在数学 中，魏尔斯特拉斯椭圆函数 （Weierstrass's elliptic functions）又称 p 函数并且以
℘
{\displaystyle \wp }
椭圆函数 ，也是雅可比椭圆函数 的特殊形式。卡尔·魏尔斯特拉斯 首先研究了这些函数。
固定
C
{\displaystyle \mathbb {C} }
Λ
=
Z
ω
1
⊕
Z
ω
2
{\displaystyle \Lambda =\mathbb {Z} \omega _{1}\oplus \mathbb {Z} \omega _{2}}
ω
1
,
ω
2
∈
C
{\displaystyle \omega _{1},\omega _{2}\in \mathbb {C} }
Q
{\displaystyle \mathbb {Q} }
℘
(
z
;
Λ
)
=
1
z
2
+
∑
(
m
,
n
)
≠
(
0
,
0
)
{
1
(
z
−
m
ω
1
−
n
ω
2
)
2
−
1
(
m
ω
1
+
n
ω
2
)
2
}
{\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\}}
显然右式只与格
Λ
{\displaystyle \Lambda \,}
ω
1
,
ω
2
{\displaystyle \omega _{1},\omega _{2}\,}
Λ
{\displaystyle \Lambda \,}
另一方面，格
Λ
{\displaystyle \Lambda \,}
C
→
C
{\displaystyle \mathbb {C} \to \mathbb {C} }
Λ
=
Z
⊕
Z
τ
{\displaystyle \Lambda =\mathbb {Z} \oplus \mathbb {Z} \tau }
τ
{\displaystyle \tau \,}
℘
(
z
;
Λ
)
=
℘
(
z
;
τ
)
=
1
z
2
+
∑
n
2
+
m
2
≠
0
1
(
z
−
n
−
m
τ
)
2
−
1
(
n
+
m
τ
)
2
{\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}}}
反之，由此亦可导出对一般的格之公式
℘
(
z
;
Z
ω
1
⊕
Z
ω
2
)
=
℘
(
z
ω
1
;
ω
2
ω
1
)
ω
1
2
(
I
m
(
ω
1
ω
2
)
>
0
)
{\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)}
在数值计算方面，
℘
{\displaystyle \wp }
Θ函数 快速地计算，方程是
℘
(
z
;
τ
)
=
π
2
ϑ
2
(
0
;
τ
)
ϑ
10
2
(
0
;
τ
)
ϑ
01
2
(
z
;
τ
)
ϑ
11
2
(
z
;
τ
)
−
π
2
3
[
ϑ
4
(
0
;
τ
)
+
ϑ
10
4
(
0
;
τ
)
]
{\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]}
在周期格中的每个点，
℘
{\displaystyle \wp }
极点 。
℘
{\displaystyle \wp }
复导函数
℘
′
{\displaystyle \wp '}
续用上节符号，模判别式
Δ
{\displaystyle \Delta }
Δ
=
g
2
3
−
27
g
3
2
.
{\displaystyle \Delta =g_{2}^{3}-27g_{3}^{2}.}
视为周期格的函数，这是权 12 之模形式 。模判别式也可以用戴德金η函数 表示。
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
![{\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]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/26f9c99244c5b95e7970a8b9d7ae2df3c3a2f432)
