t"]]]]]], "+", "EulerGamma"]]]], "/;", " ", RowBox[List[RowBox[List["Re", "[", "z", "]"]], ">", RowBox[List["-", "1"]]]]]]]]
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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> <msub> <semantics> <mi> H </mi> <annotation-xml encoding='MathML-Content'> <ci> HarmonicNumber </ci> </annotation-xml> </semantics> <mi> z </mi> </msub> <mo> ⩵ </mo> <mrow> <mrow> <msubsup> <mo> ∫ </mo> <mn> 0 </mn> <mi> ∞ </mi> </msubsup> <mrow> <mrow> <mo> ( </mo> <mrow> <mfrac> <msup> <mi> ⅇ </mi> <mrow> <mo> - </mo> <mi> t </mi> </mrow> </msup> <mi> t </mi> </mfrac> <mo> - </mo> <mfrac> <msup> <mrow> <mo> ( </mo> <mrow> <mi> t </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mrow> <mo> - </mo> <mi> z </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mi> t </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> t </mi> </mrow> </mrow> </mrow> <mo> + </mo> <semantics> <mi> ℽ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubledGamma]", Function[EulerGamma]] </annotation> </semantics> </mrow> </mrow> <mo> /; </mo> <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> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> HarmonicNumber </ci> <ci> z </ci> </apply> <apply> <plus /> <apply> <int /> <bvar> <ci> t </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <plus /> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> t </ci> </apply> </apply> <apply> <power /> <ci> t </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <apply> <plus /> <ci> t </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <ci> t </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <eulergamma /> </apply> </apply> <apply> <gt /> <apply> <real /> <ci> z </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["HarmonicNumber", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[SubsuperscriptBox["\[Integral]", "0", "\[Infinity]"], RowBox[List[RowBox[List["(", RowBox[List[FractionBox[SuperscriptBox["\[ExponentialE]", RowBox[List["-", "t"]]], "t"], "-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["t", "+", "1"]], ")"]], RowBox[List[RowBox[List["-", "z"]], "-", "1"]]], "t"]]], ")"]], RowBox[List["\[DifferentialD]", "t"]]]]]], "+", "EulerGamma"]], "/;", RowBox[List[RowBox[List["Re", "[", "z", "]"]], ">", RowBox[List["-", "1"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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