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variants of this functions
PolyGamma






Mathematica Notation

Traditional Notation









Gamma, Beta, Erf > PolyGamma[z] > Integration > Indefinite integration > Involving one direct function and elementary functions > Involving power function





http://functions.wolfram.com/06.14.21.0002.01









  


  










Input Form





Integrate[z^(\[Alpha] - 1) PolyGamma[z], z] == z^(\[Alpha] - 1)/(1 - \[Alpha]) + (z^(\[Alpha] + 1)/(\[Alpha] + 1)) Sum[(1/(k + 1)^2) HypergeometricPFQ[{1, 2, 1 + \[Alpha]}, {2, 2 + \[Alpha]}, -(z/(k + 1))], {k, 0, Infinity}] - (EulerGamma z^\[Alpha])/\[Alpha]










Standard Form





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MathML Form







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</mi> </mfrac> <mo> + </mo> <mrow> <mfrac> <msup> <mi> z </mi> <mrow> <mi> &#945; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msup> <mrow> <mi> &#945; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <mfrac> <mn> 1 </mn> <msup> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mfrac> <mo> &#8290; </mo> <semantics> <mrow> <mrow> <msub> <mo> &#8202; </mo> <mn> 3 </mn> </msub> <msub> <mi> F </mi> <mn> 2 </mn> </msub> </mrow> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 1 </mn> <mo> , </mo> <mn> 2 </mn> <mo> , </mo> <mrow> <mi> &#945; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ; </mo> <mrow> <mn> 2 </mn> <mo> , </mo> <mrow> <mi> &#945; </mi> <mo> + </mo> <mn> 2 </mn> </mrow> </mrow> <mo> ; </mo> <mrow> <mo> - </mo> <mfrac> <mi> z </mi> <mrow> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mfrac> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox[&quot;\[InvisiblePrefixScriptBase]&quot;, FormBox[&quot;3&quot;, TraditionalForm]], SubscriptBox[&quot;F&quot;, FormBox[&quot;2&quot;, TraditionalForm]]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[TagBox[RowBox[List[TagBox[&quot;1&quot;, HypergeometricPFQ, Rule[Editable, True]], &quot;,&quot;, TagBox[&quot;2&quot;, HypergeometricPFQ, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[&quot;\[Alpha]&quot;, &quot;+&quot;, &quot;1&quot;]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], &quot;;&quot;, TagBox[TagBox[RowBox[List[TagBox[&quot;2&quot;, HypergeometricPFQ, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[&quot;\[Alpha]&quot;, &quot;+&quot;, &quot;2&quot;]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], &quot;;&quot;, TagBox[RowBox[List[&quot;-&quot;, FractionBox[&quot;z&quot;, RowBox[List[&quot;k&quot;, &quot;+&quot;, &quot;1&quot;]]]]], HypergeometricPFQ, Rule[Editable, True]]]], &quot;)&quot;]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] </annotation> </semantics> </mrow> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <ci> z </ci> </bvar> <apply> <times /> <apply> <power /> <ci> z </ci> <apply> <plus /> <ci> &#945; </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> PolyGamma </ci> <ci> z </ci> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> z </ci> <apply> <plus /> <ci> &#945; </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#945; </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <eulergamma /> <apply> <power /> <ci> z </ci> <ci> &#945; </ci> </apply> <apply> <power /> <ci> &#945; </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <ci> z </ci> <apply> <plus /> <ci> &#945; </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> &#945; </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <apply> <plus /> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> HypergeometricPFQ </ci> <list> <cn type='integer'> 1 </cn> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> &#945; </ci> <cn type='integer'> 1 </cn> </apply> </list> <list> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> &#945; </ci> <cn type='integer'> 2 </cn> </apply> </list> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> z </ci> <apply> <power /> <apply> <plus /> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox["z_", RowBox[List["\[Alpha]_", "-", "1"]]], " ", RowBox[List["PolyGamma", "[", "z_", "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[SuperscriptBox["z", RowBox[List["\[Alpha]", "-", "1"]]], RowBox[List["1", "-", "\[Alpha]"]]], "+", FractionBox[RowBox[List[SuperscriptBox["z", RowBox[List["\[Alpha]", "+", "1"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", "2", ",", RowBox[List["1", "+", "\[Alpha]"]]]], "}"]], ",", RowBox[List["{", RowBox[List["2", ",", RowBox[List["2", "+", "\[Alpha]"]]]], "}"]], ",", RowBox[List["-", FractionBox["z", RowBox[List["k", "+", "1"]]]]]]], "]"]], SuperscriptBox[RowBox[List["(", RowBox[List["k", "+", "1"]], ")"]], "2"]]]]]], RowBox[List["\[Alpha]", "+", "1"]]], "-", FractionBox[RowBox[List["EulerGamma", " ", SuperscriptBox["z", "\[Alpha]"]]], "\[Alpha]"]]]]]]]










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





2001-10-29





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