4-21-5. Thomas Craig to H. Poincaré

Feb. 5 1884

American Journal of Mathematics

Johns Hopkins University

Baltimore

Dear Monsieur Poincaré,

I have just received your note informing me that I will be elected a member of the Société mathematique. I thank you very much for your kindness in presenting my name there. You are good enough to ask me to send something to the Society to be published in the Bulletin. I shall have great pleasure in doing so, but at present I am very busy with my University work and the work connected with the editing of the “Journal of Mathematics”. I am just now writing a short paper of the theta-functions with complex characteristics. Denoting as usual a $\vartheta$-function of $p$ variables by

$$\vartheta\left(\begin{array}[]{cccc}l_{1},&l_{2}&\ldots&l_{p}\\ \lambda_{1},&\lambda_{2}&\ldots&\lambda_{p}\end{array}\right)(\begin{array}[]{cccc}v_{1},&v_{2}&\ldots&v_{p}\end{array})$$

I examine the functions for which

$$\begin{array}[]{cc}l=a+ib,&\lambda=\alpha+i\beta\end{array}$$

and confine my attention to the case when $a$, $b$, $\alpha$, $\beta=0$ or 1. There are in this case $2^{4p}$ functions, the square of the corresponding number of $\vartheta$-functions.

I have also been studying recently a class of functions defined as follows (taking the simplest case).

$$f(x+w)=\phi(x)f(x)$$

where

$$\phi(x+w) =\phi(x),$$

$$f(x+nw) =[\phi(x)]^{n}f(x)\text{ etc.}$$

Can you tell me whether or not functions of this sort have ever been studied?

I take great pleasure in informing you that I have a class devoted to reading your papers on “Groupes Fuchsiens” and “Groupes Kleinéens”. It is not a part of my regular work, but I meet the young men every day at 3 P.M. in order to read and explain to them your beautiful memoirs on these subjects. I am happy to say that the students are very enthusiastic over the new and charming ideas which you have so elegantly developed in your memoirs in the “Acta Math”, the Comptes Rendus” and the “Math. Annalen”. I wish very much that you could come here for a few months and give a course of lectures on the Kleinéens and Fuchsiens. Could you not do so? say for four months. I await with much anxiety the arrival of your promised memoir for the ‘Journal of Mathematics’ and hope to receive it at an early date. Prof. Sylvester has left us and is now the “Savilian Professor of Mathematics” in the University of Oxford – his place has not yet been filled and I do not know who will fill it. It is impossible to get a Frenchman because no Frenchman will leave France. I wish you could come. I hope that I shall have the pleasure of hearing from you soon. Please let me know what you think about the class of functions which I have mentioned on page 2 of this letter, and also please let me have as soon as you can an article for the ‘Journal of Mathematics’.

I remain dear M. Poincaré,

Yours very sincerely,

Thomas Craig

ALS 4p. Collection particulière, Paris 75017.

Time-stamp: " 9.06.2022 21:27"

Notes

• 1 See Craig (1884), which was the fourth paper by Craig on theta functions in Volume 6 of the American Journal of Mathematics ((Craig, 1883a, b, a).
• 2 Simon Newcomb was appointed professor of mathematics and astronomy in 1884. He succeeded Sylvester as editor of the American Journal of Mathematics, published by Johns Hopkins University Campbell (1924).