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
- 1920: E.T. Whittaker and G.N. Watson: A Course of Modern Analysis (3rd ed.): $20.21$: Periodicity and other properties of $\wp\left({z}\right)$