Definitions

Weierstrass P function

Weierstrass's elliptic functions

In mathematics, Weierstrass's elliptic functions are elliptic functions that take a particularly simple form (cf Jacobi's elliptic functions); they are named for Karl Weierstrass. This class of functions are also referred to as P-functions and generally written using the symbol wp (a stylised letter p called Weierstrass p).
Symbol for Weierstrass P function

Definitions

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 left|f(z)right|=left|f(x+iy)right|=1;. 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 z and a lattice Lambda in the complex plane. Another is in terms of z and two complex numbers omega_1 and omega_2 defining a pair of generators, or periods, for the lattice. The third is in terms z and of a modulus tau in the upper half-plane. This is related to the previous definition by tau = omega_2/omega_1, which by the conventional choice on the pair of periods is in the upper half-plane. Using this approach, for fixed z the Weierstrass functions become modular functions of tau.

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

wp(z;omega_1,omega_2)=frac{1}{z^2}+ sum_{m^2+n^2 ne 0} left{ frac{1}{(z-momega_1-nomega_2)^2}- frac{1}{left(momega_1+nomega_2right)^2} right}.

Then Lambda=momega_1+nomega_2 are the points of the period lattice, so that

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

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

If tau is a complex number in the upper half-plane, then

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

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

wp(z;omega_1,omega_2) = wp(z/omega_1; omega_2/omega_1)/omega_1^2.

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

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)

where

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

There is 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 with respect to z, 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.

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= 60sum{}' Omega_{m,n}^{-4},qquad
g_3=140sum{}' Omega_{m,n}^{-6}.

(here Omega_{m,n}=momega_1+nomega_2 and a dashed summation refers to summation over all pairs of integers except m=n=0). The numbers g_2 and g_3 are known as the invariants — they are two terms out of the 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(ipitau) as

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

and

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

where sigma_a(k) 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 omega_1,omega_2, the invariants satisfy

g_2(omega_1,omega_2)= frac{pi^4}{12omega_1^4} left(
   theta_2(0,q)^8-theta_3(0,q)^4theta_2(0,q)^4+theta_3(0,q)^8
right) and
g_3(omega_1,omega_2)= frac{pi^6}{(2omega_1)^6} left[ frac{8}{27}left(theta_2(0,q)^{12}+theta_3(0,q)^{12}right)right.
left. -
frac{4}{9}left(theta_2(0,q)^4+theta_3(0,q)^4right)cdot
             theta_2(0,q)^4theta_3(0,q)^4
right]

where tau=omega_2/omega_1 is the half-period ratio and q=e^{pi itau} is the nome.

Special cases

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

Differential equation

With this notation, the wp function satisfies the following differential equation:
[wp'(z)]^2=4[wp(z)]^3-g_2wp(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=(2pi)^{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 {atau+b} {ctau+d}right) =
left(ctau+dright)^{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 cubic polynomial equation 4t^3-g_2t-g_3=0 with roots e_1, e_2, and e_3. If the discriminant Delta = g_{2}^{3} - 27 g_{3}^{2} is not zero, no two of these roots are equal. Since the quadratic term of this cubic polynomial is zero, the roots are related by the equation

e_1+e_2+e_3=0

The linear and constant coefficients (g2 and g3, respectively) are related to the roots by the equations

g_{2} = -4 left(e_{1} e_{2} + e_{1} e_{3} + e_{2} e_{3} right) = 2 left(e_{1}^{2} + e_{2}^{2} + e_{3}^{2} right)

g_{3} = 4 e_{1} e_{2} e_{3}

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

The half-periods ω1 and ω2 of Weierstrass' elliptic function are related to the roots

wp(omega_1)=e_1qquad wp(omega_2)=e_2qquad wp(omega_3)=e_3 where omega_3=-(omega_1+omega_2). Since the derivative of Weierstrass' elliptic function equals the above cubic polynomial of the function's value, wp'(omega_i)=0 for i=1,2,3; if the function's value equals a root of the polynomial, the derivative is zero.

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. If the roots are ordered as above (e_{1} > e_{2} > e_{3}), then the first half-period is completely real

omega_{1} = int_{e_{1}}^{infty} frac{dz}{sqrt{4z^{3} - g_{2}z - g_{3}}}

whereas the third half-period is completely imaginary

omega_{3} = i int_{-e_{3}}^{infty} frac{dz}{sqrt{4z^{3} - g_{2}z - g_{3}}}

Addition theorems

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

detbegin{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

detbegin{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}^2-2wp(z), unless 2z is a period.

The case with 1 a basic half-period

If 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-mtau)^2} - {1 over (n+mtau)^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 a double pole at each 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;ctau) = 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 by

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.

Relation to Jacobi elliptic functions

For numerical work, it is often convenient to calculate the Weierstrass elliptic function in terms of the Jacobi's elliptic functions. The basic relations are

wp(z) = e_{3} + frac{e_{1} - e_{3}}{mathrm{sn}^{2} w} = e_{2} + left(e_{1} - e_{3} right) frac{mathrm{dn}^{2} w}{mathrm{sn}^{2} w} = e_{1} + left(e_{1} - e_{3} right) frac{mathrm{cn}^{2} w}{mathrm{sn}^{2} w}

where e1-3 are the three roots described above and where the modulus k of the Jacobi functions equals

k equiv sqrt{frac{e_{2} - e_{3}}{e_{1} - e_{3}}}

and their argument w equals

w equiv z sqrt{e_{1} - e_{3}}.

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
  • Serge Lang, Elliptic Functions (1973), Addison-Wesley, ISBN 0-201-04162-6
  • E. T. Whittaker and G. N. Watson, A course of modern analysis, Cambridge University Press, 1952, chapters 20 and 21
  • Konrad Knopp, Funktionentheorie II (1947), Dover; Republished in English translation as Theory of Functions (1996), Dover ISBN 0-486-69219-1
  • Abramowitz and Stegun, chapter 18

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