Document Type
Article
Publication Date
9-29-2025
Abstract
We present this new commentary and translation anticipating the 200th anniversary of the work, commonly known as Abel's ``Paris memoir.'' This is recognized today as one of Abel's most original and influential works. It is significant mostly because it marked the first appearance of a form of a result in the theory of algebraic curves and Riemann surfaces that has come to be known as ``Abel's theorem.'' However, Abel's original understanding of the meaning and context of his result was quite different from the typical modern formulation and the development of the modern understanding has been a long and tortuous one. In the commentary, we discuss the history behind this work and its reception and we give an overview of the contents of the memoir. We compare modern understandings of Abel's discoveries with his original statements and we give an exposition of David Cox's very recent explanation of the meaning of Abel's calculation of the quantity ``gamma'' (related to the genus of a curve) in terms of the geometry of toric surfaces. The translation is essentially completely literal and it preserves the way Abel originally presented his results.
Repository Citation
Abel, Niels Henrik 1802--1829 and Little, John, "Memoir on a general property of a very extensive class of transcendental functions" (2025). Mathematics and Computer Science Department Faculty Scholarship. 25.
https://crossworks.holycross.edu/math_fac_scholarship/25
Comments
Presented to the Academy of Sciences at Paris, October 30, 1826; first published in the Mémoires présentés par divers savants, vol. VII, Paris, 1841.
Translated from the French and Provided with Commentary by John Little