Weierstrass Elliptic Function

The Weierstrass elliptic function ℘(z) is defined by the following infinite series:

               -2     ∞                  -2              -2
℘(z;ω ; ω ) = z   +   ∑'   {(z-2mω -2nω )   - (2mω -2nω )  }
     1   2          m,n=-∞        1    2          1    2

(apologies to those whose browsers can't display the symbols, the ℘ symbol is supposed to be a script-P, which has been traditionally used for this purpose). Here, ω₁ and ω₂ are the half-periods of the elliptic function (see the elliptic function node for constraints on what values these parameters are allowed to take), as all elliptic functions are doubly periodic in the complex plane. The prime next to the summation simply means that terms in the summation that would have zero denominators should be ignored. Note that it is an even function, because &weierp(-z) produces the same terms in the series, just with a different ordering, and it possesses a single second-order pole at the origin.

For convenience, we can define:

Ω_m,n = 2mω₁ - 2nω₂

so we can write the above series more compactly as:

        -2    ∞              -2    -2
℘(z) = z +   ∑'  {(z - Ω   )   - Ω    }
           m,n=-∞        m,n       m,n

An alternative series representation that converges more rapidly is

                      2          ∞     2  z-2nω      
℘(z; ω ; ω ) = (π/2ω )  { -1/3 + ∑ csc (      2   )
      1   2         1           n=-∞     ------- π
                                           2ω
                                             1
         ∞      2  nω
      -  ∑'  csc  (  2  )
        n=-∞       --- π
                    ω
                     1

A power series representation of ℘(z) can be obtained by using the invariants g₂ and g₃ of the WEF, which are related to the half-periods ω₁ and ω₂ using the Eisenstein series:

             ∞    -4
g    = 60    ∑'  Ω
 2        m,n=-∞  m,n

             ∞    -6
g    = 140   ∑'  Ω
 3        m,n=-∞  m,n

where Ω_m,n is defined as above. The power series is:

        -2    ∞     2n-2
℘(z) = z   +  ∑  c z
             n=2  n

where c₂ = g₂/20 and c₃ = g₃/28. The other power series coefficients are given by the formula:

         1      k-2
c = -----------  ∑  c c
 k  (2k+1)(k-3) m=2  m k-m

The Weierstrass elliptic function also satisfies an ordinary differential equation:

 df 2    3
(--) = 4f - g f  - g
 dz          2      3

The solution, provided that numbers ω₁ and ω₂ can be found that satisfy the Eisenstein series above, is f(z) = ℘(±z + α ; ω₁ ; &omega₂), where α depends on the initial conditions or boundary values of the differential equation.

The elliptic integral:

     ∞   3            -1/2
z = ∫ (4t  - g t - g )     dt
     ζ        2     3

that determines z in terms of ζ, where the path of integration is any curve that does not pass through a zero of 4t³ - g₂t - g₃ can be inverted to produce ℘(z), as ζ = &weierp(z), or:

     ∞      3            -1/2
z = ∫    (4t  - g t - g )     dt
     ℘(z)        2     3

The derivative of ℘(z), obtained by term-by-term differentiation of the series, consists is:

           -3       ∞             -3
P'(z) = -2z   - 2   ∑   (z - ω   )
                 m,n=-∞       m,n

which is itself an elliptic function with a pole of order 3 at the origin, and is an odd function.

The Weierstrass elliptic functions also satisfy an addition formula:

           1  ℘'(a) - ℘'(b)
℘(a + b) = - (-------------) - ℘(a) - ℘(b)
           4  ℘(a)  -  ℘(b)

and a duplication formula:

        1  ℘''(z)
℘(2z) = - (------) - 2℘(z)
        4  ℘'(z)

Another useful property of ℘(z) is that the values &weierp(ω₁)=e₁, &weierp(ω₂)=e₂, and &weierp(ω₃)=e₃ (where ω₃ = - ω₁ - &omega₂) are all unequal, and they are the roots of the cubic equation: 4t³ - g₂t - g₃ = 0.

The Weierstrass elliptic functions are highly useful in algebraic geometry and in particular the study of elliptic curves as they provide a way of transforming the topological torus-shaped region where the solutions to the curve can be found into an algebraic representation of the curve, since the doubly periodic properties of the function cause it to have topolgy of a torus too. It is, however, clumsy to work with numerically, so for more numerical applications the Jacobi elliptic functions are used more often, but for those engaged in the theoretical foundations of elliptic functions it seems the Weierstrass formulation is a better choice.

These functions are named for the German mathematician Karl Weierstrass, who first discovered them and illustrated some of their properties in the early 1840's.

Sources:

E.T. Whittaker and G.N. Watson, A Course of Modern Analysis.

Harris Hancock, Lectures on the Theory of Elliptic Functions.