## Weierstraß Elliptic Function

The Weierstraß elliptic functions are elliptic functions which, unlike the Jacobi Elliptic Functions, have a second-order Pole at . The above plots show the Weierstraß elliptic function and its derivative for invariants (defined below) of and . Weierstraß elliptic functions are denoted and can be defined by

 (1)

Write . Then this can be written
 (2)

An equivalent definition which converges more rapidly is

 (3)

is an Even Function since gives the same terms in a different order. To specify completely, its periods or invariants, written and , respectively, must also be specified.

The differential equation from which Weierstraß elliptic functions arise can be found by expanding about the origin the function .

 (4)

But and the function is even, so and
 (5)

Taking the derivatives
 (6) (7) (8) (9)

So
 (10) (11)

Plugging in,
 (12)

Define the Invariants
 (13) (14)

then
 (15)

 (16)

Now cube (15) and square (16)
 (17)

 (18)

Taking (18) minus (17) cancels out the term, giving

 (19)

 (20)

But, from (5)
 (21)

so and (20) can be written
 (22)

The Weierstraß elliptic function is analytic at the origin and therefore at all points congruent to the origin. There are no other places where a singularity can occur, so this function is an Elliptic Function with no Singularities. By Liouville's Elliptic Function Theorem, it is therefore a constant. But as , , so

 (23)

The solution to the differential equation
 (24)

is therefore given by , providing that numbers and exist which satisfy the equations defining the Invariants. Writing the differential equation in terms of its roots , , and ,
 (25)

 (26)

 (27)

 (28)

 (29)

 (30)

Now take (30) divided by 4 plus [(30) divided by 4] quantity squared,

 (31)

 (32)

The term on the right is half the Schwarzian Derivative.

The Derivative of the Weierstraß elliptic function is given by

 (33)

This is an Odd Function which is itself an elliptic function with pole of order 3 at . The Integral is given by
 (34)

A duplication formula is obtained as follows.

 (35)

A general addition theorem is obtained as follows. Given

 (36)

 (37)

with zero and where , find the third zero . Consider . This has a pole of order three at , but the sum of zeros () equals the sum of poles for an Elliptic Function, so and .
 (38)

 (39)

Combining (36), (37), and (39) gives
 (40)

so
 (41)

Defining where and gives the symmetric form
 (42)

To get the expression explicitly, start again with
 (43)

where .
 (44)

But , so
 (45)

The solutions are given by
 (46)

But the sum of roots equals the Coefficient of the squared term, so
 (47)

 (48)

 (49)

 (50)

Half-period identities include

 (51)

Multiplying through,
 (52)

 (53)

which gives
 (54)

From Whittaker and Watson (1990, p. 445),

 (55)

The function is Homogeneous,

 (56)

 (57)

To invert the function, find and of when given . Let , , and be the roots such that is not a Real Number or . Determine the Parameter from

 (58)

Now pick
 (59)

As long as , the periods are then
 (60)

 (61)

Weierstraß elliptic functions can be expressed in terms of Jacobi Elliptic Functions by

 (62)

where
 (63) (64) (65)

and the Invariants are
 (66) (67)

Here, .

An addition formula for the Weierstraß elliptic function can be derived as follows.

 (68)

Use
 (69)

so
 (70)

Use ,

 (71)

But and
 (72)

so
 (73)

The periods of the Weierstraß elliptic function are given as follows. When and are Real and , then , , and are Real and defined such that .

 (74) (75) (76)

The roots of the Weierstraß elliptic function satisfy

 (77)

 (78)

 (79)

where . The s are Roots of and are unequal so that . They can be found from the relationships
 (80)

 (81)

 (82)

References

Abramowitz, M. and Stegun, C. A. (Eds.). Weierstrass Elliptic and Related Functions.'' Ch. 18 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 627-671, 1972.

Fischer, G. (Ed.). Plates 129-131 in Mathematische Modelle/Mathematical Models, Bildband/Photograph Volume. Braunschweig, Germany: Vieweg, pp. 126-128, 1986.

Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, 1990.