Fungrim home page

Fungrim entry: 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}
Assumptions:zCandτHandrZ0andNZ1z \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{\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}

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
Absz\left|z\right| Absolute value
JacobiThetaθj ⁣(z,τ)\theta_{j}\!\left(z , \tau\right) Jacobi theta function
Powab{a}^{b} Power
ConstPiπ\pi The constant pi (3.14...)
ConstIii Imaginary unit
Sumnf ⁣(n)\sum_{n} f\!\left(n\right) Sum
Infinity\infty Positive infinity
Expez{e}^{z} Exponential function
CCC\mathbb{C} Complex numbers
HHH\mathbb{H} Upper complex half-plane
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
Source code for this entry:
Entry(ID("c574fd"),
    Formula(Where(LessEqual(Abs(Sub(Div(JacobiTheta(4, z, tau, r), Pow(Mul(Mul(2, ConstPi), ConstI), r)), Parentheses(Add(Pow(0, r), Sum(Mul(Mul(Mul(Pow(-1, n), Pow(n, r)), Pow(q, Pow(n, 2))), Add(Pow(w, Mul(2, n)), Div(Pow(-1, r), Pow(w, Mul(2, n))))), Tuple(n, 1, Sub(N, 1))))))), Cases(Tuple(Div(Mul(Mul(Mul(2, Pow(Q, Pow(N, 2))), Pow(W, Mul(2, N))), Pow(N, r)), Sub(1, alpha)), Less(alpha, 1)), Tuple(Infinity, Otherwise))), Equal(q, Exp(Mul(Mul(ConstPi, ConstI), tau))), Equal(w, Exp(Mul(Mul(ConstPi, 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, N)))))),
    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.

2019-09-19 20:12:49.583742 UTC