A doubly periodic function with periods and such that

(1) |

- 1. The number of Poles in a cell is finite.
- 2. The number of Roots in a cell is finite.
- 3. The sum of Residues in any cell is 0.
- 4. Liouville's Elliptic Function Theorem: An elliptic function with no Poles in a cell is a constant.
- 5. The number of zeros of (the ``order'') equals the number of Poles of .
- 6. The simplest elliptic function has order two, since a function of order one would have a simple irreducible Pole, which would need to have a Nonzero residue. By property (3), this is impossible.
- 7. Elliptic functions with a single Pole of order 2 with Residue 0 are called Weierstraß Elliptic Functions. Elliptic functions with two simple Poles having residues and are called Jacobi Elliptic Functions.
- 8. Any elliptic function is expressible in terms of either Weierstraß Elliptic Function or Jacobi Elliptic Functions.
- 9. The sum of the Affixes of Roots equals the sum of the Affixes of the Poles.
- 10. An algebraic relationship exists between any two elliptic functions with the same periods.

The elliptic functions are inversions of the Elliptic Integrals. The two standard forms of these
functions are known as Jacobi Elliptic Functions and Weierstraß Elliptic Functions. Jacobi Elliptic Functions arise as solutions to differential equations of the form

(2) |

(3) |

**References**

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© 1996-9

1999-05-25