# Approximations of Jacobi theta functions

$\left|\frac{i {e}^{-\pi i \tau / 4} \theta^{(r)}_{1}\!\left(z , \tau\right)}{{\left(\pi i\right)}^{r}} - \sum_{n=0}^{N - 1} {\left(-1\right)}^{n} {\left(2 n + 1\right)}^{r} {q}^{n \left(n + 1\right)} \left({w}^{2 n + 1} - \frac{{\left(-1\right)}^{r}}{{w}^{2 n + 1}}\right)\right| \le \begin{cases} \frac{2 {Q}^{N \left(N + 1\right)} {W}^{2 N + 1} {\left(2 N + 1\right)}^{r}}{1 - \alpha}, & \alpha < 1\\\infty, & \text{otherwise}\\ \end{cases}\; \text{ where } q = {e}^{\pi i \tau},\;w = {e}^{\pi i z},\;Q = \left|q\right|,\;W = \max\!\left(\left|w\right|, \frac{1}{\left|w\right|}\right),\;\alpha = {Q}^{2 N + 1} {W}^{2} \exp\!\left(\frac{r}{N + 1}\right)$
$\left|\frac{{e}^{-\pi i \tau / 4} \theta^{(r)}_{2}\!\left(z , \tau\right)}{{\left(\pi i\right)}^{r}} - \sum_{n=0}^{N - 1} {\left(2 n + 1\right)}^{r} {q}^{n \left(n + 1\right)} \left({w}^{2 n + 1} + \frac{{\left(-1\right)}^{r}}{{w}^{2 n + 1}}\right)\right| \le \begin{cases} \frac{2 {Q}^{N \left(N + 1\right)} {W}^{2 N + 1} {\left(2 N + 1\right)}^{r}}{1 - \alpha}, & \alpha < 1\\\infty, & \text{otherwise}\\ \end{cases}\; \text{ where } q = {e}^{\pi i \tau},\;w = {e}^{\pi i z},\;Q = \left|q\right|,\;W = \max\!\left(\left|w\right|, \frac{1}{\left|w\right|}\right),\;\alpha = {Q}^{2 N + 1} {W}^{2} \exp\!\left(\frac{r}{N + 1}\right)$
$\left|\frac{\theta^{(r)}_{3}\!\left(z , \tau\right)}{{\left(2 \pi i\right)}^{r}} - \left({0}^{r} + \sum_{n=1}^{N - 1} {n}^{r} {q}^{{n}^{2}} \left({w}^{2 n} + \frac{{\left(-1\right)}^{r}}{{w}^{2 n}}\right)\right)\right| \le \begin{cases} \frac{2 {Q}^{{N}^{2}} {W}^{2 N} {N}^{r}}{1 - \alpha}, & \alpha < 1\\\infty, & \text{otherwise}\\ \end{cases}\; \text{ where } q = {e}^{\pi i \tau},\;w = {e}^{\pi i z},\;Q = \left|q\right|,\;W = \max\!\left(\left|w\right|, \frac{1}{\left|w\right|}\right),\;\alpha = {Q}^{2 N + 1} {W}^{2} {e}^{r / N}$
$\left|\frac{\theta^{(r)}_{4}\!\left(z , \tau\right)}{{\left(2 \pi i\right)}^{r}} - \left({0}^{r} + \sum_{n=1}^{N - 1} {\left(-1\right)}^{n} {n}^{r} {q}^{{n}^{2}} \left({w}^{2 n} + \frac{{\left(-1\right)}^{r}}{{w}^{2 n}}\right)\right)\right| \le \begin{cases} \frac{2 {Q}^{{N}^{2}} {W}^{2 N} {N}^{r}}{1 - \alpha}, & \alpha < 1\\\infty, & \text{otherwise}\\ \end{cases}\; \text{ where } q = {e}^{\pi i \tau},\;w = {e}^{\pi i z},\;Q = \left|q\right|,\;W = \max\!\left(\left|w\right|, \frac{1}{\left|w\right|}\right),\;\alpha = {Q}^{2 N + 1} {W}^{2} {e}^{r / N}$