Euler's theorem

In complex analysis, an elliptic function is a meromorphic function that is periodic in two directions. Just as a periodic function of a real variable is defined by its values on an interval, an elliptic function is determined by its values on a fundamental parallelogram, which then repeat in a lattice. Such a doubly periodic function cannot be holomorphic, as it would then be a bounded entire function, and by Liouville's theorem every such function must be constant. In fact, an elliptic function must have at least two poles (counting multiplicity) in a fundamental parallelogram, as it is easy to show using the periodicity that a contour integral around its boundary must vanish, implying that the residues of any simple poles must cancel.

Historically, elliptic functions were first discovered by Niels Henrik Abel as inverse functions of elliptic integrals, and their theory improved by Carl Gustav Jacobi; these in turn were studied in connection with the problem of the arc length of an ellipse, whence the name derives. Jacobi's elliptic functions have found numerous applications in physics, and were used by Jacobi to prove some results in elementary number theory. A more complete study of elliptic functions was later undertaken by Karl Weierstrass, who found a simple elliptic function in terms of which all the others could be expressed. Besides their practical use in the evaluation of integrals and the explicit solution of certain differential equations, they have deep connections with elliptic curves and modular forms.

Definition

Formally, an elliptic function is a function $f$ meromorphic on $\mathbb {C}$ for which there exist two non-zero complex numbers $\omega _{1}$ and $\omega _{2}$ with ${\frac {\omega _{1}}{\omega _{2}}}\notin \mathbb {R}$ (in other words, not parallel), such that $f(z)=f(z+\omega _{1})$ and $f(z)=f(z+\omega _{2})$ for all $z\in \mathbb {C}$.

Denoting the "lattice of periods" by $\Lambda =\left\{m\omega _{1}+n\omega _{2}\mid m,n\in \mathbb {Z} \right\}$, it follows that $f(z)=f(z+\omega )$ for all $\omega \in \Lambda$.

There are two families of 'canonical' elliptic functions: those of Jacobi and those of Weierstrass. Although Jacobi's elliptic functions are older and more directly relevant to applications, modern authors mostly follow Karl Weierstrass when presenting the elementary theory, because his functions are simpler, and any elliptic function can be expressed in terms of them.

Weierstrass's elliptic functions

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With the definition of elliptic functions given above (which is due to Weierstrass) the Weierstrass elliptic function $\wp \left(z\right)$ is constructed in the most obvious way: given a lattice $\Lambda$ as above, put

$\wp \left(z\right)={\frac {1}{z^{2}}}+\sum _{\omega \in \Lambda \smallsetminus \left\{0\right\}}\left({\frac {1}{\left(z-\omega \right)^{2}}}-{\frac {1}{\omega ^{2}}}\right)$

This function is clearly invariant with respect to the transformation $z\mapsto z+\omega$ for any $\omega \in \Lambda$ and only has poles at $z=0$ and $z=\omega$. The addition of the $-{\frac {1}{\omega ^{2}}}$ terms is necessary to make the sum converge. The technical condition to ensure that an infinite sum such as this converges to a meromorphic function is that on any compact set, after omitting the finitely many terms having poles in that set, the remaining series converges normally. On any compact disk $\mathbb {D}$ defined by $\left|z\right|\leq R$, any $\omega \notin \mathbb {D}$ satisfies

$\left|{\frac {1}{\left(z-\omega \right)^{2}}}-{\frac {1}{\omega ^{2}}}\right|=\left|{\frac {2\omega z-z^{2}}{\omega ^{2}\left(\omega -z\right)^{2}}}\right|=\left|{\frac {z\left(2-{\frac {z}{\omega }}\right)}{\omega ^{3}\left(1-{\frac {z}{\omega }}\right)^{2}}}\right|\leq {\frac {10R}{\left|\omega \right|^{3}}}$

and it can be shown that the sum

$\sum _{\omega \neq 0}{\frac {1}{\left|\omega \right|^{3}}}$

converges regardless of $\Lambda$.

By writing $\wp$ as a Laurent series and explicitly comparing terms, one may verify that it satisfies the relation

$\left(\wp '\left(z\right)\right)^{2}=4\left(\wp \left(z\right)\right)^{3}-g_{2}\wp \left(z\right)-g_{3}$

where

$g_{2}=60\sum _{\omega \in \Lambda \smallsetminus \left\{0\right\}}{\frac {1}{\omega ^{4}}}$

and

$g_{3}=140\sum _{\omega \in \Lambda \smallsetminus \left\{0\right\}}{\frac {1}{\omega ^{6}}}.$

This means that the pair $\left(\wp ,\wp '\right)$ parametrize an elliptic curve.

The functions $\wp$ take different forms depending on $\Lambda$, and a rich theory is developed when one allows $\Lambda$ to vary. To this effect, put $\omega _{1}=1$ and $\omega _{2}=\tau$, with $\operatorname {Im} \left(\tau \right)>0$. (After a rotation and a scaling factor, any lattice may be put in this form.)

A holomorphic function in the upper half plane $H=\left\{z\in \mathbb {C} |Im\left(z\right)>0\right\}$ which is invariant under linear fractional transformations with integer coefficients and determinant 1 is called a modular function. That is, a holomorphic function $h:H\rightarrow \mathbb {C}$ is a modular function if

$h\left(\tau \right)=h\left({\frac {a\tau +b}{c\tau +d}}\right)$ for all $\left({\begin{matrix}a&c\\b&d\end{matrix}}\right)\in SL_{2}\left(\mathbb {Z} \right)$.

One such function is Klein's j-invariant, defined by

$j\left(\tau \right)={\frac {1728g_{2}^{3}}{g_{2}^{3}-27g_{3}^{2}}}$ where $g_{2}$ and $g_{3}$ are as above.

Jacobi's elliptic functions

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Auxiliary rectangle construction

There are twelve Jacobian elliptic functions. Each of the twelve corresponds to an arrow drawn from one corner of a rectangle to another. The corners of the rectangle are labeled, by convention, s, c, d and n. The rectangle is understood to be lying on the complex plane, so that s is at the origin, c is at the point K on the real axis, d is at the point K + iK' and n is at point iK' on the imaginary axis. The numbers K and K' are called the quarter periods. The twelve Jacobian elliptic functions are then pq, where each of p and q is one of the letters s, c, d, n.

The Jacobian elliptic functions are then the unique doubly periodic, meromorphic functions satisfying the following three properties:

• There is a simple zero at the corner p, and a simple pole at the corner q.
• The step from p to q is equal to half the period of the function pq u; that is, the function pq u is periodic in the direction pq, with the period being twice the distance from p to q. The function pq u is also periodic in the other two directions, with a period such that the distance from p to one of the other corners is a quarter period.
• If the function pq u is expanded in terms of u at one of the corners, the leading term in the expansion has a coefficient of 1. In other words, the leading term of the expansion of pq u at the corner p is u; the leading term of the expansion at the corner q is 1/u, and the leading term of an expansion at the other two corners is 1.

More generally, there is no need to impose a rectangle; a parallelogram will do. However, if K and iK' are kept on the real and imaginary axis, respectively, then the Jacobi elliptic functions pq u will be real functions when u is real.

Properties

• The set of all elliptic functions which share some two periods form a field.

• The derivative of an elliptic function is again an elliptic function, with the same periods.

• The field of elliptic functions with respect to a given lattice is generated by ℘ and its derivative ℘′.

See also

• Elliptic integral
• Modular group
• Ramanujan theta function

References

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• Template:Cartan Cartan, Henri, Elementary Theory of Analytic Functions of one or Several Complex Variables'", Dover Publications, 1995.
• Template:Abramowitz Stegun ref2 (only considers the case of real invariants).
• N. I. Akhiezer, Elements of the Theory of Elliptic Functions, (1970) Moscow, translated into English as AMS Translations of Mathematical Monographs Volume 79 (1990) AMS, Rhode Island ISBN 0-8218-4532-2
• Tom M. Apostol, Modular Functions and Dirichlet Series in Number Theory, Springer-Verlag, New York, 1976. ISBN 0-387-97127-0 (See Chapter 1.)
• E. T. Whittaker and G. N. Watson. A course of modern analysis, Cambridge University Press, 1952

External links

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• Translation of Niels Abel's Research on Elliptic Functions at Convergence