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My question is on the notation of the Digamma function.


The Factorial function $n!$ (which is met in secondary school), is conceptually seminal to the Digamma function. The Factorial function is defined as: $$0!=1,\qquad n!=\frac{(n+1)!}{n+1}$$

This concept is extended with Gauss's Pi function: $$\operatorname\Pi(z)=\frac{\operatorname\Pi(z+1)}{z+1}$$ and with a simple unit offset (see here), the more familiar Euler Gamma function: $$\operatorname\Gamma(z)=\frac{\operatorname\Gamma(z+1)}z$$

The Digamma function is a further extension: $$\psi(z)=\dfrac{\operatorname\Gamma'(z)}{\Gamma(z)}$$ and can itself be extended with the Polygamma function: $$\psi^m(z)=\frac{\mathrm d^m}{\mathrm dz^m}\psi(z) %=\frac{\mathrm d^{m+1}}{\mathrm d z^{m+1}}\ln \operatorname\Gamma(z) $$


All sources I have encountered that mention the Digamma (or Polygamma) denote it as above with the greek letter 'psi' $\psi$. This befuddles and confuses me. Archaic greek's uppercase form of digamma resembles the glyph of capital gamma $\Gamma$, but with and additional horizontal cross bar, making it appear as a 'double gamma' or a latin $\mathrm F$.

The uppercase gamma, and lowercase digamma and psi characters are available in LaTeX/MathJax, where $$\Gamma,\digamma,\psi$$ produces: $$\huge\Gamma,\digamma,\psi$$

It makes sense that 'Digamma' would be chosen as a symbol that is both visually and etymologically similar to the Gamma function, because these functions are closely related. It does not make any sense to chooses the symbol $\psi$. But what really gets me, is that the name of the function is a complete mismatch to the symbol of the function.


My question(s):

If we are going to call it a 'Digamma' function why don't we use a 'Digamma' symbol?
or conversely
If we are going to use a 'psi' symbol why don't we call it a 'psi' function?

Is there some justification for the establishment or continuation of this entrenched convention? Is this merely an historical artefact of idiotic humanity (a typographical error)? Or is there more to the story?


My best guess is that it many have something to do with typographical limitations, or overloading. But it still seems to ridiculously confusing notation to learn, (let alone to teach and propagate).

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  • $\begingroup$ I guess you know that there was a Greek letter “digamma”, fallen into desuetude by classical times. It stood for a $w$ sound, which had disappeared from the language. Since it looked like one gamma piled on top of another, it was called digamma. We would say that it looked like an $F$ with a curly bottom. $\endgroup$ – Lubin Nov 25 '17 at 2:00
  • $\begingroup$ Springer's "Encyclopedia of Mathematics"-wiki calls it "Psi-function": encyclopediaofmath.org/index.php/Psi-function All other sources I checked say "Digamma" but mostly use as symbol some variant of the letter psi. $\endgroup$ – Torsten Schoeneberg Nov 29 '17 at 3:49
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    $\begingroup$ If recalled correctly "October 13, 1729, Leonhard Euler, Letter to Goldbach" first used $\Gamma(x)$ and in Legendre's book "Traité des fonctions elliptiques et des intégrales eulériennes" both $\Gamma(x)$ and $\psi(x)$ are used. $\endgroup$ – Leucippus Nov 29 '17 at 17:07
  • $\begingroup$ Isn't it logical, if you have a polygamma, to have a trigamma and digamma? Historically the counting has gone the other way: 1,2, etc. 'Psi' has the same beginning as 'poly' and if somebody didn't want to have a pi-function, a name already taken perhaps, psi looks as a second best. $\endgroup$ – sand1 Dec 16 '17 at 21:47
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    $\begingroup$ The 'p' in 'psi' is pronounced in Greek and in most European languages. Also, lot of mathematical writing is done by hand, and the letter digamma is rather similar to 'F', which could be confusing. $\endgroup$ – sand1 Dec 17 '17 at 17:22
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I believe it’s because this function was used, and denoted “psi”, much before it got a name.

Indeed, it looks like $(\log\circ\,\Pi)'$ and $(\log\circ\,\Gamma)'$ first occur in Euler (1755, pp. 797-801; 1769, p. 17), resp. Legendre (1810, p. 502), with no special name or notation. Surveys like Brunel (1886, p. 58; 1899, p. 162) or Jensen (1916, p. 140) agree that they are denoted $\varPsi$ after Gauss (1813, p. 34), resp. $\psi$ after — unclear whom. (Clausen (1858, p. 169) or Bertrand (1870, p. 252) might be early examples.) In any case, they are still unnamed.

Instead, as one learns in Pearson (1922, p. viii), Davis (1933, p. 277; 1935, p. 9; 1935, p. 243), or Jordan (1939, p. 58), the name “digamma” and notation $\digamma$ for $\varPsi$ arose for the first time on p. 5 of

Pairman, Eleanor, Tables of the digamma and trigamma functions. (Tracts for computers, edited by Karl Pearson, Nr. 1.), Cambridge: University Press, 19 S (1919). ZBL47.0510.15:

pairman

In short, I think Pairman won her bet to name the yet-unnamed function — but failed to displace the existing “psi” notation. (The only adopters I could find are Jeffreys & Jeffreys (1946, §15.04).)


Note added: On the other hand, Legendre in his Exercices (1811, p. 19) famously divided elliptic integrals into 3 kinds to be called Nome, Epinome, Paranome: $$ Ϝ=\int\frac{d\varphi}{\Delta},\qquad \mathrm E = \int\Delta\,d\varphi,\qquad \Pi= \int\frac{d\varphi}{(1+n\sin{\!}^2\varphi)\Delta} $$ where $\Delta=\sqrt{1-c^2\sin{\!}^2\varphi}$. When these words didn’t catch on, Verhulst (1841, p. v) justified an attempt to rename them digamma, epsilon, kappa (notation $\mathrm{dig}$, $\mathrm{eps}$, $\mathrm{kap}$) with

1º The letter Ϝ, used by Legendre to denote the function $\int\frac{d\varphi}\Delta$, bears the Greek name δίγαμμα.

$$\Huge\style{font-family:Arial}{\text{Ϝ ϝ 𝟊 𝟋}}$$

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    $\begingroup$ A terrifically researched, and supremely documented answer. $\endgroup$ – Elements in Space Dec 30 '17 at 12:06
  • $\begingroup$ @ElementsinSpace Thanks. It would be interesting to answer the question left open at “unclear whom”. (I cannot make sense of what Brunel and Jensen say there.) $\endgroup$ – Francois Ziegler Dec 30 '17 at 22:12
  • $\begingroup$ What is the problem with those paragraphs in Brunel and Jensen? As far as I understand, they just collect, for reference, which symbols had been used for this function by various authors (including $\varphi$ by Legendre and lower case $\psi$ by Cauchy). $\endgroup$ – Torsten Schoeneberg Dec 31 '17 at 2:41
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    $\begingroup$ @TorstenSchoeneberg The problem is, I’m not really seeing all those $\psi$’s in “Cauchy, Gudermann and most authors since”. $\endgroup$ – Francois Ziegler Dec 31 '17 at 3:38
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According to the French wikipedia page Fonction digamma, it was James Stirling (1730) who first introduced and studied the digamma function, denoting it with the Greek letter digamma (upper case) $\digamma$. However, this claim is marked as requiring a citation. After noting its subsequent study by Legendre, Poisson and Gauss around 1810, the article then simply states "Elle est désormais le plus souvent notée par la lettre $\psi$ (psi minuscule)" - "It is now most often denoted by the letter $\psi$".

After the gamma and digamma functions were introduced they were generalised as the full sequence of polygamma functions. In the context of the polygamma functions it is obviously desirable to have a single functional denotation - i.e., $$\psi^{(n)}(z) := \frac{d^{m+1}}{dz^{(m+1)}} \text{ ln } \Gamma (z)$$

where $\psi^{(0)}$ denotes the digamma function, $\psi^{(1)}$ the trigamma function, etc... In other words, in the context of the sequence of polygamma functions, there is not reason for the digamma function to have a special designation.

If is not clear why psi was chosen, but it seems reasonable to assume that this is why the special $\digamma$ Digamma designation introduced by Stirling fell out of usage. It was simply a matter of notational elegance in the general case.

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    $\begingroup$ The Wikipedia page in question says: [réf. souhaitée]! The closest I can find to such a source is mention in Nielsen (1906, p. 17) that Stirling studied $\beta(x)=\frac12\left(\psi\left(\frac{x+1}2\right)-\psi\left(\frac{x}2\right)\right)$ in (1730). However, a glance there suggests that he had neither a name nor a symbol for that function. $\endgroup$ – Francois Ziegler Dec 30 '17 at 8:33
  • $\begingroup$ @FrancoisZiegler Thank-you for reminding my of my omission here. I have edited my answer to note the requirement of a citation. Possibly Nielsen is using the then standard $\psi$ denotation in place of what the article states is Stirling's use of $\digamma$. $\endgroup$ – Nick R Dec 30 '17 at 17:31

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