Weierstrass's Elliptic Function is Even in z

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Theorem

Let $\omega_1$ and $\omega_2$ be non-zero complex constants with $\dfrac {\omega_1} {\omega_2}$ having a positive imaginary part.

For $z \in \C \setminus \set {2 m \omega_1 + 2 n \omega_2: \tuple {n, m} \in \Z^2}$:

$\ds \map \wp {-z; \omega_1, \omega_2} = \map \wp {z; \omega_1, \omega_2}$

That is, Weierstrass's elliptic function is even in $z$.


Proof

\(\ds \map \wp {-z; \omega_1, \omega_2}\) \(=\) \(\ds \frac 1 {\paren {-z}^2} + \sum_{\tuple {n, m} \mathop \in \Z^2 \setminus \tuple {0, 0} } \paren {\frac 1 {\paren {-z - 2 m \omega_1 - 2 n \omega_2}^2} - \frac 1 {\paren {2 m \omega_1 + 2 n \omega_2}^2} }\) Definition of Weierstrass's Elliptic Function
\(\ds \) \(=\) \(\ds \frac 1 {z^2} + \paren {\sum_{\tuple {n, m} \mathop \in \Z_{\ge 0}^2 \setminus \tuple {0, 0} } + \sum_{\tuple {n, m} \mathop \in \Z_{<0}^2} } \paren {\frac 1 {\paren {z - 2 m \omega_1 - 2 n \omega_2}^2} - \frac 1 {\paren {-2 m \omega_1 - 2 n \omega_2}^2} }\)
\(\ds \) \(=\) \(\ds \frac 1 {z^2} + \paren {\sum_{\tuple {n, m} \mathop \in \Z_{\le 0}^2 \setminus \tuple {0, 0} } } + \sum_{\tuple {n, m} \mathop \in \Z_{> 0}^2} \paren {\frac 1 {\paren {z - 2 m \omega_1 - 2 n \omega_2}^2} - \frac 1 {\paren {2 m \omega_1 + 2 n \omega_2}^2} }\) letting $\tuple {n, m} \to \tuple {-n, -m}$
\(\ds \) \(=\) \(\ds \frac 1 {z^2} + {\sum_{\tuple {n, m} \mathop \in \Z^2 \setminus \tuple {0, 0} } } \paren {\frac 1 {\paren {z - 2 m \omega_1 - 2 n \omega_2}^2} - \frac 1 {\paren {2 m \omega_1 + 2 n \omega_2}^2} }\)
\(\ds \) \(=\) \(\ds \map \wp {z; \omega_1, \omega_2}\) Definition of Weierstrass's Elliptic Function


$\blacksquare$


Sources