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Visualize q-PolyGamma
QPolyGamma is defined for <1 and contains a dense subset of essential singularities on the unit circle.
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With[{b = 0.97}, 

 Plot3D[Re[QPolyGamma[1, x + I y]], {x, -b, b}, {y, -b, b}, 

  MaxRecursion -> 1, RegionFunction -> (#1^2 + #2^2 < b^2 &), 

  Mesh -> None, 

  PlotStyle -> Directive[Yellow, Specularity[White, 30]], 

  PlotPoints -> 35, Exclusions -> {{y == 0, x <= 0}}, 

  ExclusionsStyle -> {None, Red}]]
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