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) |

(2) |

(3) |

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

(4) |

(5) |

(6) | |||

(7) | |||

(8) | |||

(9) |

So

(10) | |||

(11) |

Plugging in,

(12) |

(13) | |||

(14) |

then

(15) |

(16) |

(17) |

(18) |

(19) |

(20) |

(21) |

(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) |

(24) |

(25) |

(26) |

(27) |

(28) |

(29) |

(30) |

(31) |

(32) |

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) |

(38) |

(39) |

(40) |

(41) |

(42) |

(43) |

(44) |

(45) |

(46) |

(47) |

(48) |

(49) |

(50) |

Half-period identities include

(51) |

Multiplying through,

(52) |

(53) |

(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) |

(59) |

(60) |

(61) |

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

(62) |

(63) | |||

(64) | |||

(65) |

and the Invariants are

(66) | |||

(67) |

Here, .

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

(68) |

(69) |

(70) |

Use ,

(71) |

But and

(72) |

(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) |

(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.

© 1996-9

1999-05-26