# Fungrim entry: 001234

$\left|\frac{i {e}^{-\pi i \tau / 4} \theta^{(r)}_{1}\!\left(z , \tau\right)}{{\left(\pi i\right)}^{r}} - \sum_{n=0}^{N - 1} {\left(-1\right)}^{n} {\left(2 n + 1\right)}^{r} {q}^{n \left(n + 1\right)} \left({w}^{2 n + 1} - \frac{{\left(-1\right)}^{r}}{{w}^{2 n + 1}}\right)\right| \le \begin{cases} \frac{2 {Q}^{N \left(N + 1\right)} {W}^{2 N + 1} {\left(2 N + 1\right)}^{r}}{1 - \alpha}, & \alpha < 1\\\infty, & \text{otherwise}\\ \end{cases}\; \text{ where } q = {e}^{\pi i \tau},\;w = {e}^{\pi i z},\;Q = \left|q\right|,\;W = \max\!\left(\left|w\right|, \frac{1}{\left|w\right|}\right),\;\alpha = {Q}^{2 N + 1} {W}^{2} \exp\!\left(\frac{r}{N + 1}\right)$
Assumptions:$z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H} \;\mathbin{\operatorname{and}}\; r \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; N \in \mathbb{Z}_{\ge 1}$
TeX:
\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)

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H} \;\mathbin{\operatorname{and}}\; r \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; N \in \mathbb{Z}_{\ge 1}
Definitions:
Fungrim symbol Notation Short description
Abs$\left|z\right|$ Absolute value
ConstI$i$ Imaginary unit
Exp${e}^{z}$ Exponential function
Pi$\pi$ The constant pi (3.14...)
JacobiTheta$\theta_{j}\!\left(z , \tau\right)$ Jacobi theta function
Pow${a}^{b}$ Power
Sum$\sum_{n} f(n)$ Sum
Infinity$\infty$ Positive infinity
CC$\mathbb{C}$ Complex numbers
HH$\mathbb{H}$ Upper complex half-plane
ZZGreaterEqual$\mathbb{Z}_{\ge n}$ Integers greater than or equal to n
Source code for this entry:
Entry(ID("001234"),
Formula(Where(LessEqual(Abs(Sub(Div(Mul(Mul(ConstI, Exp(Neg(Div(Mul(Mul(Pi, ConstI), tau), 4)))), JacobiTheta(1, z, tau, r)), Pow(Mul(Pi, ConstI), r)), Sum(Mul(Mul(Mul(Pow(-1, n), Pow(Add(Mul(2, n), 1), r)), Pow(q, Mul(n, Add(n, 1)))), Sub(Pow(w, Add(Mul(2, n), 1)), Div(Pow(-1, r), Pow(w, Add(Mul(2, n), 1))))), For(n, 0, Sub(N, 1))))), Cases(Tuple(Div(Mul(Mul(Mul(2, Pow(Q, Mul(N, Add(N, 1)))), Pow(W, Add(Mul(2, N), 1))), Pow(Add(Mul(2, N), 1), r)), Sub(1, alpha)), Less(alpha, 1)), Tuple(Infinity, Otherwise))), Equal(q, Exp(Mul(Mul(Pi, ConstI), tau))), Equal(w, Exp(Mul(Mul(Pi, ConstI), z))), Equal(Q, Abs(q)), Equal(W, Max(Abs(w), Div(1, Abs(w)))), Equal(alpha, Mul(Mul(Pow(Q, Add(Mul(2, N), 1)), Pow(W, 2)), Exp(Div(r, Add(N, 1))))))),
Variables(z, tau, r, N),
Assumptions(And(Element(z, CC), Element(tau, HH), Element(r, ZZGreaterEqual(0)), Element(N, ZZGreaterEqual(1)))))

## Topics using this entry

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2021-03-15 19:12:00.328586 UTC