LaplaceTransform[HarmonicNumber[t, r], t, z] == Zeta[r]/z - z^(r - 1) Sum[E^(k z) Gamma[1 - r, k z], {k, 1, Infinity}] /; Re[z] > 0 && Element[r, Integers] && r > 0

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["LaplaceTransform", "[", RowBox[List[RowBox[List["HarmonicNumber", "[", RowBox[List["t", ",", "r"]], "]"]], ",", "t", ",", "z"]], "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List["Zeta", "[", "r", "]"]], "z"], "-", RowBox[List[SuperscriptBox["z", RowBox[List["r", "-", "1"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["k", " ", "z"]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["1", "-", "r"]], ",", RowBox[List["k", " ", "z"]]]], "]"]]]]]]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Re", "[", "z", "]"]], ">", "0"]], "\[And]", RowBox[List["r", "\[Element]", "Integers"]], "\[And]", RowBox[List["r", ">", "0"]]]]]]]]

<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> &#8466; </mi> <mi> t </mi> </msub> <mo> [ </mo> <msubsup> <semantics> <mi> H </mi> <annotation-xml encoding='MathML-Content'> <ci> HarmonicNumber </ci> </annotation-xml> </semantics> <mi> t </mi> <mrow> <mo> ( </mo> <mi> r </mi> <mo> ) </mo> </mrow> </msubsup> <mo> ] </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <semantics> <mrow> <mi> &#950; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> r </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[Zeta]&quot;, &quot;(&quot;, TagBox[&quot;r&quot;, Rule[Editable, True]], &quot;)&quot;]], InterpretTemplate[Function[$CellContext`e, Zeta[$CellContext`e]]]] </annotation> </semantics> <mi> z </mi> </mfrac> <mo> - </mo> <mrow> <msup> <mi> z </mi> <mrow> <mi> r </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <msup> <mi> &#8519; </mi> <mrow> <mi> k </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> r </mi> </mrow> <mo> , </mo> <mrow> <mi> k </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &gt; </mo> <mn> 0 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mi> r </mi> <mo> &#8712; </mo> <msup> <mi> &#8469; </mi> <mo> + </mo> </msup> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> LaplaceTransform </ci> <apply> <ci> HarmonicNumber </ci> <ci> t </ci> <ci> r </ci> </apply> <ci> t </ci> <ci> z </ci> </apply> <apply> <plus /> <apply> <times /> <apply> <ci> Zeta </ci> <ci> r </ci> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <ci> z </ci> <apply> <plus /> <ci> r </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <ci> k </ci> <ci> z </ci> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> r </ci> </apply> </apply> <apply> <times /> <ci> k </ci> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <gt /> <apply> <real /> <ci> z </ci> </apply> <cn type='integer'> 0 </cn> </apply> <apply> <in /> <ci> r </ci> <apply> <ci> SuperPlus </ci> <ci> &#8469; </ci> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["LaplaceTransform", "[", RowBox[List[RowBox[List["HarmonicNumber", "[", RowBox[List["t_", ",", "r_"]], "]"]], ",", "t_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List["Zeta", "[", "r", "]"]], "z"], "-", RowBox[List[SuperscriptBox["z", RowBox[List["r", "-", "1"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["k", " ", "z"]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["1", "-", "r"]], ",", RowBox[List["k", " ", "z"]]]], "]"]]]]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Re", "[", "z", "]"]], ">", "0"]], "&&", RowBox[List["r", "\[Element]", "Integers"]], "&&", RowBox[List["r", ">", "0"]]]]]]]]]]

2001-10-29