Abel elliptic functions

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.N.H. Abel, Recherches sur les fonctions elliptiques, Journal für die reine und angewandte Mathematik, 2, 101–181 (1827). 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.N.H. Abel, Recherches sur les fonctions elliptiques, Journal für die reine und angewandte Mathematik, 3, 160–190 (1828). 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.{{citation|last=Gray|first=Jeremy|title=Real and the complex: a history of analysis in the 19th century|publication-place=Cham|page=85|isbn=978-3-319-23715-2|date=2015|language=de}} 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.

Consider the elliptic integral of the first kind in the following symmetric form:{{Cite book|url=https://www.worldcat.org/oclc/53919054|title=The legacy of Niels Henrik Abel: the Abel bicentennial, Oslo, 2002|date=2004|publisher=Springer|first1=Niels Henrik|last1=Abel|first2=Olav Arnfinn|last2=Laudal|first3=Ragni|last3=Piene|isbn=3-540-43826-2|location=Berlin|pages=106|oclc=53919054}}

: $\backslash alpha(x):=\backslash int\_\{0\}^\{x\}\backslash frac\{\backslash mathrm\; dt\}\{\backslash sqrt\{(1-c^2t^2)(1+e^2t^2)\}\}$ with $c,e\backslash in\backslash mathbb\{R\}$.

$\backslash alpha$ is an odd increasing function on the interval $\backslash bigl[\{-\backslash tfrac1c\},\backslash tfrac1c\backslash bigr]$ with the maximum:

: $\{\backslash omega\backslash over\; 2\}:=\; \backslash int\_0^\{1/c\}\; \backslash frac\{dt\}\{\backslash sqrt\{(1\; -\; c^2\; t^2)(1\; +\; e^2\; t^2)\}\}.$

That means $\backslash alpha$ is invertible: There exists a function $\backslash varphi$ such that $x=\backslash varphi(\backslash alpha(x))$, which is well-defined on the interval $\backslash bigl[\{-\backslash tfrac\backslash omega2\},\backslash tfrac\backslash omega2\backslash bigr]$.

Like the function $\backslash alpha$, it depends on the parameters $c$ and $e$ which can be expressed by writing $\backslash varphi(u;e,c)$.

Since $\backslash alpha$ is an odd function, $\backslash varphi$ is also an odd function which means $\backslash varphi(-u)=-\backslash varphi(u)$.

By taking the derivative with respect to $u$ one gets:

: $\backslash varphi\text{'}(u)\; =\; \backslash sqrt\{(1\; -\; c^2\backslash varphi^2(u))(1\; +\; e^2\backslash varphi^2(u))\}$

which is an even function, i.e., $\backslash varphi(-u)=\backslash varphi(u)$.

Abel introduced the new functions

: $f(u)\; =\; \backslash sqrt\{1\; -\; c^2\backslash varphi^2(u)\},\; \backslash ;\backslash ;\backslash ;\; F(u)\; =\; \backslash sqrt\{1\; +\; e^2\backslash varphi^2(u)\}$.

Thereby it holds that $\backslash varphi\text{'}(u)=f(u)F(u)$.

$\backslash varphi$, $f$ and $F$ are the functions known as Abel elliptic functions. They can be continued using the addition theorems.

For example adding $\backslash plusmn\backslash tfrac12\backslash omega$ one gets:

: $\backslash varphi\backslash big(u\; \backslash pm\; \{\backslash omega\backslash over\; 2\}\backslash big)\; =\; \backslash pm\; \{1\backslash over\; c\}\{f(u)\backslash over\; F(u)\},\; \backslash quad$$f\backslash big(u\; \backslash pm\; \{\backslash omega\backslash over\; 2\}\backslash big)\; =\; \backslash mp\; \backslash sqrt\{c^2\; +\; e^2\}\{\backslash varphi(u)\backslash over\; F(u)\},\; \backslash ;\backslash ;\; F\backslash big(u\; \backslash pm\; \{\backslash omega\backslash over\; 2\}\backslash big)\; =\; \{\backslash sqrt\{c^2\; +\; e^2\}\backslash over\; c\}\; \{1\backslash over\; F(u)\}$.

$\backslash varphi$ can be continued onto purely imaginary numbers by introducing the substitution $t\backslash rightarrow\; it$. One gets $xi=\backslash varphi(\backslash beta\; i)$, where

: $\backslash beta(x)=\backslash int\_\{0\}^\{x\}\backslash frac\{\backslash mathrm\; dt\}\{\backslash sqrt\{(1+c^2t^2)(1-e^2t^2)\}\}$.

$\backslash beta$ is an increasing function on the interval $\backslash bigl[\{-\backslash tfrac1e\},\backslash tfrac1e\backslash bigr]$ with the maximum{{citation|last1=Houzel|first1=Christian|last2=Laudal|first2=Olav Arnfinn|last3=Piene|first3=Ragni|title=The legacy of Niels Henrik Abel: the Abel bicentennial, Oslo, 2002|publisher=Springer|publication-place=Berlin|page=107|isbn=3-540-43826-2|date=2004|language=de}}

: $\backslash frac\{\backslash tilde\{\{\backslash omega\}\}\}\{2\}:=\backslash int\_\{0\}^\{\backslash frac1\{e\}\}\backslash frac\{\backslash mathrm\; dt\}\{\backslash sqrt\{(1+c^2t^2)(1-e^2t^2)\}\}$.

That means $\backslash varphi$, $f$ and $F$ are known along the real and imaginary axes. Using the addition theorems again they can be extended onto the complex plane.

For example for $\backslash alpha\; \backslash in\; \backslash bigl[\{-\backslash tfrac\backslash omega2\},\backslash tfrac\backslash omega2\backslash bigr]$ yields to

: $\backslash varphi(\backslash alpha+\backslash tfrac12\backslash tilde\{\backslash omega\}i)=$

\frac{\varphi(\alpha)f(\tfrac12\tilde{\omega}i)F(\tfrac12\tilde{\omega}i)+\varphi(\tfrac12\tilde{\omega}i)f(\alpha)F(\alpha)}{1+c^2e^2\varphi^2(\alpha)\varphi^2(\tfrac12\tilde{\omega}i)}=

\frac{\frac{i} {e} f(\alpha)F(\alpha)}{1+c^2e^2\varphi^2(\alpha)\frac{i^2}{e^2}}=

\frac{i} {e}\frac{f(\alpha)F(\alpha)}{1- c^2\varphi^2(\alpha)}=

\frac{i} {e}\frac{f(\alpha)F(\alpha)}{f^2(\alpha)}=

\frac{i} {e} \frac{F(\alpha)}{f(\alpha)}

.

The periodicity of $\backslash varphi$

, $f$

and $F$

can be shown by applying the addition theorems multiple times. All three functions are doubly periodic which means they have two $\backslash mathbb\{R\}$

-linear independent periods in the complex plane:{{citation|last1=Houzel|first1=Christian|last2=Laudal|first2=Olav Arnfinn|last3=Piene|first3=Ragni|title=The legacy of Niels Henrik Abel: the Abel bicentennial, Oslo, 2002|publisher=Springer|publication-place=Berlin|page=108|isbn=3-540-43826-2|date=2004|language=de}}

: $\backslash varphi(\backslash alpha+2\backslash omega)=\backslash varphi(\backslash alpha)=\backslash varphi(\backslash alpha+2\backslash tilde\{\backslash omega\}i)=\backslash varphi(\backslash alpha+\backslash omega+\backslash tilde\{\backslash omega\}i)$

: $f(\backslash alpha+2\backslash omega)=f(\backslash alpha)=f(\backslash alpha+\backslash tilde\{\backslash omega\}i)$

: $F(\backslash alpha+\backslash omega)=F(\backslash alpha)=F(\backslash alpha+2\backslash tilde\{\backslash omega\}i)$

.

The poles of the functions $\backslash varphi(\backslash alpha)$

,$f(\backslash alpha)$

and $F(\backslash alpha)$

are at{{citation|last1=Houzel|first1=Christian|last2=Laudal|first2=Olav Arnfinn|last3=Piene|first3=Ragni|title=The legacy of Niels Henrik Abel: the Abel bicentennial, Oslo, 2002|publisher=Springer|publication-place=Berlin|page=109|isbn=3-540-43826-2|date=2004|language=de}}

: $\backslash alpha=(m+\backslash tfrac12)\backslash omega+(n+\backslash tfrac12)\backslash tilde\{\backslash omega\; i\},\; \backslash quad$ for $m,n\backslash in\backslash mathbb\{Z\}$.

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

: $\backslash varphi(u;c,e)=\backslash frac1\{c\}\backslash operatorname\{sn\}(cu,k)$

: $f(u;c,e)=\backslash operatorname\{cn\}(cu,k)$

:$F(u;c,e)=\backslash operatorname\{dn\}(cu,k)$

,

where $k=\backslash frac\{ie\}\{c\}$.

For the functions $\backslash varphi$, $f$ and $F$ the following addition theorems hold:

: $\backslash varphi(\backslash alpha+\backslash beta)=\backslash frac\{\backslash varphi(\backslash alpha)f(\backslash beta)F(\backslash beta)+\backslash varphi(\backslash beta)f(\backslash alpha)F(\backslash alpha)\}\{R\}$

: $f(\backslash alpha+\backslash beta)=\backslash frac\{f(\backslash alpha)f(\backslash beta)-c^2\backslash varphi(\backslash alpha)\backslash varphi(\backslash beta)F(\backslash alpha)F(\backslash beta)\}\{R\}$

: $F(\backslash alpha+\backslash beta)=\backslash frac\{F(\backslash alpha)F(\backslash beta)+e^2\backslash varphi(\backslash alpha)\backslash varphi(\backslash beta)f(\backslash alpha)f(\backslash beta)\}\{R\}$,

where $R=1+c^2e^2\backslash varphi^2(\backslash alpha)\backslash varphi^2(\backslash beta)$.

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

• Niels Henrik Abel, [http://www.abelprize.no/nedlastning/verker/oeuvres_1881_del1/oeuvres_completes_de_abel_nouv_ed_1_kap16_opt.pdf Recherches sur le fonctions elliptiques] {{Webarchive|url=https://web.archive.org/web/20160913235716/http://www.abelprize.no/nedlastning/verker/oeuvres_1881_del1/oeuvres_completes_de_abel_nouv_ed_1_kap16_opt.pdf |date=2016-09-13 }}, first and second part in Sophus Lie and Ludwig Sylow (eds.) Collected Works, Oslo (1881).
• Christian Houzel, [http://www.abelprisen.no/c53052/binfil/download.php?tid=53200 The Work of Niels Henrik Abel], in O.A. Laudal and R. Piene, The Legacy of Niels Henrik Abel – The Abel Bicentennial, Oslo 2002, Springer Verlag, Berlin (2004). {{ISBN|3-540-43826-2}}.

Category:Elliptic functions