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I would need an approximation in terms of elementary functions for digamma(x + i y), x>1/2, smoothly interpolating between small and large |y|

Probably digamma function could be done using this formula: https://en.wikipedia.org/wiki/Digamma_function#Series_formula and for negative real part additional: https://en.wikipedia.org/wiki/Digamma_function#Reflection_formula Although different methods should be tested as well, since i don't think series formula converges too fast, and most likely depends on if input is small/big enough

@ kmdrGroch : Many thanks for your reply. In the meantime I found that the recurrence formula (x -> x + 1) works astonishingly well for y's in the range of my interest, incomparably better than the generalized Puiseux series indicated in a previous posting.

Unfortunatelly we need to do it for everyone, not just you :/ it has to be general and works for all ranges with decent time and approximation. Recurrence formula will most likely fail with Re big enough (I dunno how big Re can get tho). If it's max 20/30 we could use recurrence formula to get to value 0-1 and use series formula for it for example

I have just noticed it works another way around. You want X as big as possible for most of these series...

I guess it could work then.

I believe it would roughly work like:

function digamma(z) {
  if(z.re < 0) return digamma(reflect(z));
  
  if(z.re < 30) return digamma(z + 1) - 1 / z;
  
  return log(z) - 1 / 2z - ....;
}

@ kmdrGroch : Many thanks for your reply. In the meantime I found that the recurrence formula (x -> x + 1) works astonishingly well for y's in the range of my interest, incomparably better than the generalized Puiseux series indicated in a previous posting.

@ kmdrGroch : Thanks again, it is exactly what I meant: shift x (x -> x + 1), then use the lowest terms (log(z) - 1/(2 z) in the asymptotic expansion are sufficient.

Do you know what relative error of this approximation is? Or I should check it?

Please see examples in attchm. Relative errors psi(z) = psi(x + i y), (Re psi_approx/Re psi_exact - 1)*100 and Im psi_approx/Im psi_exact - 1)*100 versus y = Im z at ixed x = Re z .

On Fri, Dec 17, 2021 at 7:40 AM Bartosz Leoniak @.***> wrote:

Do you know what relative error of this approximation is? Or I should check it?

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@ticucubot you refer to attachments, but that is not supported by mailing a reply in the github issue. Can you explain/attach directly in github please?

I have mostly implemented this feature. I only need testing it and checking if it's providing good enough approximation. I need to handle edge cases as well. If you know any, please let me know so I can test them as well :) I used method described in my previous comment