Abel elliptic functions

In mathematics Abel elliptic functions are a special kind of elliptic functions, that were established by the Norwegian mathematician Niels Henrik Abel. He published his paper "Recherches sur les Fonctions elliptiques" in Crelle's Journal in 1827.[1] It was the first work on elliptic functions that was actually published.[2] Abel's work on elliptic functions also influenced Jacobi's studies of elliptic functions, whose 1829 published book "Fundamenta nova theoriae functionum ellipticarum" became the standard work on elliptic functions.[3]

History

Abel's starting point were the elliptic integrals which had been studied in great detail by Adrien-Marie Legendre. He began his research in 1823 when he still was a student. In particular he viewed them as complex functions which at that time were still in their infancy. In the following years Abel continued to explore these functions. He also tried to generalize them to functions with even more periods, but seemed to be in no hurry to publish his results.

But in the beginning of the year 1827 he wrote together his first, long presentation Recherches sur les fonctions elliptiques of his discoveries.[4] At the end of the same year he became aware of Carl Gustav Jacobi and his works on new transformations of elliptic integrals. Abel finishes then a second part of his article on elliptic functions and shows in an appendix how the transformation results of Jacobi would easily follow.[5][3] When he then sees the next publication by Jacobi where he makes use of elliptic functions to prove his results without referring to Abel, the Norwegian mathematician finds himself to be in a struggle with Jacobi over priority. He finishes several new articles about related issues, now for the first time dating them, but dies less than a year later in 1829.[6] In the meantime Jacobi completes his great work Fundamenta nova theoriae functionum ellipticarum on elliptic functions which appears the same year as a book. It ended up defining what would be the standard form of elliptic functions in the years that followed.[6]

Derivation from elliptic Integrals

Consider the elliptic integral of the first kind in the following symmetric form:[7]

with .

is an odd increasing function on the interval with the maximum:[2]

That means is invertible: There exists a function such that , which is well-defined on the interval .

Like the function , it depends on the parameters and which can be expressed by writing .

Since is an odd function, is also an odd function which means .

By taking the derivative with respect to one gets:

which is an even function, i.e., .

Abel introduced the new functions

.

Thereby it holds that[2] .

, and are the functions known as Abel elliptic functions. They can be continued using the addition theorems.

For example adding one gets:

.

Complex extension

can be continued onto purely imaginary numbers by introducing the substitution . One gets , where

.

is an increasing function on the interval with the maximum[8]

.

That means , and are known along the real and imaginary axes. Using the addition theorems again they can be extended onto the complex plane.

For example for yields to

.

Double periodicity and poles

The periodicity of , and can be shown by applying the addition theorems multiple times. All three functions are doubly periodic which means they have two -linear independent periods in the complex plane:[9]

.

The poles of the functions , and are at[10]

for .

Relation to Jacobi elliptic functions

Abel's elliptic functions can be expressed by the Jacobi elliptic functions, which do not depend on the parameters and but on a modulus :

,

where .

Addition Theorems

For the functions , and the following addition theorems hold:[8]

,

where .

These follow from the addition theorems for elliptic integrals that Euler already had proven.[8]

References

  1. ^ Gray, Jeremy (14 October 2015), Real and the complex: a history of analysis in the 19th century, Springer Cham, p. 73, ISBN 978-3-319-23715-2{{citation}}: CS1 maint: location missing publisher (link)
  2. ^ a b c Gray, Jeremy (14 October 2015), Real and the complex: a history of analysis in the 19th century, Springer Cham, pp. 74f, ISBN 978-3-319-23715-2{{citation}}: CS1 maint: location missing publisher (link)
  3. ^ a b Gray, Jeremy (14 October 2015), Real and the complex: a history of analysis in the 19th century, Springer Cham, pp. 84f, ISBN 978-3-319-23715-2{{citation}}: CS1 maint: location missing publisher (link)
  4. ^ N.H. Abel, Recherches sur les fonctions elliptiques, Journal für die reine und angewandte Mathematik, 2, 101–181 (1827).
  5. ^ N.H. Abel, Recherches sur les fonctions elliptiques, Journal für die reine und angewandte Mathematik, 3, 160–190 (1828).
  6. ^ a b Gray, Jeremy (2015), Real and the complex: a history of analysis in the 19th century (in German), Cham, p. 85, ISBN 978-3-319-23715-2{{citation}}: CS1 maint: location missing publisher (link)
  7. ^ Abel, Niels Henrik; Laudal, Olav Arnfinn; Piene, Ragni (2004). The legacy of Niels Henrik Abel: the Abel bicentennial, Oslo, 2002. Berlin: Springer. p. 106. ISBN 3-540-43826-2. OCLC 53919054.
  8. ^ a b c Houzel, Christian; Laudal, Olav Arnfinn; Piene, Ragni (2004), The legacy of Niels Henrik Abel: the Abel bicentennial, Oslo, 2002 (in German), Berlin: Springer, p. 107, ISBN 3-540-43826-2
  9. ^ Houzel, Christian; Laudal, Olav Arnfinn; Piene, Ragni (2004), The legacy of Niels Henrik Abel: the Abel bicentennial, Oslo, 2002 (in German), Berlin: Springer, p. 108, ISBN 3-540-43826-2
  10. ^ Houzel, Christian; Laudal, Olav Arnfinn; Piene, Ragni (2004), The legacy of Niels Henrik Abel: the Abel bicentennial, Oslo, 2002 (in German), Berlin: Springer, p. 109, ISBN 3-540-43826-2

Literature

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