2
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Here is the code:

pw[y_, x_] := -HurwitzZeta[1 - x, 1/2 + y] x
Plot3D[pw[b, p], {b, -3, 3}, {p, -3, 3}, Mesh -> {6, 5}, 
 ClippingStyle -> None, 
 PlotStyle -> Directive[Orange, Opacity[.8], Specularity[White, 20]], 
 PlotPoints -> 100, AxesLabel -> Automatic]

Here is what I get:

enter image description here

How can I plot this function at $b<-1$? Numerically it evaluates well.

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  • $\begingroup$ I get a nice plot with PlotPoints -> Automatic Imo it's not a good idea to use ` ClippingStyle -> None` here $\endgroup$ – eldo Nov 29 '15 at 22:47
  • $\begingroup$ @eldo I used the both of your pieces of advice, added PlotPoints -> Automatic and removed ClippingStyle -> None and still it cannot plot anything at b<-1 $\endgroup$ – Anixx Nov 29 '15 at 22:50
  • $\begingroup$ b must be >= 0. With negative b-values there are branch cuts and singularities (see doc). You can plot f.e Plot3D[pw[b, p], {b, 0, 3}, {p, -6, 6}] $\endgroup$ – eldo Nov 29 '15 at 23:04
  • $\begingroup$ @eldo numerically it evaluates well. $\endgroup$ – Anixx Nov 29 '15 at 23:34
  • $\begingroup$ No, it doesn't. Try Table[pw[b, p], {b, -3, 3, 1.}, {p, -3, 3, 1.}]. Error messages and indeterminate values! $\endgroup$ – eldo Nov 29 '15 at 23:44
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Clear@pw

pw[x_, y_] := -HurwitzZeta[1 - y, 1/2 + x] y

With negative x-values you get many complex numbers and indeterminate values. For example:

pw[-3., -2.5] // N

29.052 + 28.9903 I

pw[-3., 0.] // N

Indeterminate

Avoiding negative x-values gives the following plot

Plot3D[pw[x, y], {x, 0, 3}, {y, -3, 3}]

enter image description here

If you want to plot negative x-values you must skip the 0 by choosing a plot range of f.e. {-3.01, 3.01} and you can convert the complex numbers with Abs

Plot3D[Abs @ pw[b, p], {b, -3.01, 3.01}, {p, -3.01, 3.01}]

enter image description here

From the documentation:

Unlike Zeta, HurwitzZeta has singularities at a==-n for non-negative integers n ... HurwitzZeta has branch cut discontinuities in the complex a plane running from 0 to -Infinity.

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