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variants of this functions
PolyGamma






Mathematica Notation

Traditional Notation









Gamma, Beta, Erf > PolyGamma[nu,z] > Differentiation > Fractional integro-differentiation > With respect to z





http://functions.wolfram.com/06.15.20.0007.01









  


  










Input Form





D[PolyGamma[\[Nu], z], {z, \[Alpha]}] == (-FDPowerConstant[z, -1, \[Alpha] + \[Nu]]) z^(-1 - \[Nu] - \[Alpha]) + z^(1 - \[Nu] - \[Alpha]) Sum[(1/k^2) Hypergeometric2F1Regularized[1, 2, 2 - \[Nu] - \[Alpha], -(z/k)], {k, 1, Infinity}] - (EulerGamma z^(-\[Nu] - \[Alpha]))/Gamma[1 - \[Nu] - \[Alpha]]










Standard Form





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MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mfrac> <mrow> <msup> <mo> &#8706; </mo> <mi> &#945; </mi> </msup> <mrow> <msup> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mi> &#957; </mi> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mrow> <mo> &#8706; </mo> <msup> <mi> z </mi> <mi> &#945; </mi> </msup> </mrow> </mfrac> <mo> &#10869; </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mrow> <msubsup> <mi> &#8497;&#119966; </mi> <mi> exp </mi> <mrow> <mo> ( </mo> <mrow> <mi> &#957; </mi> <mo> + </mo> <mi> &#945; </mi> </mrow> <mo> ) </mo> </mrow> </msubsup> <mo> ( </mo> <mrow> <mi> z </mi> <mo> , </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mrow> <mo> - </mo> <mi> &#945; </mi> </mrow> <mo> - </mo> <mi> &#957; </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> </mrow> <mo> - </mo> <mfrac> <mrow> <semantics> <mi> &#8509; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubledGamma]&quot;, Function[EulerGamma]] </annotation> </semantics> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mrow> <mo> - </mo> <mi> &#945; </mi> </mrow> <mo> - </mo> <mi> &#957; </mi> </mrow> </msup> </mrow> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> &#945; </mi> <mo> - </mo> <mi> &#957; </mi> </mrow> <mo> ) </mo> </mrow> </mfrac> <mo> + </mo> <mrow> <msup> <mi> z </mi> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> &#945; </mi> <mo> - </mo> <mi> &#957; </mi> </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> <mfrac> <mn> 1 </mn> <msup> <mi> k </mi> <mn> 2 </mn> </msup> </mfrac> <mo> &#8290; </mo> <semantics> <mrow> <mrow> <msub> <mo> &#8202; </mo> <mn> 2 </mn> </msub> <msub> <mover> <mi> F </mi> <mo> ~ </mo> </mover> <mn> 1 </mn> </msub> </mrow> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 1 </mn> <mo> , </mo> <mn> 2 </mn> </mrow> <mo> ; </mo> <mrow> <mn> 2 </mn> <mo> - </mo> <mi> &#945; </mi> <mo> - </mo> <mi> &#957; </mi> </mrow> <mo> ; </mo> <mrow> <mo> - </mo> <mfrac> <mi> z </mi> <mi> k </mi> </mfrac> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox[&quot;\[InvisiblePrefixScriptBase]&quot;, FormBox[&quot;2&quot;, TraditionalForm]], SubscriptBox[OverscriptBox[&quot;F&quot;, &quot;~&quot;], FormBox[&quot;1&quot;, TraditionalForm]]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[TagBox[RowBox[List[TagBox[&quot;1&quot;, Hypergeometric2F1Regularized, Rule[Editable, True]], &quot;,&quot;, TagBox[&quot;2&quot;, Hypergeometric2F1Regularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], &quot;;&quot;, TagBox[TagBox[TagBox[RowBox[List[&quot;2&quot;, &quot;-&quot;, &quot;\[Alpha]&quot;, &quot;-&quot;, &quot;\[Nu]&quot;]], Hypergeometric2F1Regularized, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], &quot;;&quot;, TagBox[RowBox[List[&quot;-&quot;, FractionBox[&quot;z&quot;, &quot;k&quot;]]], Hypergeometric2F1Regularized, Rule[Editable, True]]]], &quot;)&quot;]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1Regularized] </annotation> </semantics> </mrow> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <partialdiff /> <bvar> <ci> z </ci> <degree> <ci> &#945; </ci> </degree> </bvar> <apply> <ci> PolyGamma </ci> <ci> &#957; </ci> <ci> z </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> &#8497;&#119966; </ci> <ci> exp </ci> </apply> <apply> <plus /> <ci> &#957; </ci> <ci> &#945; </ci> </apply> </apply> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <ci> z </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#945; </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <eulergamma /> <apply> <power /> <ci> z </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#945; </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> </apply> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#945; </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <ci> z </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#945; </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <ci> k </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> Hypergeometric2F1Regularized </ci> <cn type='integer'> 1 </cn> <cn type='integer'> 2 </cn> <apply> <plus /> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#945; </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; 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Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["z_", ",", "\[Alpha]_"]], "}"]]]]], RowBox[List["PolyGamma", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["-", RowBox[List["FDPowerConstant", "[", RowBox[List["z", ",", RowBox[List["-", "1"]], ",", RowBox[List["\[Alpha]", "+", "\[Nu]"]]]], "]"]]]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "1"]], "-", "\[Nu]", "-", "\[Alpha]"]]]]], "+", RowBox[List[SuperscriptBox["z", RowBox[List["1", "-", "\[Nu]", "-", "\[Alpha]"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], FractionBox[RowBox[List["Hypergeometric2F1Regularized", "[", RowBox[List["1", ",", "2", ",", RowBox[List["2", "-", "\[Nu]", "-", "\[Alpha]"]], ",", RowBox[List["-", FractionBox["z", "k"]]]]], "]"]], SuperscriptBox["k", "2"]]]]]], "-", FractionBox[RowBox[List["EulerGamma", " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "\[Nu]"]], "-", "\[Alpha]"]]]]], RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Nu]", "-", "\[Alpha]"]], "]"]]]]]]]]]










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





2001-10-29