weierstrass's elliptic functions

In mathematics, Weierstrass introduced some particular elliptic functions that have become the basis for the most standard notations used.

Definitions

Consider two complex numbers

\omega_1

and

\omega_2

defining a lattice. There are some significant choices of convention, and the literature is not consistent in its usage. It is common to name these constants so that

\omega_2/\omega_1

has a positive imaginary part. As defined below, the two numbers serve as half-periods. Compare the trigonometric usage of 2π. Then Weierstrass's elliptic function is an elliptic function with periods

2\omega_1

and

2\omega_2

is defined as

\wp(z;\omega_1,\omega_2)=\frac{1}{z^2}+ \sum{}' \left\{ \frac{1}{(z-2m\omega_1-2n\omega_2)^2}- \frac{1}{\left(2m\omega_1+2n\omega_2\right)^2} \right\} where

\sum{}'

represents the sum over all pairs of integers

m

and

n

except

m=n=0

. It is usual to write

\Omega_{m,n}=2m\omega_1+2n\omega_1

, the points of the period lattice, so that

\wp(z;\omega_1,\omega_2)=

z^{-2}+\sum{}'\left\{(z-\Omega_{m,n})^{-2}-\Omega_{m,n}^{-2} \right\}. There is thus a second order pole at each point of the period lattice (including the origin). With these definitions,

\wp(z)

is an even function and its derivative

\wp'

an odd function. Further development of the theory of elliptic functions shows that the condition on Weierstrass's function (correctly called pe) is determined up to addition of a constant and multiplication by a non-zero constant by the condition on the poles alone, amongst all meromorphic functions with the given period lattice. It can be shown that

\wp(z;\omega_1,\omega_2)= \left(\frac{\pi}{2\omega_1}\right)^2\leftcosec}^2\left(\frac{z-2n\omega_2}{2\omega_1}\pi\right)- \sum_{n=-\infty}^{n=+\infty}{}'{\rm cosec}^2\frac{n\omega_2}{\omega_1}\pi\right, which converges faster than the other formula given above.

Invariants

If points close to the origin are considered the appropriate Laurent series is

\wp(z;\omega_1,\omega_2)=z^{-2}+\frac{1}{20}g_2z^2+\frac{1}{28}g_3z^4+O(z^6) where

g_2= 60\sum{}' \Omega_{m,n}^{-4},\qquad

        g_3=140\sum{}' \Omega_{m,n}^{-6}.

The numbers

g_2

and

g_3

are known as the invariants — they are special cases of Eisenstein series. (Abramowitz and Stegun restrict themselves to the case of real

g_2

and

g_3

, stating that this case "seems to cover most applications"; this may be true from the point of view of applied mathematics. If

\omega_1

is real and

\omega_2

pure imaginary, or if

\omega_1=\overline{\omega_2}

, the invariants are real). Note that

g_2

and

g_3

are homogeneous functions of degree -4 and -6; that is,

g_2(\lambda \omega_1, \lambda \omega_2) = \lambda^{-4} g_2(\omega_1, \omega_2)

and

g_3(\lambda \omega_1, \lambda \omega_2) = \lambda^{-6} g_3(\omega_1, \omega_2)

Thus, by convention, one frequently writes

g_2

and

g_3

in terms of the half-period ratio

\tau=\omega_2/\omega_1

and take

\tau

to lie in the upper half plane. Thus,

g_2(\tau)=g_2(1, \omega_2/\omega_1)

and

g_3(\tau)=g_3(1, \omega_2/\omega_1)

. The Fourier series for

g_2

and

g_3

can be written in terms of the square of the nome

q=\exp(i\pi\tau)

g_2(\tau)=\frac{4\pi^4}{3} \left 240\sum_{k=1}^\infty \sigma_3(k) q^{2k} \right

and

g_3(\tau)=\frac{8\pi^6}{27} \left 504\sum_{k=1}^\infty \sigma_5(k) q^{2k} \right

where

\sigma_a(k)

is the divisor function. In pratical calculations, these are best re-written as Lambert series.

Differential equation

With this notation, the

\wp

function satisfies the following differential equation:

\wp'(z)^2=4\wp(z)^3-g_2\wp(z)-g_3, where dependence on

\omega_1

and

\omega_2

is suppressed.

Integral equation

The Weierstrass elliptic function can be given as the inverse of an elliptic integral. Let

u = \int_y^\infty \frac {ds} {\sqrt{4s^3 - g_2s -g_3}}

Here, g₂ and g₃ are taken as constants. Then one has

y=\wp(u)

The above follows directly by integrating the differential equation.

Modular discriminant

The modular discriminant

\Delta

is defined as

\Delta=g_2^3-27g_3^2. This is studied in its own right, as a cusp form, in modular form theory (that is, as a function of the period lattice). Note that

\Delta=(2\pi)^{12}\eta^{24}

where

\eta

is the Dedekind eta function. The discriminant is a modular form of weight 12. That is, under the action of the modular group, it transforms as

\Delta \left( \frac {a\tau+b} {c\tau+d}\right) =

\left(c\tau+d\right)^{12} \Delta(\tau) with τ being the half-period ratio, and a,b,c and d being integers, with ad-bc=1.

The constants e₁, e₂ and e₃

Consider the algebraic equation

4t^3-g_2t-g_3=0

, and name its roots

e_1

e_2

, and

e_3

. It can be shown from the non-vanishing of the discriminant that no two of these three are equal. (When all three are real, it is conventional to name them so that

e_1>e_2>e_3

; from the reality of the invariants it follows only that one is real.) We also have

\wp(\omega_1)=e_1\qquad \wp(\omega_2)=e_2\qquad \wp(\omega_3)=e_3 where

\omega_3=-\omega_1-\omega_2

. Also,

\wp'(\omega_i)=0

for

i=1,2,3

. If

g_2

and

g_3

are real and

\Delta>0

, the

e_i

are all real, and

\wp()

is real on the perimeter of the rectangle with corners

0

\omega_3

\omega_1+\omega_3

, and

\omega_1

Addition theorems

The Weierstrass elliptic functions have several properties that may be proved:

\det\begin{bmatrix} \wp(z) & \wp'(z) & 1\\ \wp(y) & \wp'(y) & 1\\ \wp(z+y) & -\wp'(z+y) & 1 \end{bmatrix}=0 (a symmetrical version would be

\det\begin{bmatrix} \wp(u) & \wp'(u) & 1\\ \wp(v) & \wp'(v) & 1\\ \wp(w) & -\wp'(w) & 1 \end{bmatrix}=0 where

u+v+w=0

). Also

\wp(z+y)=\frac{1}{4} \left\{ \frac{\wp'(z)-\wp'(y)}{\wp(z)-\wp(y)} \right\}^2 -\wp(z)-\wp(y). and the duplication formula

\wp(2z)= \frac{1}{4}\left\{ \frac{\wp''(z)}{\wp'(z)}\right\}^2-2\wp(z), unless

2z

is a period.

The case with 1 a basic half-period

\omega_1=1

, much of the above theory becomes simpler; it is then conventional to write

\tau

for

\omega_2

. For a fixed τ in the upper half plane, so that the imaginary part of τ is positive, we define the Weierstrass $\wp$ function by:

\wp(z;\tau) =\frac{1}{z^2} + \sum_{n^2+m^2 \ne 0}{1 \over (z-n-m\tau)^2} - {1 \over (n+m\tau)^2}

The sum extends over the lattice {n+mτ : n and m in Z} with the origin omitted. Here we regard τ as fixed and

\wp

as a function of

z

; fixing

z

and letting τ vary leads into the area of elliptic modular functions.

General theory

\wp

is a meromorphic function in the complex plane with poles at the lattice points. It is doubly periodic with periods 1 and τ; this means that

\wp

satisfies

\wp(z+1) = \wp(z+\tau) = \wp(z)

The above sum is homogeneous of degree minus two, and if

c

is any non-zero complex number,

\wp(cz;c\tau) = \wp(z;\tau)/c^2

from which we may define the Weierstrass

\wp

function for any pair of periods. We also may take the derivative (of course, with respect to z) and obtain a function algebraically related to

\wp

\wp'^2 = \wp^3 - g_2 \wp - g_3

where

g_2

and

g_3

depend only on τ, being modular forms. The equation

Y^2 = X^3 - g_2 X - g_3

defines an elliptic curve, and we see that (

\wp

\wp'

) is a parametrization of that curve. The totality of meromorphic doubly periodic functions with given periods defines an algebraic function field, associated to that curve. It can be shown that this field is

\Bbb{C}(\wp, \wp')

so that all such functions are rational functions in the Weierstrass function and its derivative. We can also wrap a single period parallelogram into a torus, or donut-shaped Riemann surface, and regard the elliptic functions associated to a given pair of periods to be functions defined on that Riemann surface. The roots

e_1

e_2

, and

e_3

of the equation

X^3 - g_2 X - g_3

depend on τ and can be expressed in terms of theta functions; we have

e_1(\tau) = {\pi^2 \over {3}}(\vartheta^4(0;\tau) + \vartheta_{01}^4(0;\tau))

e_2(\tau) = -{\pi^2 \over {3}}(\vartheta^4(0;\tau) + \vartheta_{10}^4(0;\tau))

e_3(\tau) = {\pi^2 \over {3}}(\vartheta_{10}^4(0;\tau) - \vartheta_{01}^4(0;\tau))

Since

g_2 = -4(e_1e_2+e_2e_3+e_3e_1)

and

g_3 = 4e_1e_2e_3

we have these in terms of theta functions also. We may also express

\wp

in terms of theta functions; because these converge very rapidly, this is a more expeditious way of computing

\wp

than the series we used to define it.

\wp(z; \tau) = \pi^2 \vartheta^2(0;\tau) \vartheta_{10}^2(0;\tau){\vartheta_{01}^2(z;\tau) \over \vartheta_{11}^2(z;\tau)} + e_2(\tau)

The function

\wp

has two zeroes (modulo periods) and the function

\wp'

has three. The zeroes of

\wp'

are easy to find: since

\wp'

is an odd function they must be at the half-period points. On the other hand it is very difficult to express the zeroes of

\wp

by closed formula, except for special values of the modulus (e.g. when the period lattice is the Gaussian integers). An expression was found, by Zagier and Eichler. The Weierstrass theory also includes the Weierstrass zeta-function, which is an indefinite integral of

\wp

and not doubly-periodic, and a theta function called the Weierstrass sigma-function, of which his zeta-function is the log-derivative. The sigma-function has zeroes at all the period points (only), and can be expressed in terms of Jacobi's functions. This gives one way to convert between Weierstrass and Jacobi notations. The Weierstrass sigma-function is an entire function; it played the role of 'typical' function in a theory of random entire functions of J. E. Littlewood.

References

Naum Illyich 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, Second Edition (1990), Springer, New York ISBN 0-387-97127-0 (See chapter 1.)
K. Chandrasekharan, Elliptic functions (1980), Springer-Verlag ISBN 0-387-15295-4
Abramowitz and Stegun, chapter 18

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