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variants of this functions
HarmonicNumber






Mathematica Notation

Traditional Notation









Gamma, Beta, Erf > HarmonicNumber[z] > Complex characteristics > Real part





http://functions.wolfram.com/06.16.19.0001.01









  


  










Input Form





Re[HarmonicNumber[x + I y]] == RootSum[(1 + #1) ((1 + x)^2 + y^2 + 2 (1 + x) #1 + #1^2) & , -((PolyGamma[-#1] ((1 + x)^2 + y^2 + (1 + x) (-1 + #1) - #1))/ ((1 + x)^2 + y^2 + #1 (2 + 3 #1) + (1 + x) (2 + 4 #1))) & ]










Standard Form





Cell[BoxData[RowBox[List[RowBox[List["Re", "[", RowBox[List["HarmonicNumber", "[", RowBox[List["x", "+", RowBox[List["\[ImaginaryI]", " ", "y"]]]], "]"]], "]"]], "\[Equal]", RowBox[List["RootSum", "[", RowBox[List[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["1", "+", "#1"]], ")"]], " ", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "x"]], ")"]], "2"], "+", SuperscriptBox["y", "2"], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List["1", "+", "x"]], ")"]], " ", "#1"]], "+", SuperscriptBox["#1", "2"]]], ")"]]]], "&"]], ",", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["-", "#1"]], "]"]], " ", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "x"]], ")"]], "2"], "+", SuperscriptBox["y", "2"], "+", RowBox[List[RowBox[List["(", RowBox[List["1", "+", "x"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "#1"]], ")"]]]], "-", "#1"]], ")"]]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "x"]], ")"]], "2"], "+", SuperscriptBox["y", "2"], "+", RowBox[List["#1", " ", RowBox[List["(", RowBox[List["2", "+", RowBox[List["3", " ", "#1"]]]], ")"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["1", "+", "x"]], ")"]], " ", RowBox[List["(", RowBox[List["2", "+", RowBox[List["4", " ", "#1"]]]], ")"]]]]]]]]], "&"]]]], "]"]]]]]]










MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <msub> <semantics> <mi> H </mi> <annotation-xml encoding='MathML-Content'> <ci> HarmonicNumber </ci> </annotation-xml> </semantics> <mrow> <mi> x </mi> <mo> + </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> y </mi> </mrow> </mrow> </msub> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mi> RootSum </mi> <mo> [ </mo> <mrow> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> #1 </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mi> x </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> #1 </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> x </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <msup> <mi> y </mi> <mn> 2 </mn> </msup> <mo> + </mo> <msup> <mi> #1 </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> &amp; </mo> </mrow> <mo> , </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mrow> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mo> - </mo> <mi> #1 </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mi> x </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> #1 </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> x </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <msup> <mi> y </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mi> #1 </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mi> x </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> #1 </mi> </mrow> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> x </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <msup> <mi> y </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mrow> <mi> #1 </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mi> #1 </mi> </mrow> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mfrac> </mrow> <mo> &amp; </mo> </mrow> </mrow> <mo> ] </mo> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <real /> <apply> <ci> HarmonicNumber </ci> <apply> <plus /> <ci> x </ci> <apply> <times /> <imaginaryi /> <ci> y </ci> </apply> </apply> </apply> </apply> <apply> <ci> RootSum </ci> <apply> <ci> Function </ci> <apply> <times /> <apply> <plus /> <apply> <ci> Slot </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <apply> <power /> <apply> <plus /> <ci> x </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> Slot </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <ci> x </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <power /> <ci> y </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <ci> Slot </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <apply> <ci> Function </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <ci> PolyGamma </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Slot </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <power /> <apply> <plus /> <ci> x </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <apply> <plus /> <apply> <ci> Slot </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <plus /> <ci> x </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <power /> <ci> y </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Slot </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <power /> <apply> <plus /> <ci> x </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <ci> Slot </ci> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <ci> x </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <power /> <ci> y </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <apply> <ci> Slot </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <ci> Slot </ci> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Re", "[", RowBox[List["HarmonicNumber", "[", RowBox[List["x_", "+", RowBox[List["\[ImaginaryI]", " ", "y_"]]]], "]"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List["RootSum", "[", RowBox[List[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["1", "+", "#1"]], ")"]], " ", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "x"]], ")"]], "2"], "+", SuperscriptBox["y", "2"], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List["1", "+", "x"]], ")"]], " ", "#1"]], "+", SuperscriptBox["#1", "2"]]], ")"]]]], "&"]], ",", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["-", "#1"]], "]"]], " ", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "x"]], ")"]], "2"], "+", SuperscriptBox["y", "2"], "+", RowBox[List[RowBox[List["(", RowBox[List["1", "+", "x"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "#1"]], ")"]]]], "-", "#1"]], ")"]]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "x"]], ")"]], "2"], "+", SuperscriptBox["y", "2"], "+", RowBox[List["#1", " ", RowBox[List["(", RowBox[List["2", "+", RowBox[List["3", " ", "#1"]]]], ")"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["1", "+", "x"]], ")"]], " ", RowBox[List["(", RowBox[List["2", "+", RowBox[List["4", " ", "#1"]]]], ")"]]]]]]]]], "&"]]]], "]"]]]]]]










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





2001-10-29





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