Abel elliptic functions

Graphical illustration of an elliptic function where its values are indicated by colours. These are periodically repeated in the two directions of the complex plane.

Abel elliptic functions are holomorphic functions of one complex variable and with two periods. They were first established by Niels Henrik Abel and are a generalization of trigonometric functions. Since they are based on elliptic integrals, they were the first examples of elliptic functions. Similar functions were shortly thereafter defined by Carl Gustav Jacobi. In spite of the Abel functions having several theoretical advantages, the Jacobi elliptic functions have become the standard. This can have to do with the fact that Abel died only two years after he presented them while Jacobi could continue with his exploration of them throughout his lifetime. Both the elliptic functions of Abel and of Jacobi can be derived from a more general formulation which was later given by Karl Weierstrass based on their double periodicity.

History

The first elliptic functions were found by Carl Friedrich Gauss around 1796 in connection with his calculation of the lemniscate arc length, but first published after his death. These are special cases of the general, elliptic functions which were first investigated by Abel in 1823 when he still was a student. His starting point were the elliptic integrals which had been studied in great detail by Adrien-Marie Legendre. The year after Abel could report that his new functions had two periods. Especially this property made them more interesting than the normal trigonometric functions which have only one period. In particular it meant that they had to be 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. 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. 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 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.

Properties

Abel considered a elliptic integral of the first kind in the following symmetric form:

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

In the special case c = 1 and e = 0 the integral gives the arc length of a circle, while for c = e = 1 it leads to the arc length of the lemniscate. He could thus make contact both with the trigonometric functions and the lemniscatic functions which Gauss had hinted at in his Disquisitiones Arithmeticae.

$\alpha$ is an odd increasing function on the intervall $[-{\frac {1}{c}},{\frac {1}{c}}]$ with the maximum:

${\omega \over 2}:=\int _{0}^{1/c}{\frac {dt}{\sqrt {(1-c^{2}t^{2})(1+e^{2}t^{2})}}}.$

That means $\alpha$ is invertible and there exists a function $\varphi$ such that $x=\varphi (\alpha (x))$, which is well defined on the intervall $[-{\frac {\omega }{2}},{\frac {\omega }{2}}]$. It has the special values $\varphi \left(\pm {\frac {\omega }{2}}\right)=\pm {\frac {1}{c}}$. Like the function $\alpha$ it depends on the paramters $c$ and $e$ what can be expressed by writing $\varphi (u;e,c)$ but for now we will do without.

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

By taking the derivative with respect to $u$ we get:

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

which now is an even function φ' (u) = φ' (−u) with the values $\varphi '\left(\pm {\frac {\omega }{2}}\right)=0$ and $\varphi '(0)=1$.

For the two square roots which here appear Abel introduced the new functions

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

Thereby we get $\varphi '(u)=f(u)F(u)$.

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

For example adding $\pm {\frac {\omega }{2}}$, it gives

$\varphi {\big (}u\pm {\omega \over 2}{\big )}=\pm {1 \over c}{f(u) \over F(u)}$

and similarly for the two other functions,

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

With u = ω/2 one thus has φ(ω) = 0 so that the functions will be defined in the whole interval $[-\omega ,\omega ]$.

Complex extension

We can continue $\varphi$ onto purly imaginary numbers by introducing the substitution $t\rightarrow it$. What we get is $xi=\varphi (\beta i)$, where

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

$\beta$ is an increasing function on the interval $[-{\frac {1}{e}},{\frac {1}{e}}]$ with the maximum

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

That means $\varphi$, $f$ and $F$ are known along the real and imaginary axes. Using the addition theorems again we can extend the functions onto the complex plane.

For example we get for $\alpha \in [-{\frac {\omega }{2}},{\frac {\omega }{2}}]$

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

Double periodicity and poles

The periodicity of $\varphi$, $f$ and $F$ can be shown by applying the addition theorems multiple times. All three functions are double periodic which means they have two distinct periods in the complex plane:

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

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

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

The poles of the functions $\varphi (\alpha )$,$f(\alpha )$ and $F(\alpha )$ are at

$\alpha =(m+{\frac {1}{2}})\omega +(n+{\frac {1}{2}}){\tilde {\omega i}}$, for $m,n\in \mathbb {Z}$.

Relation to Jacobi elliptic functions

From the defining integrals one sees that Abel's elliptic functions can be expressed by the Jacobi elliptic functions. Theses functions do not depend on the parameters $c$ and $e$ but on a modulus $k$. The precise relation between these functions can be found by a change of the integration variable and is

$\varphi (u;c,e)={\frac {1}{c}}\operatorname {sn} (cu,{\frac {ie}{c}})$

$f(u;c,e)=\operatorname {cn} (cu,{\frac {ie}{c}})$

$F(u;c,e)=\operatorname {dn} (cu,{\frac {ie}{c}})$

Addition Theorems

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

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

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

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

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

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

Literature

• Niels Henrik Abel, Recherhes sur le fonctions elliptiques, first and second part in Sophus Lie and Ludwig Sylow (eds.) Collected Works, Oslo (1881).
• Christian Houzel, 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.