Elliptic function

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 all simple poles must cancel.

Historically, elliptic functions were first discovered by Niels Henrik Abel as inverse functions of elliptic integrals, and their theory was 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 $ℂ$ for which there exist two non-zero complex numbers $ω 1$ and $ω 2$ with $ω 1 / ω 2 \notin ℝ$ , such that $f (z) = f (z + ω 1)$ and $f (z) = f (z + ω 2)$ for all $z \in ℂ$ .

Denoting the "lattice of periods" by $Λ = {mω 1 + nω 2 | m, n \in ℤ}$ , it follows that $f (z) = f (z + ω)$ for all $ω \in Λ$ .

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 Weierstrass when presenting the elementary theory, because his functions are simpler, and any elliptic function can be expressed in terms of them.

Weierstrass' elliptic functions

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

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

This function is clearly invariant with respect to the transformation $z \mapsto z + ω$ for any $ω \in Λ$ . The addition of the $- 1 / ω 2$ 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 defined by $| z | \leq R$ , and for any $| ω | > 2 R$ , one has

\left|{\frac {1}{\left(z-\omega \right)^{2}}}-{\frac {1}{\omega ^{2}}}\right|=\left|{\frac {2\omega z-z^{2}}{\omega ^{2}(\omega -z)^{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 $Λ$ .^[1]

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

{\bigl (}\wp '(z){\bigr )}^{2}=4{\bigl (}\wp (z){\bigr )}^{3}-g_{2}\wp (z)-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 $(℘,℘')$ parametrize an elliptic curve.

The functions $℘$ take different forms depending on $Λ$ , and a rich theory is developed when one allows $Λ$ to vary. To this effect, put $ω 1 = 1$ and $ω 2 = τ$ , with $Im(τ) > 0$ . (After a rotation and a scaling factor, any lattice may be put in this form.)

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

h\left(\tau \right)=h\left({\frac {a\tau +b}{c\tau +d}}\right)\quad {\text{for all }}\left({\begin{matrix}a&c\\b&d\end{matrix}}\right)\in {\text{SL}}_{2}(\mathbb {Z} )

.

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.

Properties

The set of all elliptic functions which share the same 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 $℘'$ .

Jacobi's elliptic functions

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 $p$ and $q$ are two different letters in $s, c, d, n$ .

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.

Abel's elliptic functions

Elliptic integrals had been studied in great detail by Legendre who had reduced them to three fundamental types. Abel wrote an integral of the first kind as

u=\int _{0}^{x}{\frac {dt}{\sqrt {\left(1-c^{2}t^{2}\right)\left(1+e^{2}t^{2}\right)}}}

where $c$ and $e$ are two parameters.^[2] This is a generalization of the integral which gives the arc length of the lemniscate corresponding to the special values $c = e = 1$ and investigated by Carl Friedrich Gauss. The arc length of the circle would result from setting $c = 1$ and $e = 0$ .

The value $u$ of the integral is an increasing function of the upper limit for $0 < x < 1 / c$ and reaches a maximum

{\frac {\omega }{2}}=\int _{0}^{\frac {1}{c}}{\frac {dt}{\sqrt {\left(1-c^{2}t^{2}\right)\left(1+e^{2}t^{2}\right)}}}

Abel's stroke of genius was now to consider the inverse function $x = φ (u)$ which now is well-defined in the interval $0 \leq u \leq ω / 2$ . Since the defining integral is an odd function of $x$ , the function $φ (u)$ is also odd with the special values $φ (0) = 0$ and $φ (\pm ω / 2) = \pm 1 / c$ . The derivative of the function $φ'(u) = dφ / du$ follows also from the integral as

{\frac {d\varphi }{du}}={\sqrt {\left(1-c^{2}\varphi ^{2}\right)\left(1+e^{2}\varphi ^{2}\right)}}

and is an even function. The two square roots can be considered to be new, even functions of the argument $u$ . Abel defined them as

{\begin{aligned}f(u)&={\sqrt {1-c^{2}\varphi ^{2}(u)}}\\F(u)&={\sqrt {1+e^{2}\varphi ^{2}(u)}}\end{aligned}}

In this way the derivative can be written on the more compact form $φ'(u) = f (u) F (u)$ . These new functions have the derivatives $f'(u) = - c 2 φ (u) F (u)$ and $F'(u) = e 2 φ (u) f (u)$ . All three elliptic functions depend on the parameters $c$ and $e$ although this dependence is usually not explicitly written out.

As for the trigonometric functions Abel could show that these new functions satisfied addition theorems in agreement with what Euler had previously found from such integrals.^[2] They enable the functions to be continued into the whole interval $- ω \leq u \leq ω$ and show that they are periodic with period $2 ω$ . Furthermore, by letting $t \to it$ in the integral the functions can also be defined for complex values of the argument. By this extension the parameters $c$ and $e$ are exchanged and implies that the functions also have the imaginary period $2 iω'$ with

{\frac {\omega '}{2}}=\int _{0}^{\frac {1}{e}}{\frac {dt}{\sqrt {\left(1-e^{2}t^{2}\right)\left(1+c^{2}t^{2}\right)}}}.

The elliptic functions thus have a double periodicity. Equivalently one can say that they have two complex periods $ω 1,2 = ω \pm iω'$ . Their zeros and poles will thus form a regular, two-dimensional lattice. Corresponding properties of the lemniscatic elliptic functions had also been established by Gauss, but not published before after his death.^[3]

References

↑ Cartan, Henri (1995). Elementary Theory of Analytic Functions of One or Several Complex Variables. Dover Publications. p. 154. ISBN 9780486685434.
1 2 J. Gray, The Real and the Complex: A History of Analysis in the 19th Century, Springer, Heidelberg (2015). ISBN 978-3-319-23714-5.
↑ J. Stillwell, Mathematics and Its History, Springer, New York (2010). ISBN 978-1441960528.

Literature

Abramowitz, Milton; Stegun, Irene Ann, eds. (1983) [June 1964]. "Chapter 16". Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Applied Mathematics Series. 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications. pp. 567, 627. ISBN 978-0-486-61272-0. LCCN 64-60036. MR 0167642. LCCN 65-12253. See also chapter 18. (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

Hazewinkel, Michiel, ed. (2001) [1994], "Elliptic function", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
MAA, Translation of Abel's paper on elliptic functions.
Elliptic Functions and Elliptic Integrals on YouTube, lecture by William A. Schwalm (4 hours)
Johansson, Fredrik (2018). "Numerical Evaluation of Elliptic Functions, Elliptic Integrals and Modular Forms". arXiv:1806.06725 [cs.NA].

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[Cartan-1] Cartan, Henri (1995). Elementary Theory of Analytic Functions of One or Several Complex Variables. Dover Publications. p. 154. ISBN 9780486685434.

[Gray-2] 1 2 J. Gray, The Real and the Complex: A History of Analysis in the 19th Century, Springer, Heidelberg (2015). ISBN 978-3-319-23714-5.

[Stillwell-3] J. Stillwell, Mathematics and Its History, Springer, New York (2010). ISBN 978-1441960528.