## 4-22-5. 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.11endnote: 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). 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

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

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

I remain dear M. Poincaré,

Yours very sincerely,

Thomas Craig

ALS 4p. Collection particulière, Paris 75017.

Time-stamp: “ 4.05.2019 00:49”

### 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).

## References

• W. W. Campbell (1924) Biographical memoir Simon Newcomb 1835–1909. Memoirs of the National Academy of Sciences 17, pp. 1–18. Cited by: endnote 2.
• T. Craig (1883a) On quadruple theta-functions. American Journal of Mathematics 6 (1), pp. 14–59. Cited by: endnote 1.
• T. Craig (1883b) On quadruple theta-functions. American Journal of Mathematics 6 (1), pp. 183–204. Cited by: endnote 1.
• T. Craig (1884) On theta-functions with complex characteristics. American Journal of Mathematics 6 (1), pp. 337–358. Cited by: endnote 1.