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Fungrim entry: 001234

∣ie−πiτ/4θ1(r) ⁣(z,τ)(πi)r−∑n=0N−1(−1)n(2n+1)rqn(n+1)(w2n+1−(−1)rw2n+1)∣≤{2QN(N+1)W2N+1(2N+1)r1−α,α<1∞,otherwise   where q=eπiτ,  w=eπiz,  Q=∣q∣,  W=max⁡ ⁣(∣w∣,1∣w∣),  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)
Assumptions:z∈C  and⁡  τ∈H  and⁡  r∈Z≥0  and⁡  N∈Z≥1z \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∣z∣\left|z\right| Absolute value
ConstIii Imaginary unit
Expez{e}^{z} Exponential function
Piπ\pi The constant pi (3.14...)
JacobiThetaθj ⁣(z,τ)\theta_{j}\!\left(z , \tau\right) Jacobi theta function
Powab{a}^{b} Power
Sum∑nf(n)\sum_{n} f(n) Sum
Infinity∞\infty Positive infinity
CCC\mathbb{C} Complex numbers
HHH\mathbb{H} Upper complex half-plane
ZZGreaterEqualZ≥n\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