# 魏爾斯特拉斯橢圓函數

## 定義

$\wp(z; \Lambda)=\frac{1}{z^2}+ \sum_{(m,n) \ne (0,0)} \left\{ \frac{1}{(z-m\omega_1-n\omega_2)^2}- \frac{1}{\left(m\omega_1+n\omega_2\right)^2} \right\}$

$\wp(z; \Lambda) = \wp(z;\tau) =\frac{1}{z^2} + \sum_{n^2+m^2 \ne 0}{1 \over (z-n-m\tau)^2} - {1 \over (n+m\tau)^2}$

$\wp(z; \mathbb{Z}\omega_1 \oplus \mathbb{Z}\omega_2) = \frac{\wp(\frac{z}{\omega_1}; \frac{\omega_2}{\omega_1})}{\omega_1^2} \quad (\mathrm{Im}(\frac{\omega_1}{\omega_2}) > 0)$

$\wp(z; \tau) = \pi^2 \vartheta^2(0;\tau) \vartheta_{10}^2(0;\tau){\vartheta_{01}^2(z;\tau) \over \vartheta_{11}^2(z;\tau)}-{\pi^2 \over {3}}\left[\vartheta^4(0;\tau) + \vartheta_{10}^4(0;\tau)\right]$
• 在週期格中的每個點，$\wp$ 有二階極點
• $\wp$ 是偶函數。
• 複導函數 $\wp'$ 是奇函數。

## 加法定理

$\det\begin{pmatrix} \wp(z) & \wp'(z) & 1\\ \wp(y) & \wp'(y) & 1\\ \wp(z+y) & -\wp'(z+y) & 1 \end{pmatrix}=0$

$\det\begin{pmatrix} \wp(u) & \wp'(u) & 1\\ \wp(v) & \wp'(v) & 1\\ \wp(w) & \wp'(w) & 1 \end{pmatrix}=0$

$\wp(z+y)=\frac{1}{4} \left\{ \frac{\wp'(z)-\wp'(y)}{\wp(z)-\wp(y)} \right\}^2 -\wp(z)-\wp(y).$

$\wp(2z)= \frac{1}{4}\left\{ \frac{\wp''(z)}{\wp'(z)}\right\}^2-2\wp(z),$

## 微分方程與積分方程

$g_2 := 60 \sum_{w \in \Lambda}' w^{-4}$
$g_3 := 120 \sum_{w \in \Lambda}' w^{-6}$

$\wp'(z)^2=4\wp(z)^3-g_2\wp(z)-g_3$

$z \mapsto (\wp(z),\wp'(z))$ 給出了從複環面 $\mathbb{C}/\Lambda$ 映至三次複射影曲線 $y^2 = 4x^3 - g_2 x -g_3$ 的全純映射；可證明這是同構。

$z_1 - z_2 = \int_{\wp(z_1)}^{\wp(z_2)} \frac {ds} {\sqrt{4s^3 - g_2s -g_3}}$

## 模判別式

$\Delta=g_2^3-27g_3^2.$

## 文獻

• Naum Illyich Akhiezer, Elements of the Theory of Elliptic Functions, (1970) Moscow, translated into English as AMS Translations of Mathematical Monographs Volume 79 (1990) AMS, Rhode Island ISBN 0-8218-4532-2
• Tom M. Apostol, Modular Functions and Dirichlet Series in Number Theory, Second Edition (1990), Springer, New York ISBN 0-387-97127-0 (See chapter 1.)
• K. Chandrasekharan, Elliptic functions (1980), Springer-Verlag ISBN 0-387-15295-4
• Serge Lang, Elliptic Functions (1973), Addison-Wesley, ISBN 0-201-04162-6
• E. T. Whittaker and G. N. Watson, A course of modern analysis (1952), Cambridge University Press, chapters 20 and 21