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variants of this functions
Log






Mathematica Notation

Traditional Notation









Elementary Functions > Log[z] > Differentiation > Fractional integro-differentiation





http://functions.wolfram.com/01.04.20.0024.01









  


  










Input Form





D[z^a Log[z]^3, {z, \[Alpha]}] == Piecewise[{{(-1)^(-a + \[Alpha]) Gamma[1 + a] Gamma[-a + \[Alpha]] (Pi^2 - 3 (Log[z] + PolyGamma[1 + a] - PolyGamma[-a + \[Alpha]])^2 - 3 PolyGamma[1, 1 + a] - 3 PolyGamma[1, -a + \[Alpha]]) z^(a - \[Alpha]), Element[\[Alpha] - a, Integers] && \[Alpha] - a > 0}, {(((-1)^(a - 1) z^(a - \[Alpha]))/(60 (-1 - a)! Gamma[1 + a - \[Alpha]])) (15 Log[z]^4 + 60 Log[z]^3 (PolyGamma[-a] - PolyGamma[1 + a - \[Alpha]]) + 30 Log[z]^2 (Pi^2 + 3 PolyGamma[-a]^2 - 6 PolyGamma[-a] PolyGamma[1 + a - \[Alpha]] + 3 PolyGamma[1 + a - \[Alpha]]^2 - 3 PolyGamma[1, -a] - 3 PolyGamma[1, 1 + a - \[Alpha]]) + 60 Log[z] (PolyGamma[-a]^3 - 3 PolyGamma[-a]^2 PolyGamma[1 + a - \[Alpha]] - PolyGamma[1 + a - \[Alpha]]^3 + PolyGamma[-a] (Pi^2 + 3 PolyGamma[1 + a - \[Alpha]]^2 - 3 PolyGamma[1, -a] - 3 PolyGamma[1, 1 + a - \[Alpha]]) + PolyGamma[1 + a - \[Alpha]] (-Pi^2 + 3 PolyGamma[1, -a] + 3 PolyGamma[1, 1 + a - \[Alpha]]) + PolyGamma[2, -a] - PolyGamma[2, 1 + a - \[Alpha]]) + 7 Pi^4 + 15 PolyGamma[-a]^4 - 60 PolyGamma[-a]^3 PolyGamma[1 + a - \[Alpha]] + 15 PolyGamma[1 + a - \[Alpha]]^4 - 30 Pi^2 PolyGamma[1, -a] + 45 PolyGamma[1, -a]^2 + 30 PolyGamma[1 + a - \[Alpha]]^2 (Pi^2 - 3 PolyGamma[1, -a] - 3 PolyGamma[1, 1 + a - \[Alpha]]) + 30 PolyGamma[-a]^2 (Pi^2 + 3 PolyGamma[1 + a - \[Alpha]]^2 - 3 PolyGamma[1, -a] - 3 PolyGamma[1, 1 + a - \[Alpha]]) - 30 Pi^2 PolyGamma[1, 1 + a - \[Alpha]] + 90 PolyGamma[1, -a] PolyGamma[1, 1 + a - \[Alpha]] + 45 PolyGamma[1, 1 + a - \[Alpha]]^2 - 60 PolyGamma[1 + a - \[Alpha]] (PolyGamma[2, -a] - PolyGamma[2, 1 + a - \[Alpha]]) - 60 PolyGamma[-a] (PolyGamma[1 + a - \[Alpha]]^3 + PolyGamma[1 + a - \[Alpha]] (Pi^2 - 3 PolyGamma[1, -a] - 3 PolyGamma[1, 1 + a - \[Alpha]]) - PolyGamma[2, -a] + PolyGamma[2, 1 + a - \[Alpha]]) - 15 PolyGamma[3, -a] - 15 PolyGamma[3, 1 + a - \[Alpha]]), Element[-a, Integers] && -a > 0}}, (Gamma[1 + a]/Gamma[1 + a - \[Alpha]]) (Log[z]^3 + 3 Log[z]^2 (PolyGamma[1 + a] - PolyGamma[1 + a - \[Alpha]]) + 3 Log[z] (PolyGamma[1 + a]^2 - 2 PolyGamma[1 + a] PolyGamma[1 + a - \[Alpha]] + PolyGamma[1 + a - \[Alpha]]^2 + PolyGamma[1, 1 + a] - PolyGamma[1, 1 + a - \[Alpha]]) + PolyGamma[1 + a]^3 - 3 PolyGamma[1 + a]^2 PolyGamma[1 + a - \[Alpha]] - PolyGamma[1 + a - \[Alpha]]^3 - 3 PolyGamma[1 + a - \[Alpha]] (PolyGamma[1, 1 + a] - PolyGamma[1, 1 + a - \[Alpha]]) + 3 PolyGamma[1 + a] (PolyGamma[1 + a - \[Alpha]]^2 + PolyGamma[1, 1 + a] - PolyGamma[1, 1 + a - \[Alpha]]) + PolyGamma[2, 1 + a] - PolyGamma[2, 1 + a - \[Alpha]]) z^(a - \[Alpha])]










Standard Form





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MathML Form







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</mo> <mrow> <msup> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mn> 1 </mn> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mrow> <mi> &#945; </mi> <mo> - </mo> <mi> a </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mi> a </mi> <mo> - </mo> <mi> &#945; </mi> </mrow> </msup> </mrow> </mtd> <mtd> <mrow> <mrow> <mi> &#945; </mi> <mo> - </mo> <mi> a </mi> </mrow> <mo> &#8712; </mo> <msup> <mi> &#8469; </mi> <mo> + </mo> </msup> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> a </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 15 </mn> <mo> &#8290; </mo> <mrow> <msup> <mi> log </mi> <mn> 4 </mn> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 60 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mo> - </mo> <mi> a </mi> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mi> &#945; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msup> <mi> log </mi> <mn> 3 </mn> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 30 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <msup> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mo> - </mo> <mi> a </mi> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 6 </mn> <mo> &#8290; </mo> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mi> &#945; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mo> - </mo> <mi> a </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <msup> <mi> &#960; </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <msup> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mi> &#945; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mrow> <msup> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mn> 1 </mn> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mrow> <mo> - </mo> <mi> a </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mrow> <msup> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mn> 1 </mn> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mi> &#945; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msup> <mi> log </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 60 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mo> - </mo> <mi> a </mi> </mrow> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> <mo> - </mo> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mi> &#945; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mo> - </mo> <mi> a </mi> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <msup> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mi> &#945; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <msup> <mi> &#960; </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mrow> <msup> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mn> 1 </mn> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mrow> <mo> - </mo> <mi> a </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mrow> <msup> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mn> 1 </mn> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mi> &#945; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mo> - </mo> <mi> a </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <msup> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mi> &#945; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> <mo> + </mo> <mrow> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mi> &#945; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mrow> <msup> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mn> 1 </mn> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mrow> <mo> - </mo> <mi> a </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <msup> <mi> &#960; </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mrow> <msup> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mn> 1 </mn> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mi> &#945; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <msup> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mn> 2 </mn> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mrow> <mo> - </mo> <mi> a </mi> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <msup> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mn> 2 </mn> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mi> &#945; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 7 </mn> <mo> &#8290; </mo> <msup> <mi> &#960; </mi> <mn> 4 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 15 </mn> <mo> &#8290; </mo> <msup> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mo> - </mo> <mi> a </mi> </mrow> <mo> ) </mo> </mrow> <mn> 4 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 15 </mn> <mo> &#8290; </mo> <msup> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mi> &#945; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 4 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 45 </mn> <mo> &#8290; </mo> <msup> <mrow> <msup> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mn> 1 </mn> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mrow> <mo> - </mo> <mi> a </mi> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 45 </mn> <mo> &#8290; </mo> <msup> <mrow> <msup> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mn> 1 </mn> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mi> &#945; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 60 </mn> <mo> &#8290; </mo> <msup> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mo> - </mo> <mi> a </mi> </mrow> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> <mo> &#8290; </mo> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mi> &#945; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 30 </mn> <mo> &#8290; </mo> <msup> <mi> &#960; </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mrow> <msup> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mn> 1 </mn> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mrow> <mo> - </mo> <mi> a </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 30 </mn> <mo> &#8290; </mo> <msup> <mi> &#960; </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mrow> <msup> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mn> 1 </mn> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mi> &#945; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 90 </mn> <mo> &#8290; </mo> <mrow> <msup> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mn> 1 </mn> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mrow> <mo> - </mo> <mi> a </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msup> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mn> 1 </mn> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mi> &#945; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 30 </mn> <mo> &#8290; </mo> <msup> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mi> &#945; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 3 </mn> </mrow> <mo> &#8290; </mo> <mrow> <msup> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mn> 1 </mn> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mrow> <mo> - </mo> <mi> a </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <msup> <mi> &#960; </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mrow> <msup> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mn> 1 </mn> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mi> &#945; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 30 </mn> <mo> &#8290; </mo> <msup> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mo> - </mo> <mi> a </mi> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <msup> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mi> &#945; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <msup> <mi> &#960; </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mrow> <msup> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mn> 1 </mn> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mrow> <mo> - </mo> <mi> a </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mrow> <msup> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mn> 1 </mn> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mi> &#945; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 60 </mn> <mo> &#8290; </mo> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mi> &#945; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mn> 2 </mn> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mrow> <mo> - </mo> <mi> a </mi> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <msup> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mn> 2 </mn> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mi> &#945; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 60 </mn> <mo> &#8290; </mo> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mo> - </mo> <mi> a </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mi> &#945; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 3 </mn> </mrow> <mo> &#8290; </mo> <mrow> <msup> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mn> 1 </mn> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mrow> <mo> - </mo> <mi> a </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <msup> <mi> &#960; </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mrow> <msup> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mn> 1 </mn> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mi> &#945; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mi> &#945; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <msup> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mn> 2 </mn> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mrow> <mo> - </mo> <mi> a </mi> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <msup> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mn> 2 </mn> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mi> &#945; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 15 </mn> <mo> &#8290; </mo> <mrow> <msup> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mn> 3 </mn> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mrow> <mo> - </mo> <mi> a </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 15 </mn> <mo> &#8290; </mo> <mrow> <msup> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mn> 3 </mn> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mi> &#945; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mn> 60 </mn> <mo> &#8290; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> a </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mi> &#945; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mi> a </mi> <mo> - </mo> <mi> &#945; </mi> </mrow> </msup> </mrow> </mtd> <mtd> <mrow> <mrow> <mo> - </mo> <mi> a </mi> </mrow> <mo> &#8712; </mo> <msup> <mi> &#8469; </mi> <mo> + </mo> </msup> </mrow> </mtd> </mtr> <mtr> <mtd> <mfrac> <mrow> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mi> log </mi> <mn> 3 </mn> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mi> &#945; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msup> <mi> log </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mi> &#945; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <msup> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mi> &#945; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> + </mo> <mrow> <msup> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mn> 1 </mn> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <msup> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mn> 1 </mn> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mi> &#945; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <msup> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> <mo> - </mo> <msup> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mi> &#945; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> <mo> - </mo> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <msup> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mi> &#945; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mi> &#945; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mn> 1 </mn> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <msup> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mn> 1 </mn> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mi> &#945; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mi> &#945; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> + </mo> <mrow> <msup> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mn> 1 </mn> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <msup> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mn> 1 </mn> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mi> &#945; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <msup> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mn> 2 </mn> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <msup> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mn> 2 </mn> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mi> &#945; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mi> a </mi> <mo> - </mo> <mi> &#945; </mi> </mrow> </msup> </mrow> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mi> &#945; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mfrac> </mtd> <mtd> <semantics> <mi> True </mi> <annotation encoding='Mathematica'> TagBox[&quot;True&quot;, &quot;PiecewiseDefault&quot;, Rule[AutoDelete, False], Rule[DeletionWarning, True]] </annotation> </semantics> </mtd> </mtr> </mtable> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <partialdiff /> <bvar> <ci> z </ci> <degree> <ci> &#945; 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Date Added to functions.wolfram.com (modification date)





2007-05-02





© 1998-2014 Wolfram Research, Inc.