# $\mathrm{\wp}$-function

Let $L$ be a lattice on $\u2102$. The Weierstrass $\mathrm{\wp}$ function^{} associated to $L$ is given by

$$\mathrm{\wp}(z)=\frac{1}{{z}^{2}}+\sum _{w\in L\setminus \{0\}}\left(\frac{1}{{(z-w)}^{2}}-\frac{1}{{w}^{2}}\right).$$ |

The $\mathrm{\wp}$ function is meromorphic and analytic on $\u2102\setminus L$, whereas at each $w\in L$, it has an order $2$ pole. It is also an even function^{}, because $\mathrm{\wp}(z)=\mathrm{\wp}(-z)$.

Its derivative

$${\mathrm{\wp}}^{\prime}(z)=-2\sum _{w\in L}\frac{1}{{(z-w)}^{3}}$$ |

is also an odd, meromorphic, and elliptic function^{}, analytic at $\u2102\setminus L$ and having order $3$ poles at each $w\in L$.

The functions $\mathrm{\wp}$ and ${\mathrm{\wp}}^{\prime}$ form together a generator set for the field of elliptic functions associated to the lattice $L$.

Title | $\mathrm{\wp}$-function |
---|---|

Canonical name | wpfunction |

Date of creation | 2013-03-22 14:51:54 |

Last modified on | 2013-03-22 14:51:54 |

Owner | drini (3) |

Last modified by | drini (3) |

Numerical id | 10 |

Author | drini (3) |

Entry type | Definition |

Classification | msc 33E05 |

Synonym | $\mathrm{\wp}$ |

Synonym | Weierstrass $\mathrm{\wp}$ function |

Synonym | Weierstrass p-function |

Synonym | p-Weierstrass |

Synonym | Weierstrass $\mathrm{\wp}$-function |

Related topic | EllipticCurve |

Related topic | EllipticFunction |

Related topic | ExamplesOfEllipticFunctions |