1. Do you think the letter $\wp$ has a name? It may depend on community - the language, region, speciality, etc, so if you don't mind, please be specific about yours. (Mainly I'd like to know the English names, if any, but other information is welcome.) If yes, when and how did you come to know it? When, how, and how often do you mention it? (See below.)
  2. What's the origin of the letter?

In computing, various names, many of which are bad, have been given to $\wp$. See my answer.

Background: (Sorry for being a bit chatty.)

Originally I raised a related question at Wikipedia. The user Momotaro answered that in math community it's called "Weierstrass-p". Momotaro also gave a nice reference to the book The Brauer-Hasse-Noether Theorem in Historical Perspective by Peter Roquette. The author's claim supports Momotaro. (The episode in the book about the use of $\wp$ by Hasse and Emmy Noether is very interesting - history amuses - but it's off topic. Read the above link to Wikipedia. :)

However I'm not completely sure yet, because the occasions on which the letter's name becomes a topic must be quite limited. For example perhaps in the classroom a professor draws $\wp$, and students giggle by witnessing such a weird symbol and mastery of handwriting it; then the professor solemnly announces "this letter is called Weierstrass-p", like that? And "Weierstrass-p" is never an alias of the p-function?

After reading Momotaro's comment, I think I've read somewhere that the letter was invented by Weierstrass himself, but my memory about it is quite vague. Does anyone know something about it? Is it a mere folklore, or any reference?

I don't think mathoverflow is a place for votes, but if it were, I'd like one: "Have you ever heard of the name of the letter $\wp$?

Slightly off-topic, about the p-function's name in Japanese; In Japanese, the names of the Latin alphabets are mostly of English origin, エー, ビー, シー... (eh, bee, cee, etc.) But $\wp$-function is called ペー (peh), indicating its German origin. See e.g. 岩波 数学公式 III, p34, footnote 2 I don't know the name of the letter in Japanese. (In fact, most non-English European languages read "p" as "peh"...)

EDIT Typography in some early literature (off-topic, but interesting):

  • In an era where all mathematics is written in English, it is easy to forget that the use of German script letters for mathematical notation was once common throughout the German mathematical literature. – Michael Renardy Aug 7 '17 at 18:01
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    I never heard the letter called "the Weierstrass p"; I only ever heard the function called "the Weierstrass p-function." I've heard the letter called "curly p." – Gerry Myerson Aug 8 '17 at 2:59
  • @GerryMyerson: It's a very good point now, but there is the subtlety that 19th century people would overwhelmingly call $\wp(u)$, not $\wp$, the function. With rare exceptions, they were queasy about $\wp$ alone being anything more than a letter. So a name for $\wp$ (which I'm not saying that had, but maybe Pe) would initially be a letter name, not a function name. – Francois Ziegler Aug 8 '17 at 5:36
up vote 26 down vote accepted

Apparently first introduced by Weierstrass in Winter 1862/63 lectures published by H. A. Schwarz (1881, 1885, 1892, 1893), §9:

Mit der Sigma-Function $\mathfrak Su$ ist die Pe-Function $\wp u=\wp(u\mid\omega,\omega')=\wp(u;g_2,g_3)$ durch die Gleichung $$ \wp u=-\frac{d^2}{du^2}\log\mathfrak S u=\frac{(\mathfrak S'u)^2-\mathfrak S u\mathfrak S''u}{\mathfrak S^2u} $$ verbunden. (...)

The letter and a reference to Schwarz's notes also appear on the first page of Weierstrass's paper Zur Theorie der elliptischen Functionen (1882). Attribution in e.g. (Schwarz student) H. Hancock's Lectures on the Theory of Elliptic Functions (1910), p. 309:

(...) the function which we thus have was called by Weierstrass the Pe-function and denoted by $$ \wp(u)\qquad\text{or more simply}\qquad\wp u $$

or R. Godement's Analysis I (2004), p. 181:

(...) the famous function $$ \wp(u)=1/u^2+\sum_{\omega\ne0}\left[1/(u-\omega)^2-1/\omega^2\right] $$ of Weierstrass (it already appeared in Eisenstein), with a $p$ which smacks of the gothic, of the italic and of the cursive, chosen by the inventor65 and retained by posterity. (...)


65 His biography in the DSB tells us that in the course of his fourteen years of high-school teaching he had to teach mathematics, physics, German, botany, geography, history, gymnastics “and even calligraphy”.

Note added: While I don’t know of a handwritten specimen by Weierstrass himself (asked about in comments by @NateEldredge and @ManfredWeis), there are a few in lecture notes of S. Pincherle who had studied with Weierstrass in Berlin: (1899-1900, Chap. XXII).

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    $\mathfrak S_\Lambda(z),z \in \mathbb{C}$ is his sigma function, the Weierstrass product associated to a lattice $\Lambda$ – reuns Aug 7 '17 at 6:13
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    It might be interesting to ask what was the sequence that lead him to (regularization terms of) the Weierstrass product, if it was the points of a lattice or if he was interested in something else. – reuns Aug 7 '17 at 6:18
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    Let me add that in German "pe" is "how to pronounce 'P'", i.e. the equivalent of the English ['pi] (in phonetic transcription). Therefore, at the time of Weierstrass, $\wp$ was nothing more than a nicely calligraphied hand-written letter "P", something like $\mathcal P$ or $\mathscr P$. It was only later that it became a special symbol by itself. – Alex M. Aug 7 '17 at 9:34
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    Did Weierstrass write this $\wp$ significantly differently from how he wrote other instances of the letter $p$? – Nate Eldredge Aug 7 '17 at 14:32

The Weierstrass p has a strong similarity to p in the Sütterlin alphabet, which had been developed in Prussia and, as Weierstrass worked in Berlin (the capital of Prussia), it may well be that that is the origin of the letter.

Books were printed in Fraktur, where the p looks quite normal, i.e., quite different from a handwritten Sütterlin p which could explain, why it hasn't been replaced in the publication of Amandus Schwarz.

But I am not a historian, so this answer is a bit speculative, albeit reasonable.

  • Though similarity is a subjective matter, I'm afraid the $\wp$ letter doesn't look to me like Sütterlin. However I found an example of Sütterlin-like lowercase p (but published in Paris in French!) Search for "Sütterlin" in my edited question. – teika kazura Aug 15 '17 at 4:25
  • @teikakazura No, not Sütterlin but Kurrent (I saw it in my father's handwriting when a child). – Duchamp Gérard H. E. Aug 17 '17 at 3:15
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    @DuchampGérardH.E. in Kurrent writing of the capital letter P starts from upper left and there is no inflection point before entering the loop; the capital letter K however starts from the right and has an inflection point before the loop. This isn't meant to rule out the explanation that the origin of $\wp$ is Kurrent - it only is meant to indicate that questions still remain. What is desparately missing is a handwritten text from Weierstrass himself. – Manfred Weis Aug 17 '17 at 6:20
  • If one looks at a Kurrent alphabet from 1865, one sees, that the small letter $p$ is written with two loops, the top one first and then the bottom one. It might be that Weierstrass switched that order (maybe because that allowed faster writing). – Manfred Weis Aug 17 '17 at 6:30
  • Or someone not familiar with Kurrent mixed up the order of the loops, thus 'inventing' the $\wp$ – Manfred Weis Aug 17 '17 at 6:38

The Sütterlin alphabeth was introduced by the graphic designer Sütterlin in the 1920s and was the handwriting in German schools 1935-1941. It was a slightly simplified version of Kurrent. Before that time the usual handwriting style in Germany (since the 1500s) in german language was Kurrent (also known as Kurrentschrift, Alte Deutsche Schrift ("old German script") and German cursive) which was used also in Switzerland till 1900s and in Austria till 1941 (and still taught in schools after 1945; I learned it besides the Ausgangsschrift in elementary school 1955-1959).

The letter $\wp$ is, originally (in my eyes, and as I learned in lecture classes), just a capital P in Kurrent, written in a slightly elaborate way. I would say that the Weierstrass-P symbol $\wp$ was introduced by Donald Knuth as a special math TeX-symbol, :) . I would have used $\mathfrak P$ before learning about $\wp$ in the question.

See here for six letters by Gauss written in Kurrent.

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    Hmm, the Weierstrass -P looks notably different from the Kurrent P in most parts of the letter. Also the references by Francois Ziegler show the use of that very P long before TeX. – Dirk Aug 15 '17 at 7:12
  • @Dirk No, I think it looks like Kurrent (I saw it in my father's handwriting when a child). – Duchamp Gérard H. E. Aug 17 '17 at 3:13

Eugenio Calabi once said in a complex analysis course that it was just Weierstrass's personal way to write lower-case p. He did not say where he learned that.

  • You mean you were there and heard it? – teika kazura Aug 15 '17 at 4:25

My own answer, about the name of the letter $\wp$. Mostly in computing, with little math. (Lengthy)

Summary

In computing, the letter $\wp$ has been plagued with a plethora of inappropriate names. I guess "Weierstrass p" is one of such names that some standard created, and has been copied without much scrutiny. I suspect that ISO-8879 in 1986 was the root of all evil.

Names and references that survive in latest standards are:

  • \wp: TeX
  • script capital p (wrong) and weierstrass elliptic function (alias): Unicode
  • ℘: HTML (and XML?)

Unicode

Unicode is not directly related to my question, but it is important behind the scenes, so a good point to start.

Basics

In Unicode the letter $\wp$ is given the codepoint U+2118 in the block "letterlike symbols", named "script capital p". But in fact it's lowercase. There's also an official alias "weierstrass elliptic function".

Unicode Technical Note #27 regrets this misnomer:

Should have been called calligraphic small p or Weierstrass elliptic function symbol, which is what it is used for. It is not a capital "P" at all. A formal alias correcting this to WEIERSTRASS ELLIPTIC FUNCTION has been defined.

The first version of Technical Note #27 in 2006 was a bit more emphatic

Should have been called calligraphic small p or perhaps even Weierstrass elliptic function symbol, which is what it is used for. It's not a capital "P" at all.

(Boldification is done by me.)

Unicode 1.0 (1991)

Unicode 1.0 (1991) defined the letter, and the name was "script p". It was also called "weierstrass elliptic function". (See Footnote)

Unicode 2 (1996) screwed it up

They gave wrong names. As seen later, this change matters for us. From the file Nameslist:

2118  SCRIPT CAPITAL P
      = per
      = power set
      = Weierstrass elliptic function

(All of them were legitimate names. See also the file index.)
Alas, power set; $\wp$ is lowercase. (But what's "per"? According to Wiktionary's entry "per" the preposition "per" used to be written sometimes with a script letter "p". An example can be found in 1911 Encyclopædia Britannica. See Wikisource or archive.org)

Teminology note: The file "NamesList" maps from the codepoint (letter itself) to their names, the main one and aliases, if any. "Index" is the inverse mapping.

(Paranoiac detail: In Unicode 2.1, they changed its writing direction from "Other neutral" to "Left-to-right" See here. For writing directionos see this.

Unicode 3.0 (1999)

In 3.0, two wrong names "per" and "power set" can't be seen in Nameslist, but they still exist in Index. Wow. They still remained in 3.2 Index

But not sure if this removal was officially announced

Unicode never changes the main names once they are given, so "script capital p" will stay for ever.

Unicode 4.0 (2003)

In Unicode 4.0, it seems the index file was abolished. So it's finally settled to be "script capital p" and "weierstrass elliptic function".

Unicode 6.1 (2012)

Finally in Unicode 6.1, the meaning of aliases were clarified. "weierstrass elliptic function" of U+2118 was defined to be "correction".

Knuth's book (1986)

Computer Modern Typeface (google books link) by Don Knuth was published in 1986.

In p 26, there's a line there are special symbols like Weierstrass's ‘p’ (℘). However, in the index in p 580, there's only the entry "Weierstrass, Karl Theodor Wilhelm", pointing to pp 26, 233 and 235, and the entry "Weierstrass'p" is not there. So I guess Knuth did not mean that "Weierstrass'p" was the name of the symbol.

Old ISO, MathML, XML... true culprit?

Section summary: "Weierstrass p" was introduced sometime, but not sure exactly when. It has becoming obsolete, but not completely yet.

In HTML, you can write $\wp$ by escaping as ℘. Dunno other markup languages.

In MathML 1.01 specification (1998), $\wp$ is given the name "weierp", of which description is "Weierstrass p" and the alias "wp". See here. The letter derived from "ISOamso", but I can't be sure if the name came from ISOamso, too. The status of the latest MathML, 3.0, will be stated at the end of this section.

ISOamso ("Added Math Symbols: Ordinary", has nothing to do with AMS :) is a part of SGML = ISO 8879:1986 in 1986, according to this. When the first MathML was under construction, ISO 10646 (roughly speaking Unicode itself, but its counterpart in ISO) were apparently not there. See this. They had to gather letters from various standards.

HTML is worse: I couldn't find any entity specification about $\wp$ in HTML 1 - 3. (In very rough translation, entity = letter. See this intro) In HTML4 (1999), it is defined as

<!ENTITY weierp CDATA "&#8472;" -- script capital P = power set = Weierstrass p, U+2118 ISOamso -->

Because HTML4 has been used so long (it's still in use), this wrong sepcification has spreaded widely. See below.

In HTML5 (2014), the word "entity" was abolished. "Character entity references" was change to "Character references", and the term "Named entities" was replaced with "Named character references". See this. "Weierp" seems to have lost peculiarity up to HTML4; it's simply U+2118, and that's all. See this (Work on HTML5 started back in 2004, but it took long time until final publication.)

There's also "XML entity definitions". Although XML is rather old, the official specification of entities seems to have had been lacking for long. In its latest version (2nd ed) in 2014, $\wp$ is nothing but "weierp". See chap 2. ISOamso is now one of legacy entity sets. In its first version in 2010 however, it's classified as ISOamso

Anyway in ISOamso pages (ver 2 and ver 1) its description is "/wp - Weierstrass p", with the alias "wp". It seems the slash in "/wp" corresponds to LaTex backslash. (Cf. /hbar, /ell, /Re etc)

Unfortunately in MathML 3.0, (the latest, released in 2010) it still depends on XML's "legacy" entity sets, and it refers to ISOamso. "Weierstrass'p" is not yet dead.

"Weierstrass-p" spreaded

The word "Weierstrass p" appears in e.g. Encyclopedia Machintosh (1990), MacUser magazine (1992)

I googled for "weierstrass's p" - "weierstrass's p function" limiting to pre-1986 instances. There had not been any to mean the symbol $\wp$. The result of dropping "'s" is similar. Date limitation in Google search is not so reliable thus it does not prove anything.

Wrong HTML4's specification is now popular. Microsoft's document HTML Character sets reads "Office 2003", although its exact date and scope are not clear. It says $\wp$ is: "script capital P, =power set, =Weierstrass p, U2118 ISOamso". XML in a Nutshell (2002) and Beginning HTML, XHTML, CSS, and JavaScript (2011) say Weierp is "Script capital P, power set, Weierstrass p"

Acknowledgments

The books "Encyclopedia Machintosh", "MacUser magazine", "Computer Modern Typeface" and "XML in a Nutshell" were pointed out in Wikipedia Users Momotaro and The Man in Question. I thank them.

Footnote

In Unicode 1.0.0 (1991), it was NOT "script capital p", the current name. In Unicode Name Index (pdf), names "script p" and "weierstrass elliptic function" are found. See pdf pages 22 and 27. This did not change in Unicode 1.0.1 (1992).
Unicode 1.0 implicitly, but clearly meant it was capital. All lowercase letters were named "small".

It seems that in Unicode 1.1 (1993) the name was changed to "script capital p" See UnicodeData-1.1.5

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