Approximations of Jacobi theta functions

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)))))

dac0aa

\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)

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{{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)

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
`Exp`	${e}^{z}$	Exponential function
`Pi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit
`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("dac0aa"),
    Formula(Where(LessEqual(Abs(Sub(Div(Mul(Exp(Neg(Div(Mul(Mul(Pi, ConstI), tau), 4))), JacobiTheta(2, z, tau, r)), Pow(Mul(Pi, ConstI), r)), Sum(Mul(Mul(Pow(Add(Mul(2, n), 1), r), Pow(q, Mul(n, Add(n, 1)))), Add(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)))))

24a793

\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}

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{\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}

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
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	Jacobi theta function
`Pow`	${a}^{b}$	Power
`Pi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit
`Sum`	$\sum_{n} f(n)$	Sum
`Infinity`	$\infty$	Positive infinity
`Exp`	${e}^{z}$	Exponential function
`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("24a793"),
    Formula(Where(LessEqual(Abs(Sub(Div(JacobiTheta(3, z, tau, r), Pow(Mul(Mul(2, Pi), ConstI), r)), Parentheses(Add(Pow(0, r), Sum(Mul(Mul(Pow(n, r), Pow(q, Pow(n, 2))), Add(Pow(w, Mul(2, n)), Div(Pow(-1, r), Pow(w, Mul(2, n))))), For(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(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, N)))))),
    Variables(z, tau, r, N),
    Assumptions(And(Element(z, CC), Element(tau, HH), Element(r, ZZGreaterEqual(0)), Element(N, ZZGreaterEqual(1)))))

c574fd

\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:

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{\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
`Abs`	$\left\|z\right\|$	Absolute value
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	Jacobi theta function
`Pow`	${a}^{b}$	Power
`Pi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit
`Sum`	$\sum_{n} f(n)$	Sum
`Infinity`	$\infty$	Positive infinity
`Exp`	${e}^{z}$	Exponential function
`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("c574fd"),
    Formula(Where(LessEqual(Abs(Sub(Div(JacobiTheta(4, z, tau, r), Pow(Mul(Mul(2, Pi), 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))))), For(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(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, N)))))),
    Variables(z, tau, r, N),
    Assumptions(And(Element(z, CC), Element(tau, HH), Element(r, ZZGreaterEqual(0)), Element(N, ZZGreaterEqual(1)))))

Approximation by truncated Fourier series