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Approximations of Jacobi theta functions

Table of contents: Approximation by truncated Fourier series

See Jacobi theta functions for an introduction to these functions.

Approximation by truncated Fourier series

001234
ieπiτ/4θ1(r) ⁣(z,τ)(πi)rn=0N1(1)n(2n+1)rqn(n+1)(w2n+1(1)rw2n+1){2QN(N+1)W2N+1(2N+1)r1α,α<1,otherwise   where q=eπiτ,w=eπiz,Q=q,W=max ⁣(w,1w),α=Q2N+1W2exp ⁣(rN+1)\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)
dac0aa
eπiτ/4θ2(r) ⁣(z,τ)(πi)rn=0N1(2n+1)rqn(n+1)(w2n+1+(1)rw2n+1){2QN(N+1)W2N+1(2N+1)r1α,α<1,otherwise   where q=eπiτ,w=eπiz,Q=q,W=max ⁣(w,1w),α=Q2N+1W2exp ⁣(rN+1)\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)
24a793
θ3(r) ⁣(z,τ)(2πi)r(0r+n=1N1nrqn2(w2n+(1)rw2n)){2QN2W2NNr1α,α<1,otherwise   where q=eπiτ,w=eπiz,Q=q,W=max ⁣(w,1w),α=Q2N+1W2er/N\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}
c574fd
θ4(r) ⁣(z,τ)(2πi)r(0r+n=1N1(1)nnrqn2(w2n+(1)rw2n)){2QN2W2NNr1α,α<1,otherwise   where q=eπiτ,w=eπiz,Q=q,W=max ⁣(w,1w),α=Q2N+1W2er/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}

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2019-09-19 20:12:49.583742 UTC