PolyGamma[1, z] == 1/(2 z^2) + 1/z + (1/(2 z^2)) (1/(3 z + 3/(5 z + 18/(7 z + 60/(9 z + 150/(11 z + 315/(13 z + 588/(15 z + 1008/(17 z + \[Ellipsis]))))))))) /; Re[z] > 0

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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <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> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </mfrac> <mo> + </mo> <mfrac> <mn> 1 </mn> <mi> z </mi> </mfrac> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </mfrac> <mo> &#8290; </mo> <mfrac> <mn> 1 </mn> <mrow> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> + </mo> <mstyle scriptlevel='0'> <mfrac> <mn> 3 </mn> <mrow> <mrow> <mn> 5 </mn> <mo> &#8290; </mo> <mstyle scriptlevel='0'> <mi> z </mi> </mstyle> </mrow> <mstyle scriptlevel='0'> <mo> + </mo> </mstyle> <mstyle scriptlevel='0'> <mfrac> <mn> 18 </mn> <mrow> <mrow> <mn> 7 </mn> <mo> &#8290; </mo> <mstyle scriptlevel='0'> <mi> z </mi> </mstyle> </mrow> <mstyle scriptlevel='0'> <mo> + </mo> </mstyle> <mstyle scriptlevel='0'> <mfrac> <mn> 60 </mn> <mrow> <mrow> <mn> 9 </mn> <mo> &#8290; </mo> <mstyle scriptlevel='0'> <mi> z </mi> </mstyle> </mrow> <mstyle scriptlevel='0'> <mo> + </mo> </mstyle> <mstyle scriptlevel='0'> <mfrac> <mn> 150 </mn> <mrow> <mrow> <mn> 11 </mn> <mo> &#8290; </mo> <mstyle scriptlevel='0'> <mi> z </mi> </mstyle> </mrow> <mstyle scriptlevel='0'> <mo> + </mo> </mstyle> <mstyle scriptlevel='0'> <mfrac> <mn> 315 </mn> <mrow> <mrow> <mn> 13 </mn> <mo> &#8290; </mo> <mstyle scriptlevel='0'> <mi> z </mi> </mstyle> </mrow> <mstyle scriptlevel='0'> <mo> + </mo> </mstyle> <mstyle scriptlevel='0'> <mfrac> <mn> 588 </mn> <mrow> <mrow> <mn> 15 </mn> <mo> &#8290; </mo> <mstyle scriptlevel='0'> <mi> z </mi> </mstyle> </mrow> <mstyle scriptlevel='0'> <mo> + </mo> </mstyle> <mstyle scriptlevel='0'> <mfrac> <mn> 1008 </mn> <mrow> <mrow> <mn> 17 </mn> <mo> &#8290; </mo> <mstyle scriptlevel='0'> <mi> z </mi> </mstyle> </mrow> <mstyle scriptlevel='0'> <mo> + </mo> </mstyle> <mstyle scriptlevel='0'> <mo> &#8230; </mo> </mstyle> </mrow> </mfrac> </mstyle> </mrow> </mfrac> </mstyle> </mrow> </mfrac> </mstyle> </mrow> </mfrac> </mstyle> </mrow> </mfrac> </mstyle> </mrow> </mfrac> </mstyle> </mrow> </mfrac> </mstyle> </mrow> </mfrac> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &gt; </mo> <mn> 0 </mn> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> PolyGamma </ci> <cn type='integer'> 1 </cn> <ci> z </ci> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 3 </cn> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 5 </cn> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> 18 </cn> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 7 </cn> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> 60 </cn> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 9 </cn> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> 150 </cn> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 11 </cn> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> 315 </cn> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 13 </cn> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> 588 </cn> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 15 </cn> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> 1008 </cn> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 17 </cn> <ci> z </ci> </apply> <ci> &#8230; </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <gt /> <apply> <real /> <ci> z </ci> </apply> <cn type='integer'> 0 </cn> </apply> </apply> </annotation-xml> </semantics> </math>

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2001-10-29