Abstract
The classical single-valued trigonometric functions could be viewed as the inverses of specific integrals of algebraic functions of degree two. Abel and Jacobi created the theory of elliptic functions of a complex variable which were inverses of elliptic integrals, and these turned out to be doublyperiodic functions on the complex plane with specific kinds of singularities. These functions became models for the general theory of meromorphic functions in the late nineteenth century.
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Notes
- 1.
“I propose, in this memoir, to consider the inverse function, that is to say the function \(\varphi \alpha \) determined by the equations”.
- 2.
Abel doesn’t distinguish between the upper limit of the integral and the variable of integration, but we do to clarify the discussion.
- 3.
The inverse function \(\varphi (\alpha )\) and its related functions \(f(\alpha )\) and \(F(\alpha )\) are well defined locally near \(\alpha =0\) by the inverse function theorem. The extension to the full real line is discussed later in this section.
- 4.
“Several properties of these functions are deducible immediately from the known properties of the elliptic function [elliptic integral] of the first kind, but others are more hidden. For example, one can show that the equations \(\varphi \alpha =0, f\alpha =0 , Fa=0\) have an infinite number of roots, where one can find all of them. One of the most remarkable properties is that one can express rationally \(\varphi (m\alpha ), f(m\alpha ), F(m\alpha )\) (\(m\) an integer) in \(\varphi \alpha , f\alpha , Fa\). Also, nothing is more simple than to find \(\varphi (m\alpha ), f(m\alpha ), F(m\alpha )\), when one knows \(\varphi \alpha , f\alpha , F\alpha \); but the inverse problem, to know how to determine \(\varphi \alpha , f\alpha , F\alpha \) in \(\varphi (m\alpha ), f(m\alpha ), F(m\alpha )\), is more difficult, since it depends on an equation of high degree (more specifically of degree \(m^2\)).
The solution of this equation is the principal object of this memoir. At first one can see how one can find all the roots, by means of the functions \(\varphi , f, F\). Then one treats the algebraic solution of the equation in question, and one comes to this remarkable result, that \(\varphi \frac{\alpha }{m}, f\frac{\alpha }{m}, F\frac{\alpha }{m}\) can be expressed in \(\varphi \alpha ,f\alpha , F\alpha \), by a formula, which, with respect to \(\alpha \), doesn’t contain any irrationality except radicals. This gives a very general class of equations which are solvable algebraically.”
- 5.
Of course this proof depends on knowing what the right-hand side of such an addition formula looks like, and this knowledge stems from the work of Euler and Legendre.
- 6.
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Wells, R.O. (2017). Elliptic Functions. In: Differential and Complex Geometry: Origins, Abstractions and Embeddings. Springer, Cham. https://doi.org/10.1007/978-3-319-58184-2_8
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DOI: https://doi.org/10.1007/978-3-319-58184-2_8
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