Weierstrass's elliptic functions

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In mathematics, Weierstrass's elliptic functions are a standard type of elliptic functions (the other is the Jacobi's elliptic functions). They are named for Karl Weierstrass.

Contents

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Definitions

Image:WeierstrassP.png

Weierstrass P function defined over a subset of the complex plane using a standard visualization technique in which white corresponds to a pole, black to a zero, and maximal saturation to <math>\left|f(z)\right|=\left|f(x+iy)\right|=1\;.</math> Note the regular lattice of poles, and two interleaving lattices of zeroes.

The Weierstrass elliptic function can be defined in three closely related ways, each of which possesses certain advantages. One is as a function of a complex variable <math>z</math> and a lattice <math>\Lambda</math> in the complex plane. Another is in terms of <math>z</math> and two complex numbers <math>\omega_1</math> and <math>\omega_2</math> defining a pair of generators, or periods, for the lattice. The third is in terms <math>z</math> and of a modulus <math>\tau</math> in the upper half-plane. This is related to the previous definition by <math>\tau = \omega_2/\omega_1</math>, which by the conventional choice on the pair of periods is in the upper half-plane. Using this approach, for fixed <math>z</math> the Weierstrass functions become modular functions of <math>\tau</math>.

In terms of the two periods, Weierstrass's elliptic function is an elliptic function with periods <math>\omega_1</math> and <math>\omega_2</math> defined as

<math>

\wp(z;\omega_1,\omega_2)=\frac{1}{z^2}+ \sum_{m^2+n^2 \ne 0} \left\{ \frac{1}{(z-m\omega_1-n\omega_2)^2}- \frac{1}{\left(m\omega_1+n\omega_2\right)^2} \right\} </math>

Then <math>\Lambda=m\omega_1+n\omega_2</math> are the points of the period lattice, so that

<math>\wp(z;\Lambda)=\wp(z;\omega_1,\omega_2)</math>

for any pair of generators of the lattice defines the Weierstrass function as a function of a complex variable and a lattice.

If <math>\tau</math> is a complex number in the upper half-plane, then

<math>\wp(z;\tau) = \wp(z;1,\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}.</math>

The above sum is homogeneous of degree minus two, from which we may define the Weierstrass <math>\wp</math> function for any pair of periods, as

<math>\wp(z;\omega_1,\omega_2) = \wp(z/\omega_1; \omega_2/\omega_1)/\omega_1^2</math>.

We may compute <math>\wp</math> very rapidly in terms of theta functions; because these converge so quickly, this is a more expeditious way of computing <math>\wp</math> than the series we used to define it. The formula here is

<math>\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)</math>

where

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

There is a second order pole at each point of the period lattice (including the origin). With these definitions, <math>\wp(z)</math> is an even function and its derivative with respect to <math>z</math>, <math>\wp'</math>, 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.

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Invariants

Image:Gee three real.jpeg Image:Gee three imag.jpeg

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

<math>

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

<math>g_2= 60\sum{}' \Omega_{m,n}^{-4},\qquad 
      g_3=140\sum{}' \Omega_{m,n}^{-6}.</math>

(here a dashed summation refers to summation over all pairs of integers except <math>m=n=0</math> and <math>\Omega_{m,n}=z-m\omega_1-n\omega_2</math>). The numbers <math>g_2</math> and <math>g_3</math> are known as the invariants — they are two terms out of the Eisenstein series. (Abramowitz and Stegun restrict themselves to the case of real <math>g_2</math> and <math>g_3</math>, stating that this case "seems to cover most applications"; this may be true from the point of view of applied mathematics. If <math>\omega_1</math> is real and <math>\omega_2</math> pure imaginary, or if <math>\omega_1=\overline{\omega_2}</math>, the invariants are real).

Note that <math>g_2</math> and <math>g_3</math> are homogeneous functions of degree -4 and -6; that is,

<math>g_2(\lambda \omega_1, \lambda \omega_2) = \lambda^{-4} g_2(\omega_1, \omega_2)</math>

and

<math>g_3(\lambda \omega_1, \lambda \omega_2) = \lambda^{-6} g_3(\omega_1, \omega_2)</math>.

Thus, by convention, one frequently writes <math>g_2</math> and <math>g_3</math> in terms of the half-period ratio <math>\tau=\omega_2/\omega_1</math> and take <math>\tau</math> to lie in the upper half-plane. Thus, <math>g_2(\tau)=g_2(1, \omega_2/\omega_1)</math> and <math>g_3(\tau)=g_3(1, \omega_2/\omega_1)</math>.

The Fourier series for <math>g_2</math> and <math>g_3</math> can be written in terms of the square of the nome <math>q=\exp(i\pi\tau)</math> as

<math>g_2(\tau)=\frac{4\pi^4}{3} \left[ 1+ 240\sum_{k=1}^\infty \sigma_3(k) q^{2k} \right] </math>

and

<math>g_3(\tau)=\frac{8\pi^6}{27} \left[ 1- 504\sum_{k=1}^\infty \sigma_5(k) q^{2k} \right] </math>

where <math>\sigma_a(k)</math> is the divisor function. This formula may be re-written in terms of Lambert series.

The invariants may be expressed in terms of Jacobi's theta functions. This method is very convenient for numerical calculation: the theta functions converge very quickly. In the notation of Abramowitz and Stegun, but denoting the primitive half-periods by <math>\omega_1,\omega_2</math>, the invariants satisfy

<math>

g_2(\omega_1,\omega_2)= \frac{\pi^4}{12\omega_1^4} \left( 

   \theta_2(0,q)^8-\theta_3(0,q)^4\theta_2(0,q)^4+\theta_3(0,q)^8

\right) </math> and

<math>

g_3(\omega_1,\omega_2)= \frac{\pi^6}{(2\omega_1)^6} \left[ 

  \frac{8}{27}\left(\theta_2(0,q)^{12}+\theta_3(0,q)^{12}\right)\right.

</math>

<math>\left. - 
  \frac{4}{9}\left(\theta_2(0,q)^4+\theta_3(0,q)^4\right)\cdot
             \theta_2(0,q)^4\theta_3(0,q)^4

\right] </math> where <math>\tau=\omega_2/\omega_1</math> is the half-period ratio and <math>q=e^{\pi i\tau}</math> is the nome.

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Special cases

If the invariants are <math>g_2=0</math>, <math>g_3=1</math>, then this is known as the equianharmonic case; <math>g_2=1</math>, <math>g_3=0</math> is the lemniscatic case.

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Differential equation

With this notation, the <math>\wp</math> function satisfies the following differential equation:

<math>

[\wp'(z)]^2=4[\wp(z)]^3-g_2\wp(z)-g_3,</math> where dependence on <math>\omega_1</math> and <math>\omega_2</math> is suppressed.

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Integral equation

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

<math>u = \int_y^\infty \frac {ds} {\sqrt{4s^3 - g_2s -g_3}}</math>.

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

<math>y=\wp(u)</math>.

The above follows directly by integrating the differential equation.

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Modular discriminant

Image:Discriminant real part.jpeg

The modular discriminant <math>\Delta</math> is defined as

<math>

\Delta=g_2^3-27g_3^2.</math>

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 <math>\Delta=(2\pi)^{12}\eta^{24}</math> where <math>\eta</math> 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

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

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

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The constants e1, e2 and e3

Consider the algebraic equation <math>4t^3-g_2t-g_3=0</math>, and name its roots <math>e_1</math>, <math>e_2</math>, and <math>e_3</math>. It can be shown from the non-vanishing of the discriminant that no two of these three are equal.

Algebraic considerations show that <math>e_1+e_2+e_3=0</math>.

In the case of real invariants, the sign of <math>\Delta</math> determines the nature of the roots. If <math>\Delta>0</math>, all three are real and it is conventional to name them so that <math>e_1>e_2>e_3</math>. If <math>\Delta<0</math>, it is conventional to write <math>e_1=-\alpha+\beta i</math> (where <math>\alpha\geq 0</math>, <math>\beta>0</math>), whence <math>e_3=\overline{e_1}</math> and <math>e_2</math> is real and non-negative. We also have

<math>

\wp(\omega_1)=e_1\qquad \wp(\omega_2)=e_2\qquad \wp(\omega_3)=e_3 </math> where <math>\omega_3=-\omega_1-\omega_2</math>. Also, <math>\wp'(\omega_i)=0</math> for <math>i=1,2,3</math>.

If <math>g_2</math> and <math>g_3</math> are real and <math>\Delta>0</math>, the <math>e_i</math> are all real, and <math>\wp()</math> is real on the perimeter of the rectangle with corners <math>0</math>, <math>\omega_3</math>, <math> \omega_1+\omega_3</math>, and <math>\omega_1</math>.

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Addition theorems

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

<math>

\det\begin{bmatrix} \wp(z) & \wp'(z) & 1\\ \wp(y) & \wp'(y) & 1\\ \wp(z+y) & -\wp'(z+y) & 1 \end{bmatrix}=0</math>

(a symmetrical version would be

<math>

\det\begin{bmatrix} \wp(u) & \wp'(u) & 1\\ \wp(v) & \wp'(v) & 1\\ \wp(w) & \wp'(w) & 1 \end{bmatrix}=0</math> where <math>u+v+w=0</math>).

Also

<math>

\wp(z+y)=\frac{1}{4} \left\{ \frac{\wp'(z)-\wp'(y)}{\wp(z)-\wp(y)} \right\}^2 -\wp(z)-\wp(y).</math>

and the duplication formula

<math>

\wp(2z)= \frac{1}{4}\left\{ \frac{\wp(z)}{\wp'(z)}\right\}^2-2\wp(z),</math> unless <math>2z</math> is a period.

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The case with 1 a basic half-period

If <math>\omega_1=1</math>, much of the above theory becomes simpler; it is then conventional to write <math>\tau</math> for <math>\omega_2</math>. For a fixed τ in the upper half-plane, so that the imaginary part of τ is positive, we define the Weierstrass <math>\wp</math> function by:

<math>\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}</math>

The sum extends over the lattice {n+mτ : n and m in Z} with the origin omitted. Here we regard τ as fixed and <math>\wp</math> as a function of <math>z</math>; fixing <math>z</math> and letting τ vary leads into the area of elliptic modular functions.

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General theory

<math>\wp</math> 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 <math>\wp</math> satisfies

<math>\wp(z+1) = \wp(z+\tau) = \wp(z)</math>

The above sum is homogeneous of degree minus two, and if <math>c</math> is any non-zero complex number,

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

from which we may define the Weierstrass <math>\wp</math> 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 <math>\wp</math> by

<math>\wp'^2 = \wp^3 - g_2 \wp - g_3</math>

where <math>g_2</math> and <math>g_3</math> depend only on τ, being modular forms. The equation

<math>Y^2 = X^3 - g_2 X - g_3</math>

defines an elliptic curve, and we see that (<math>\wp</math>,<math>\wp'</math>) 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

<math>\Bbb{C}(\wp, \wp')</math>,

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 <math>e_1</math>, <math>e_2</math>, and <math>e_3</math> of the equation <math>X^3 - g_2 X - g_3</math> depend on τ and can be expressed in terms of theta functions; we have

<math>e_1(\tau) = {\pi^2 \over {3}}(\vartheta^4(0;\tau) + \vartheta_{01}^4(0;\tau))</math>
<math>e_2(\tau) = -{\pi^2 \over {3}}(\vartheta^4(0;\tau) + \vartheta_{10}^4(0;\tau))</math>
<math>e_3(\tau) = {\pi^2 \over {3}}(\vartheta_{10}^4(0;\tau) - \vartheta_{01}^4(0;\tau))</math>

Since <math>g_2 = -4(e_1e_2+e_2e_3+e_3e_1)</math> and <math>g_3 = 4e_1e_2e_3</math> we have these in terms of theta functions also.

We may also express <math>\wp</math> in terms of theta functions; because these converge very rapidly, this is a more expeditious way of computing <math>\wp</math> than the series we used to define it.

<math>\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)</math>

The function <math>\wp</math> has two zeroes (modulo periods) and the function <math>\wp'</math> has three. The zeroes of <math>\wp'</math> are easy to find: since <math>\wp'</math> 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 <math>\wp</math> 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 <math>\wp</math> 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.

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References

it:Funzioni ellittiche di Weierstrass

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