# 魏爾斯特拉斯橢圓函數

## 定義

${\displaystyle \wp (z;\Lambda )={\frac {1}{z^{2}}}+\sum _{(m,n)\neq (0,0)}\left\{{\frac {1}{(z-m\omega _{1}-n\omega _{2})^{2}}}-{\frac {1}{\left(m\omega _{1}+n\omega _{2}\right)^{2}}}\right\}}$

${\displaystyle \wp (z;\Lambda )=\wp (z;\tau )={\frac {1}{z^{2}}}+\sum _{n^{2}+m^{2}\neq 0}{1 \over (z-n-m\tau )^{2}}-{1 \over (n+m\tau )^{2}}}$

${\displaystyle \wp (z;\mathbb {Z} \omega _{1}\oplus \mathbb {Z} \omega _{2})={\frac {\wp ({\frac {z}{\omega _{1}}};{\frac {\omega _{2}}{\omega _{1}}})}{\omega _{1}^{2}}}\quad (\mathrm {Im} ({\frac {\omega _{1}}{\omega _{2}}})>0)}$

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

## 加法定理

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

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

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

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

## 微分方程與積分方程

${\displaystyle g_{2}:=60\sum _{w\in \Lambda }'w^{-4}}$
${\displaystyle g_{3}:=120\sum _{w\in \Lambda }'w^{-6}}$

${\displaystyle \wp '(z)^{2}=4\wp (z)^{3}-g_{2}\wp (z)-g_{3}}$

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

${\displaystyle z_{1}-z_{2}=\int _{\wp (z_{1})}^{\wp (z_{2})}{\frac {ds}{\sqrt {4s^{3}-g_{2}s-g_{3}}}}}$

## 模判別式

${\displaystyle \Delta =g_{2}^{3}-27g_{3}^{2}.}$

## 文獻

• Stein. Complex Analysis.
• 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