Niels Henrik Abel’s Theory Of Elliptic Function

Elliptic function is one of the most important mathematical achievements in 19 th century. In the process of its founding and development, the Norwegian mathematician Niels Henrik Abel had made outstanding contributions. He began to study elliptic integral in the early 1820 s. With his acute insight, he achieved elliptic function. He had made significant results in the work on elliptic function’s basic properties and various expressions. His work promoted contemporary mathematicians to turn to study elliptic function. This also made the theory of elliptic function rapidly develop.This dissertation studies the important achievement Abel got, during the period of the originating and development of the theory on the basis of the study of documents.Connected with the developing history of elliptic function, the dissertation explores the thinking method how Abel solved the problem. The main results were as follows:1. This dissertation summarizes the prehistory of elliptic function. Before Abel originated the theory, Euler and Legendre achieved lots of important results about elliptic integral. And Gauss studied the lemniscate function, but it didn’t exert any influence at that time in the absence of publication. Those all provided conditions for Abel to be the founder of elliptic function.2. This dissertation explores the contribution Abel had made during the founding period of the theory of elliptic function. Abel’s research methods is investigated detailedly,through the essay which he published about elliptic function in 1820’s. This dissertation also analyzes the work he had done. All the work had important significance for the follow development of elliptic function.3. This dissertation expounds the development of elliptic function with the work of other mathematicians in 19 th century. Jacobi defined the theta-functions, and all parts of the theory of elliptic functions could be investigated by them. Liouville developed an entire theory for doubly periodic functions. Weierstrass introduced his function as the solution to the differential equation. This prompted the theory of elliptic function to develop rapidly.