engleski

On growth rates of Weierstrass $\wp'(z)$ and $\wp(z)$

Non--negative functions $L, R$ are given such that $L(z) \le |\wp'(z)|\le R(z)$, where $L(z) = \mathcal O(H(2|z|)\delta_z^{; ; -4}; ; ), \, R(z) = \mathcal O (\delta_z^{; ; -3}; ; )$ and $\delta_z := \inf_{; ; \mathbb Z^2}; ; |z-\mathbb Z^2|, \, z \in \mathbb C$. Here $$H(r):= \frac{; ; \min\{; ; r^2- [r^2], [r^2] +1-r^2\}; ; }; ; {; ; 2r+1/\sqrt{; ; 2}; ; }; ; \qquad (r \ge 0), $$ with $[a]$ being the integer part of $a$. By this results growth rate are deduced for $|\wp(z)|$.

Bounding inequality, Jacobi $\theta$, Weierstrass $\mathfrak g_2, \mathfrak g_3$, Weierstrass $\wp'(z), \wp(z), \sigma$

nije evidentirano