# Karl Gustav Jacob Jacobi

Disclaimer: This page is not being actively maintained, but I will try to make small changes as I find time.  The translations on this site date to my graduate student days, when I was first learning about elliptic functions.  Incidentally, the translations came directly from the versions published in Crelle's Journal, and not from Jacobi's Collected Works.  (The Collected Works weren't readily available to me at the time.)  It shouldn't be difficult to find errors in these translations -- I would certainly appreciate hearing about any that you find.
Enjoy,
Eric

1. A biography of Carl Gustav Jacob Jacobi (1804-1851). (MacTutor web site)  Also on the MacTutor site:
2. [May 21, 2000] Search the catalog of the Biblioteque Nationale de France (BnF). With a small amount of searching -- in French of course -- you should have no trouble in finding an online Acrobat (.pdf) photocopy version of Jacobi's collected works.
3. gives some help in the use of the BnF site. As I find time, I may be able to provide Jacobi-specific help here. (I thank Mr. Bifet for providing me these links.)

Jacobi communicated work in progress to others in seminars. Among those influenced by Jacobi's seminars was Bernhard Riemann. Riemann, a student of Gauss, was reportedly frustrated by Gauss's secretiveness about his research and attended Jacobi's seminars to see mathematics research in progress. (Among Riemann's collected works is an appendix to section 40 of Jacobi's Fundamenta Nova.)

4. Jacobi's Four Square Theorem. (Also available in postscript format [11 pages].) [CONSTRUCTION IN PROGRESS]

5. [April 25, 2000] A simple program to generate four-square representations and some sample output to test Jacobi's version of the Four Square Theorem. (C++ program foursq.C and sample output foursq.txt). The program finds all representations by brute force (in this case brute force is essentially a depth first search). The search is optimized somewhat by using congruences pare the searches for two-square and three-square representations. (It is easy to verify that a two-square representation does not exist for integers congruent to 3 modulo 4, and that a three-square representation does not exist for integers congruent to 7 modulo 8.)

[April 28, 2000] A simple program to generate r-square representations. (C++ program rsq.C and sample output rsq.txt). Essentially the same algorithm is used, except that negatives and positives are handled separately.  A few additional trivial cases are eliminated using congruences.  The number of representations goes up rapidly as the number of squares r increases.  The program will calculate all representations of 100 as a sum of 10 squares, but expect to wait a while.

[April 28, 2000] A Mathematica program to count the number of representations of (small) integers as sums of r squares. For example, the program results show that 36 has more than 100,000,000 representations as 11 squares. (See the very last entry in the sample run for details.) This number was computed exactly without producing any representations of 36 as a sum of squares. Both Mathematica program nsq.m and sample output are available for your amusement. In addition, I used the program to show that there are exactly 1,282,320,348 representations of 100 as a sus of ten squares. The computation (nine convolutions) took about two seconds. With a little thought, you should see that I could have done this using only four convolutions. (Mathematica input and a transcript of the program run).

6. [REVISED 28 Jun 98] A short exercise from section 39 in the Fundamenta Nova. In this exercise, you are asked to formally transform an infinite product (equation 1 section 39) into a Fourier series expansion (equation 6 section 39). The techniques are straightforward -- most are are taught in a typical first year calculus sequence. (I suggest you not try this on your first year calculus students without their consent!) Analysts will probably want to assume that q<1 and fill in details about convergence.
7. [REVISED 27 Apr 97] The table of contents (postscript or Acrobat formats) for Jacobi's Fundamenta Nova Theoriae Functionum Ellipticarum translated from Latin into English. Pages numbers refer to the version printed in volume 1 of his collected works, available from the American Mathematical Society. (1 page in postscript format. Keywords: Jacobi elliptic functions, theta functions, elliptic integrals, algebraic combinatorics.)
8. [MINOR REVISIONS 28 Jun 98] C. Jacobi, A new application of elliptic functions to algebra, originally in Crelle's Journal 7(1832), 41-43, translated from French into English. (3 pages in postscript format. Keywords: Jacobi elliptic functions, continued fractions.)
9. [REVISED 28 Jun 98] C. Jacobi, Note sur les fonctions elliptiques, originally published in Crelle's Journal 3(1828), 251-254, translated from French into English. (4 pages in postscript format. Keywords: Jacobi elliptic functions, theta functions.)
10. [Corrected Jan 2004] C. Jacobi, Suite des notices sur les fonctions elliptiques, originally published in Crelle's Journal 3(1828), 255-263, translated from French into English. (9 pages in postscript format. Keywords: Jacobi elliptic functions, theta functions, sums of squares, modular transformations.) (Note: The source file was corrupt. The corrections were applied to a backup copy.) Also available as a PDF file.

### Elliptic functions and integrals

1. (MacTutor) Elliptic functions and integrals.  A brief early survey from Wallis to Jacob Bernoulli.