Talk:Digamma function

Computing Digamma

It looks like link to Cephes is incorrect (I was not able to find digamma there). On the other hand GNU Scientific Library and Boost Math Library provide implementations of digamma for C and C++ respectively. Perhaps somebody with better knowledge could create section on numerical approximations of digamma (see also remark by linas)? Rzolau (talk) 23:51, 4 April 2009 (UTC)

Silly Question

Does anyone know why a function named after one Greek letter is usually denoted by a different greek letter? Why say $\digamma$ but write $\psi$? 130.94.162.64 08:43, 25 November 2005 (UTC)

Huh? Ohh. Seems that wiki-TeX is rendering \digamma as a captial F instead of \psi with a prime on it. Strange. I will ask at Wikipedia talk:WikiProject Mathematics. linas 16:22, 25 November 2005 (UTC)

Oh, right. Yes, I have seen this as an alternate notation for the digamma. Its rare, I think, but not so rare that TeX didn't decide to create a special symbol for it. As you know, in math, there just aren't enough symbols. linas 17:01, 25 November 2005 (UTC)

Note that the name of the archaic Greek letter digamma, (="double gamma"), that was the Greek version of "F", is due to the its shape $\digamma ,$ that is formed by two capital letters "Γ", one over the other. I would be very curious to know who introduced the name digamma for the logarithmic derivative of the Gamma function. Certainly he alluded to the shape of the fraction ${\frac {\Gamma '}{\Gamma }},$ and it is too bad that this philological joke has been a bit spoiled by the choice of the letter Ψ, that has nothing to do with $\digamma$, not to speak about the dumb neologism "polygamma". I think the TeX \digamma renders a true digamma, as it has to be, primarily because D.Knuth is a fine scholar. --pma (talk) 15:55, 6 December 2009 (UTC)

I don't remember seeing the Greek letter digamma used as a sign for the digamma function, so I put a tag. Being rather similar to 'F', it could have been a source of confusion, as a lot of mathematics is done in handwriting.

To explain the 'psi', perhaps one should think about the one vs. many: digamma is the first from the series of p-olygamma functions and if one does not want an other pi-function, psi is a second best choice.80.72.94.18 (talk) 17:43, 18 December 2017 (UTC)

Rapidly convergent series needed

I need a a series expression for the digamma that is rapidly convergent, so that it can be used for high-precision numeric calculations. This article lacks such a beast...

Maybe the Borwein Tchebeysheff-polynomial trick for the rapid calculation of the Riemann zeta can be extended to the Hurwitz zeta. Then the digamma can be obtained from there ... anyone have a reference for this? linas 20:12, 22 April 2006 (UTC)

Variables m, a_k and b_k in "Evaluation of sums of rational functions"

In this equation, for one,

$u_{n}={\frac {p(n)}{q(n)}}=\sum _{k=1}^{m}{\frac {a_{k}}{n+b_{k}}}.$

m, a_k b_k appear throughout the section. Is there some way to introduce them when ${\frac {p(n)}{q(n)}}$ first appears?

Is m the order of the rational function, which is assumed in proper form, and is it also assumed that a_k are the coefficients of p while the b_k are those of q, and that somehow it's guaranteed that expressions involving a_k are okay to treat as if a_k = 0 when k > the degree of p?

I mean that's just all guesses I made working backwards, and then looking in the Rational_function article. (No such unexplained variables show up there.)

Or is it just that, "Well, yeah, p, q, a, b, and m-- it goes without saying?"

SteveWitham (talk) 06:01, 22 March 2020 (UTC)

Generalization of sum formula (multiplication theorem)

A small generalization of the sum formula (or multiplication theorem) is:

$\sum _{p=0}^{q-1}\psi (a+p/q)=q\psi (qa)-q\ln(q)$

and may be seen to be consistent with the original (current) formula by setting:

$a=1/q$,

for q a natural number.

Hair Commodore 19:58, 28 October 2006 (UTC)

Web Calculators

Last night, Arthur Rubin removed a link to an web calculator providing useful free services to users of exponential integrals. Many wikipedia articles on topics that have calculational aspects provide links to web calculators. This was the only calculator link on the article, so the link to that useful service is now gone. Web calcualtors for such relative obscrure functions are rare. Please cite an official policy justiifcation, explain your actions in light of these points, or engage in a conversation as to why you believe this information to be inappropriate. Otherwise I plan to revert the change. Ichbin-dcw (talk) 19:48, 25 May 2010 (UTC)

Colours in graphs

The article includes two graphs of the digamma function, very attractively coloured throughout a rectangular region of the complex plane. However, there is absolutely no explanation of the colours used! What do they mean? yoyo (talk) 04:57, 27 June 2015 (UTC)

I did a search for images created by the uploarder (Image Search), and found a description of the coloring as domain coloring. Specifically, on a similarly colored plot of Klein's J-invariant function (image), there is the link Visualizing complex-valued functions with Matplotlib and Mayavi, showing how to do this in Python. Jimw338 (talk) 16:56, 7 January 2016 (UTC)

Notation

"The digamma function, often denoted also as ψ₀(x), ψ⁰(x) or $\digamma$ (after the shape of the archaic Greek letter Ϝ digamma)..."

Right, but then why do we use an unadorned psi without the zero? I can understand not wanting to use a real digamma, given that it is probably not the most common usage, but this seems a bit self-contradictory without accompanying explanation. ψ is after all the general symbol for all the polygamma functions: is it just that ψ⁰ is the most common and is hence understood? Double sharp (talk) 09:27, 24 September 2015 (UTC)

Application

It would be nice to know what the Digamma function is used for. — Preceding unsigned comment added by Chrisranderson (talk • contribs) 18:24, 12 July 2016 (UTC)

Please mention the poles of the digamma function

Hey. The page does not mention the poles of the digamma function at all (not that they exist, where they are, what their residues are etc.) It is, however, mentioned on the "Polygamma function"-article. Could we mention the poles, or at least have a reference to the Polygamma-article ? — Preceding unsigned comment added by Amusing numbers (talk • contribs) 18:25, 19 May 2022 (UTC)