Carl Jacobi and the Elliptic Functions

Complex graph of the <span property=

Jacobi elliptic function sn u Image by Wikimedia User Fibonacci” width=”402″ height=”401″ /> Complex graph of the Jacobi elliptic function sn u
Image by Wikimedia User Fibonacci

On December 10, 1804, German mathematician Carl Gustav Jacob Jacobi was born. He made fundamental contributions to elliptic functions, dynamics, differential equations, and number theory.

Carl Gustav Jacob Jacobi
(1804 – 1851)

 

Carl Jacobi was the son of a banker and grew up in a rather wealthy family. His brother, Moritz Jacobi became a famous physicist. Carl received his early education from his mother and entered the Gymnasium in Potsdam at about 12 years. However, his previous education was so good and he was so talented that the young Jacobi was put into the final class during his first year. This means that Carl Jacobi reached the necessary standard to enroll at a university when he was 12 years old. Unfortunately, the University of Berlin did not accept students below the age of 16 and Jacobi had to remain in the same class at the Gymnasium in Potsdam until the spring of 1821. [1]

In the meantime, Carl Jacobi received awards in Latin, Greek, and history. Also, he studied advanced mathematics including Euler’s Introductio in analysin infinitorum and even began researching on his own in the field of mathematics, which was not always liked by his teachers. [2] However, by the time, Jacobi finally antered the university, he was still not aware, which field he wanted to focus on and attended seminars in philosophy as well as mathematics. When he decided so devote his life to mathematics, the student began reading the works of Lagrange. [1]

Jacobi submitted his dissertation at the age of 19 and began teaching at a grammar school in Berlin next to working on his habilitation thesis. The rights of citizenship and freedom in Germany for jews from 1812 was revoked in 1822 and all jews were officially taken from the civil services. Jacobi converted to Christianity in 1825 and became privatdozent. Jacobi was quite influenced by Gauss’ research on quadratic and biquadratic residues and studied cubic residues and informed Gauss of his discoveries, who was quite impressed. [3]

By that time, Jacobi already made several major discoveries in the field of number theory and was privatdozent at Königsberg. In this period, Jacobi also summarized his new ideas about elliptic functions and wrote a letter to Legendre, who was back then the leading expert on this topic. Legendre could have been mad or jealous that Jacobi (and also Abel) had made fundamental advances in his favorite topic, however, Legendre helped Jacobi to be promoted as associate professor in 1827. Legendre wrote a letter to Jacobi in this year:

It gives me great satisfaction to see two young mathematicians such as you and [Abel] cultivate with such success a branch of analysis which for such a long time has been my favourite topic of study but which had not been received in my own country as well as it deserves. By your works you place yourselves in the ranks of the best analysts of our era.

About two years later, Jacobi met Legendre as well as other mathematicians such as Fourier, Poisson, and Gauss, increasing his reputation in mathematics. The fundamental work on Jacobi’s theory of elliptic functions that impressed Legendre so much was based on four theta functions. Especially notable in this field is his paper Fundamenta nova theoria functionum ellipticarum, published in 1829. [1]

At yovisto, you may enjoy the video lecture “History of Mathematics in 50 Minutes” by John Dersch.

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