Mathematical Notation – The Letter \wp; Name & Origin

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  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):

Best Answer

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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