Jacobi theta functions

ed4ce5

\theta_1\!\left(z, \tau\right) = -i \sum_{n=-\infty}^{\infty} {\left(-1\right)}^{n} {e}^{\pi i \left({\left(n + 1 / 2\right)}^{2} \tau + \left(2 n + 1\right) z\right)}

Assumptions:

z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \tau \in \mathbb{H}

TeX:

\theta_1\!\left(z, \tau\right) = -i \sum_{n=-\infty}^{\infty} {\left(-1\right)}^{n} {e}^{\pi i \left({\left(n + 1 / 2\right)}^{2} \tau + \left(2 n + 1\right) z\right)}

z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \tau \in \mathbb{H}

Definitions:

Fungrim symbol	Notation	Short description
`JacobiTheta1`	$\theta_1\!\left(z, \tau\right)$	Jacobi theta function
`ConstI`	$i$	Imaginary unit
`Pow`	${a}^{b}$	Power
`Exp`	${e}^{z}$	Exponential function
`ConstPi`	$\pi$	The constant pi (3.14...)
`Infinity`	$\infty$	Positive infinity
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("ed4ce5"),
    Formula(Equal(JacobiTheta1(z, tau), Mul(Neg(ConstI), Sum(Mul(Pow(-1, n), Exp(Mul(Mul(ConstPi, ConstI), Add(Mul(Pow(Add(n, Div(1, 2)), 2), tau), Mul(Add(Mul(2, n), 1), z))))), Tuple(n, Neg(Infinity), Infinity))))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH))))

7cb651

\theta_2\!\left(z, \tau\right) = \sum_{n=-\infty}^{\infty} {e}^{\pi i \left({\left(n + 1 / 2\right)}^{2} \tau + \left(2 n + 1\right) z\right)}

Assumptions:

z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \tau \in \mathbb{H}

TeX:

\theta_2\!\left(z, \tau\right) = \sum_{n=-\infty}^{\infty} {e}^{\pi i \left({\left(n + 1 / 2\right)}^{2} \tau + \left(2 n + 1\right) z\right)}

z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \tau \in \mathbb{H}

Definitions:

Fungrim symbol	Notation	Short description
`JacobiTheta2`	$\theta_2\!\left(z, \tau\right)$	Jacobi theta function
`Exp`	${e}^{z}$	Exponential function
`ConstPi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit
`Pow`	${a}^{b}$	Power
`Infinity`	$\infty$	Positive infinity
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("7cb651"),
    Formula(Equal(JacobiTheta2(z, tau), Sum(Exp(Mul(Mul(ConstPi, ConstI), Add(Mul(Pow(Add(n, Div(1, 2)), 2), tau), Mul(Add(Mul(2, n), 1), z)))), Tuple(n, Neg(Infinity), Infinity)))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH))))

580ba0

\theta_3\!\left(z, \tau\right) = \sum_{n=-\infty}^{\infty} {e}^{\pi i \left({n}^{2} \tau + 2 n z\right)}

Assumptions:

z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \tau \in \mathbb{H}

TeX:

\theta_3\!\left(z, \tau\right) = \sum_{n=-\infty}^{\infty} {e}^{\pi i \left({n}^{2} \tau + 2 n z\right)}

z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \tau \in \mathbb{H}

Definitions:

Fungrim symbol	Notation	Short description
`JacobiTheta3`	$\theta_3\!\left(z, \tau\right)$	Jacobi theta function
`Exp`	${e}^{z}$	Exponential function
`ConstPi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit
`Pow`	${a}^{b}$	Power
`Infinity`	$\infty$	Positive infinity
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("580ba0"),
    Formula(Equal(JacobiTheta3(z, tau), Sum(Exp(Mul(Mul(ConstPi, ConstI), Add(Mul(Pow(n, 2), tau), Mul(Mul(2, n), z)))), Tuple(n, Neg(Infinity), Infinity)))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH))))

27c319

\theta_4\!\left(z, \tau\right) = \sum_{n=-\infty}^{\infty} {\left(-1\right)}^{n} {e}^{\pi i \left({n}^{2} \tau + 2 n z\right)}

Assumptions:

z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \tau \in \mathbb{H}

TeX:

\theta_4\!\left(z, \tau\right) = \sum_{n=-\infty}^{\infty} {\left(-1\right)}^{n} {e}^{\pi i \left({n}^{2} \tau + 2 n z\right)}

z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \tau \in \mathbb{H}

Definitions:

Fungrim symbol	Notation	Short description
`JacobiTheta4`	$\theta_4\!\left(z, \tau\right)$	Jacobi theta function
`Pow`	${a}^{b}$	Power
`Exp`	${e}^{z}$	Exponential function
`ConstPi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit
`Infinity`	$\infty$	Positive infinity
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("27c319"),
    Formula(Equal(JacobiTheta4(z, tau), Sum(Mul(Pow(-1, n), Exp(Mul(Mul(ConstPi, ConstI), Add(Mul(Pow(n, 2), tau), Mul(Mul(2, n), z))))), Tuple(n, Neg(Infinity), Infinity)))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH))))

2ba423

\theta_1\!\left(z, \tau\right) = 2 \sum_{n=0}^{\infty} {\left(-1\right)}^{n} {e}^{\pi i {\left(n + 1 / 2\right)}^{2} \tau} \sin\!\left(\left(2 n + 1\right) \pi z\right)

Assumptions:

z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \tau \in \mathbb{H}

TeX:

\theta_1\!\left(z, \tau\right) = 2 \sum_{n=0}^{\infty} {\left(-1\right)}^{n} {e}^{\pi i {\left(n + 1 / 2\right)}^{2} \tau} \sin\!\left(\left(2 n + 1\right) \pi z\right)

z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \tau \in \mathbb{H}

Definitions:

Fungrim symbol	Notation	Short description
`JacobiTheta1`	$\theta_1\!\left(z, \tau\right)$	Jacobi theta function
`Pow`	${a}^{b}$	Power
`Exp`	${e}^{z}$	Exponential function
`ConstPi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit
`Sin`	$\sin\!\left(z\right)$	Sine
`Infinity`	$\infty$	Positive infinity
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("2ba423"),
    Formula(Equal(JacobiTheta1(z, tau), Mul(2, Sum(Mul(Mul(Pow(-1, n), Exp(Mul(Mul(Mul(ConstPi, ConstI), Pow(Add(n, Div(1, 2)), 2)), tau))), Sin(Mul(Mul(Add(Mul(2, n), 1), ConstPi), z))), Tuple(n, 0, Infinity))))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH))))

06633e

\theta_2\!\left(z, \tau\right) = 2 \sum_{n=0}^{\infty} {e}^{\pi i {\left(n + 1 / 2\right)}^{2} \tau} \cos\!\left(\left(2 n + 1\right) \pi z\right)

Assumptions:

z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \tau \in \mathbb{H}

TeX:

\theta_2\!\left(z, \tau\right) = 2 \sum_{n=0}^{\infty} {e}^{\pi i {\left(n + 1 / 2\right)}^{2} \tau} \cos\!\left(\left(2 n + 1\right) \pi z\right)

z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \tau \in \mathbb{H}

Definitions:

Fungrim symbol	Notation	Short description
`JacobiTheta2`	$\theta_2\!\left(z, \tau\right)$	Jacobi theta function
`Exp`	${e}^{z}$	Exponential function
`ConstPi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit
`Pow`	${a}^{b}$	Power
`Infinity`	$\infty$	Positive infinity
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("06633e"),
    Formula(Equal(JacobiTheta2(z, tau), Mul(2, Sum(Mul(Exp(Mul(Mul(Mul(ConstPi, ConstI), Pow(Add(n, Div(1, 2)), 2)), tau)), Cos(Mul(Mul(Add(Mul(2, n), 1), ConstPi), z))), Tuple(n, 0, Infinity))))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH))))

f3e75c

\theta_3\!\left(z, \tau\right) = 1 + 2 \sum_{n=1}^{\infty} {e}^{\pi i {n}^{2} \tau} \cos\!\left(2 n \pi z\right)

Assumptions:

z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \tau \in \mathbb{H}

TeX:

\theta_3\!\left(z, \tau\right) = 1 + 2 \sum_{n=1}^{\infty} {e}^{\pi i {n}^{2} \tau} \cos\!\left(2 n \pi z\right)

z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \tau \in \mathbb{H}

Definitions:

Fungrim symbol	Notation	Short description
`JacobiTheta3`	$\theta_3\!\left(z, \tau\right)$	Jacobi theta function
`Exp`	${e}^{z}$	Exponential function
`ConstPi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit
`Pow`	${a}^{b}$	Power
`Infinity`	$\infty$	Positive infinity
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("f3e75c"),
    Formula(Equal(JacobiTheta3(z, tau), Add(1, Mul(2, Sum(Mul(Exp(Mul(Mul(Mul(ConstPi, ConstI), Pow(n, 2)), tau)), Cos(Mul(Mul(Mul(2, n), ConstPi), z))), Tuple(n, 1, Infinity)))))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH))))

8a34d1

\theta_4\!\left(z, \tau\right) = 1 + 2 \sum_{n=1}^{\infty} {\left(-1\right)}^{n} {e}^{\pi i {n}^{2} \tau} \cos\!\left(2 n \pi z\right)

Assumptions:

z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \tau \in \mathbb{H}

TeX:

\theta_4\!\left(z, \tau\right) = 1 + 2 \sum_{n=1}^{\infty} {\left(-1\right)}^{n} {e}^{\pi i {n}^{2} \tau} \cos\!\left(2 n \pi z\right)

z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \tau \in \mathbb{H}

Definitions:

Fungrim symbol	Notation	Short description
`JacobiTheta4`	$\theta_4\!\left(z, \tau\right)$	Jacobi theta function
`Pow`	${a}^{b}$	Power
`Exp`	${e}^{z}$	Exponential function
`ConstPi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit
`Infinity`	$\infty$	Positive infinity
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("8a34d1"),
    Formula(Equal(JacobiTheta4(z, tau), Add(1, Mul(2, Sum(Mul(Mul(Pow(-1, n), Exp(Mul(Mul(Mul(ConstPi, ConstI), Pow(n, 2)), tau))), Cos(Mul(Mul(Mul(2, n), ConstPi), z))), Tuple(n, 1, Infinity)))))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH))))

154c44

\mathop{\operatorname{zeros}\,}\limits_{z \in \mathbb{C}} \theta_1\!\left(z, \tau\right) = \left\{ m + n \tau : m \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, n \in \mathbb{Z} \right\}

Assumptions:

\tau \in \mathbb{H}

TeX:

\mathop{\operatorname{zeros}\,}\limits_{z \in \mathbb{C}} \theta_1\!\left(z, \tau\right) = \left\{ m + n \tau : m \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, n \in \mathbb{Z} \right\}

\tau \in \mathbb{H}

Definitions:

Fungrim symbol	Notation	Short description
`JacobiTheta1`	$\theta_1\!\left(z, \tau\right)$	Jacobi theta function
`CC`	$\mathbb{C}$	Complex numbers
`SetBuilder`	$\left\{ f\!\left(x\right) : P\!\left(x\right) \right\}$	Set comprehension
`ZZ`	$\mathbb{Z}$	Integers
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("154c44"),
    Formula(Equal(Zeros(JacobiTheta1(z, tau), z, Element(z, CC)), SetBuilder(Add(m, Mul(n, tau)), Tuple(m, n), And(Element(m, ZZ), Element(n, ZZ))))),
    Variables(tau),
    Assumptions(Element(tau, HH)))

ad1eaf

\mathop{\operatorname{zeros}\,}\limits_{z \in \mathbb{C}} \theta_2\!\left(z, \tau\right) = \left\{ \left(m + \frac{1}{2}\right) + n \tau : m \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, n \in \mathbb{Z} \right\}

Assumptions:

\tau \in \mathbb{H}

TeX:

\mathop{\operatorname{zeros}\,}\limits_{z \in \mathbb{C}} \theta_2\!\left(z, \tau\right) = \left\{ \left(m + \frac{1}{2}\right) + n \tau : m \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, n \in \mathbb{Z} \right\}

\tau \in \mathbb{H}

Definitions:

Fungrim symbol	Notation	Short description
`JacobiTheta2`	$\theta_2\!\left(z, \tau\right)$	Jacobi theta function
`CC`	$\mathbb{C}$	Complex numbers
`SetBuilder`	$\left\{ f\!\left(x\right) : P\!\left(x\right) \right\}$	Set comprehension
`ZZ`	$\mathbb{Z}$	Integers
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("ad1eaf"),
    Formula(Equal(Zeros(JacobiTheta2(z, tau), z, Element(z, CC)), SetBuilder(Add(Parentheses(Add(m, Div(1, 2))), Mul(n, tau)), Tuple(m, n), And(Element(m, ZZ), Element(n, ZZ))))),
    Variables(tau),
    Assumptions(Element(tau, HH)))

caf10a

\mathop{\operatorname{zeros}\,}\limits_{z \in \mathbb{C}} \theta_3\!\left(z, \tau\right) = \left\{ \left(m + \frac{1}{2}\right) + \left(n + \frac{1}{2}\right) \tau : m \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, n \in \mathbb{Z} \right\}

Assumptions:

\tau \in \mathbb{H}

TeX:

\mathop{\operatorname{zeros}\,}\limits_{z \in \mathbb{C}} \theta_3\!\left(z, \tau\right) = \left\{ \left(m + \frac{1}{2}\right) + \left(n + \frac{1}{2}\right) \tau : m \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, n \in \mathbb{Z} \right\}

\tau \in \mathbb{H}

Definitions:

Fungrim symbol	Notation	Short description
`JacobiTheta3`	$\theta_3\!\left(z, \tau\right)$	Jacobi theta function
`CC`	$\mathbb{C}$	Complex numbers
`SetBuilder`	$\left\{ f\!\left(x\right) : P\!\left(x\right) \right\}$	Set comprehension
`ZZ`	$\mathbb{Z}$	Integers
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("caf10a"),
    Formula(Equal(Zeros(JacobiTheta3(z, tau), z, Element(z, CC)), SetBuilder(Add(Parentheses(Add(m, Div(1, 2))), Mul(Add(n, Div(1, 2)), tau)), Tuple(m, n), And(Element(m, ZZ), Element(n, ZZ))))),
    Variables(tau),
    Assumptions(Element(tau, HH)))

926b2c

\mathop{\operatorname{zeros}\,}\limits_{z \in \mathbb{C}} \theta_4\!\left(z, \tau\right) = \left\{ m + \left(n + \frac{1}{2}\right) \tau : m \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, n \in \mathbb{Z} \right\}

Assumptions:

\tau \in \mathbb{H}

TeX:

\mathop{\operatorname{zeros}\,}\limits_{z \in \mathbb{C}} \theta_4\!\left(z, \tau\right) = \left\{ m + \left(n + \frac{1}{2}\right) \tau : m \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, n \in \mathbb{Z} \right\}

\tau \in \mathbb{H}

Definitions:

Fungrim symbol	Notation	Short description
`JacobiTheta4`	$\theta_4\!\left(z, \tau\right)$	Jacobi theta function
`CC`	$\mathbb{C}$	Complex numbers
`SetBuilder`	$\left\{ f\!\left(x\right) : P\!\left(x\right) \right\}$	Set comprehension
`ZZ`	$\mathbb{Z}$	Integers
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("926b2c"),
    Formula(Equal(Zeros(JacobiTheta4(z, tau), z, Element(z, CC)), SetBuilder(Add(m, Mul(Add(n, Div(1, 2)), tau)), Tuple(m, n), And(Element(m, ZZ), Element(n, ZZ))))),
    Variables(tau),
    Assumptions(Element(tau, HH)))

59f8e1

\theta_1\!\left(-z, \tau\right) = -\theta_1\!\left(z, \tau\right)

Assumptions:

z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \tau \in \mathbb{H}

TeX:

\theta_1\!\left(-z, \tau\right) = -\theta_1\!\left(z, \tau\right)

z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \tau \in \mathbb{H}

Definitions:

Fungrim symbol	Notation	Short description
`JacobiTheta1`	$\theta_1\!\left(z, \tau\right)$	Jacobi theta function
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("59f8e1"),
    Formula(Equal(JacobiTheta1(Neg(z), tau), Neg(JacobiTheta1(z, tau)))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH))))

fb55cb

\theta_2\!\left(-z, \tau\right) = \theta_2\!\left(z, \tau\right)

Assumptions:

z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \tau \in \mathbb{H}

TeX:

\theta_2\!\left(-z, \tau\right) = \theta_2\!\left(z, \tau\right)

z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \tau \in \mathbb{H}

Definitions:

Fungrim symbol	Notation	Short description
`JacobiTheta2`	$\theta_2\!\left(z, \tau\right)$	Jacobi theta function
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("fb55cb"),
    Formula(Equal(JacobiTheta2(Neg(z), tau), JacobiTheta2(z, tau))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH))))

380076

\theta_3\!\left(-z, \tau\right) = \theta_3\!\left(z, \tau\right)

Assumptions:

z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \tau \in \mathbb{H}

TeX:

\theta_3\!\left(-z, \tau\right) = \theta_3\!\left(z, \tau\right)

z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \tau \in \mathbb{H}

Definitions:

Fungrim symbol	Notation	Short description
`JacobiTheta3`	$\theta_3\!\left(z, \tau\right)$	Jacobi theta function
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("380076"),
    Formula(Equal(JacobiTheta3(Neg(z), tau), JacobiTheta3(z, tau))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH))))

4f939e

\theta_4\!\left(-z, \tau\right) = \theta_4\!\left(z, \tau\right)

Assumptions:

z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \tau \in \mathbb{H}

TeX:

\theta_4\!\left(-z, \tau\right) = \theta_4\!\left(z, \tau\right)

z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \tau \in \mathbb{H}

Definitions:

Fungrim symbol	Notation	Short description
`JacobiTheta4`	$\theta_4\!\left(z, \tau\right)$	Jacobi theta function
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("4f939e"),
    Formula(Equal(JacobiTheta4(Neg(z), tau), JacobiTheta4(z, tau))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH))))

2faeb9

\theta_1\!\left(z + 2 n, \tau\right) = \theta_1\!\left(z, \tau\right)

Assumptions:

z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \tau \in \mathbb{H} \,\mathbin{\operatorname{and}}\, n \in \mathbb{Z}

TeX:

\theta_1\!\left(z + 2 n, \tau\right) = \theta_1\!\left(z, \tau\right)

z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \tau \in \mathbb{H} \,\mathbin{\operatorname{and}}\, n \in \mathbb{Z}

Definitions:

Fungrim symbol	Notation	Short description
`JacobiTheta1`	$\theta_1\!\left(z, \tau\right)$	Jacobi theta function
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane
`ZZ`	$\mathbb{Z}$	Integers

Source code for this entry:

Entry(ID("2faeb9"),
    Formula(Equal(JacobiTheta1(Add(z, Mul(2, n)), tau), JacobiTheta1(z, tau))),
    Variables(z, tau, n),
    Assumptions(And(Element(z, CC), Element(tau, HH), Element(n, ZZ))))

b46534

\theta_2\!\left(z + 2 n, \tau\right) = \theta_2\!\left(z, \tau\right)

Assumptions:

z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \tau \in \mathbb{H} \,\mathbin{\operatorname{and}}\, n \in \mathbb{Z}

TeX:

\theta_2\!\left(z + 2 n, \tau\right) = \theta_2\!\left(z, \tau\right)

z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \tau \in \mathbb{H} \,\mathbin{\operatorname{and}}\, n \in \mathbb{Z}

Definitions:

Fungrim symbol	Notation	Short description
`JacobiTheta2`	$\theta_2\!\left(z, \tau\right)$	Jacobi theta function
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane
`ZZ`	$\mathbb{Z}$	Integers

Source code for this entry:

Entry(ID("b46534"),
    Formula(Equal(JacobiTheta2(Add(z, Mul(2, n)), tau), JacobiTheta2(z, tau))),
    Variables(z, tau, n),
    Assumptions(And(Element(z, CC), Element(tau, HH), Element(n, ZZ))))

e56f77

\theta_3\!\left(z + 2 n, \tau\right) = \theta_3\!\left(z, \tau\right)

Assumptions:

z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \tau \in \mathbb{H} \,\mathbin{\operatorname{and}}\, n \in \mathbb{Z}

TeX:

\theta_3\!\left(z + 2 n, \tau\right) = \theta_3\!\left(z, \tau\right)

z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \tau \in \mathbb{H} \,\mathbin{\operatorname{and}}\, n \in \mathbb{Z}

Definitions:

Fungrim symbol	Notation	Short description
`JacobiTheta3`	$\theta_3\!\left(z, \tau\right)$	Jacobi theta function
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane
`ZZ`	$\mathbb{Z}$	Integers

Source code for this entry:

Entry(ID("e56f77"),
    Formula(Equal(JacobiTheta3(Add(z, Mul(2, n)), tau), JacobiTheta3(z, tau))),
    Variables(z, tau, n),
    Assumptions(And(Element(z, CC), Element(tau, HH), Element(n, ZZ))))

4448f1

\theta_4\!\left(z + 2 n, \tau\right) = \theta_3\!\left(z, \tau\right)

Assumptions:

z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \tau \in \mathbb{H} \,\mathbin{\operatorname{and}}\, n \in \mathbb{Z}

TeX:

\theta_4\!\left(z + 2 n, \tau\right) = \theta_3\!\left(z, \tau\right)

z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \tau \in \mathbb{H} \,\mathbin{\operatorname{and}}\, n \in \mathbb{Z}

Definitions:

Fungrim symbol	Notation	Short description
`JacobiTheta4`	$\theta_4\!\left(z, \tau\right)$	Jacobi theta function
`JacobiTheta3`	$\theta_3\!\left(z, \tau\right)$	Jacobi theta function
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane
`ZZ`	$\mathbb{Z}$	Integers

Source code for this entry:

Entry(ID("4448f1"),
    Formula(Equal(JacobiTheta4(Add(z, Mul(2, n)), tau), JacobiTheta3(z, tau))),
    Variables(z, tau, n),
    Assumptions(And(Element(z, CC), Element(tau, HH), Element(n, ZZ))))

43fa0e

\theta_1\!\left(z + m + n \tau, \tau\right) = {\left(-1\right)}^{m + n} {e}^{-\pi i \left(\tau {n}^{2} + 2 n z\right)} \theta_1\!\left(z, \tau\right)

Assumptions:

z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \tau \in \mathbb{H} \,\mathbin{\operatorname{and}}\, m \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, n \in \mathbb{Z}

TeX:

\theta_1\!\left(z + m + n \tau, \tau\right) = {\left(-1\right)}^{m + n} {e}^{-\pi i \left(\tau {n}^{2} + 2 n z\right)} \theta_1\!\left(z, \tau\right)

z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \tau \in \mathbb{H} \,\mathbin{\operatorname{and}}\, m \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, n \in \mathbb{Z}

Definitions:

Fungrim symbol	Notation	Short description
`JacobiTheta1`	$\theta_1\!\left(z, \tau\right)$	Jacobi theta function
`Pow`	${a}^{b}$	Power
`Exp`	${e}^{z}$	Exponential function
`ConstPi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane
`ZZ`	$\mathbb{Z}$	Integers

Source code for this entry:

Entry(ID("43fa0e"),
    Formula(Equal(JacobiTheta1(Add(z, Add(m, Mul(n, tau))), tau), Mul(Mul(Pow(-1, Add(m, n)), Exp(Neg(Mul(Mul(ConstPi, ConstI), Add(Mul(tau, Pow(n, 2)), Mul(Mul(2, n), z)))))), JacobiTheta1(z, tau)))),
    Variables(z, tau, m, n),
    Assumptions(And(Element(z, CC), Element(tau, HH), Element(m, ZZ), Element(n, ZZ))))

d29148

\theta_2\!\left(z + m + n \tau, \tau\right) = {\left(-1\right)}^{m} {e}^{-\pi i \left(\tau {n}^{2} + 2 n z\right)} \theta_2\!\left(z, \tau\right)

Assumptions:

z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \tau \in \mathbb{H} \,\mathbin{\operatorname{and}}\, m \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, n \in \mathbb{Z}

TeX:

\theta_2\!\left(z + m + n \tau, \tau\right) = {\left(-1\right)}^{m} {e}^{-\pi i \left(\tau {n}^{2} + 2 n z\right)} \theta_2\!\left(z, \tau\right)

z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \tau \in \mathbb{H} \,\mathbin{\operatorname{and}}\, m \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, n \in \mathbb{Z}

Definitions:

Fungrim symbol	Notation	Short description
`JacobiTheta2`	$\theta_2\!\left(z, \tau\right)$	Jacobi theta function
`Pow`	${a}^{b}$	Power
`Exp`	${e}^{z}$	Exponential function
`ConstPi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane
`ZZ`	$\mathbb{Z}$	Integers

Source code for this entry:

Entry(ID("d29148"),
    Formula(Equal(JacobiTheta2(Add(z, Add(m, Mul(n, tau))), tau), Mul(Mul(Pow(-1, m), Exp(Neg(Mul(Mul(ConstPi, ConstI), Add(Mul(tau, Pow(n, 2)), Mul(Mul(2, n), z)))))), JacobiTheta2(z, tau)))),
    Variables(z, tau, m, n),
    Assumptions(And(Element(z, CC), Element(tau, HH), Element(m, ZZ), Element(n, ZZ))))

2e4da0

\theta_3\!\left(z + m + n \tau, \tau\right) = {e}^{-\pi i \left(\tau {n}^{2} + 2 n z\right)} \theta_3\!\left(z, \tau\right)

Assumptions:

z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \tau \in \mathbb{H} \,\mathbin{\operatorname{and}}\, m \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, n \in \mathbb{Z}

TeX:

\theta_3\!\left(z + m + n \tau, \tau\right) = {e}^{-\pi i \left(\tau {n}^{2} + 2 n z\right)} \theta_3\!\left(z, \tau\right)

z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \tau \in \mathbb{H} \,\mathbin{\operatorname{and}}\, m \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, n \in \mathbb{Z}

Definitions:

Fungrim symbol	Notation	Short description
`JacobiTheta3`	$\theta_3\!\left(z, \tau\right)$	Jacobi theta function
`Exp`	${e}^{z}$	Exponential function
`ConstPi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit
`Pow`	${a}^{b}$	Power
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane
`ZZ`	$\mathbb{Z}$	Integers

Source code for this entry:

Entry(ID("2e4da0"),
    Formula(Equal(JacobiTheta3(Add(z, Add(m, Mul(n, tau))), tau), Mul(Exp(Neg(Mul(Mul(ConstPi, ConstI), Add(Mul(tau, Pow(n, 2)), Mul(Mul(2, n), z))))), JacobiTheta3(z, tau)))),
    Variables(z, tau, m, n),
    Assumptions(And(Element(z, CC), Element(tau, HH), Element(m, ZZ), Element(n, ZZ))))

8d6a1d

\theta_4\!\left(z + m + n \tau, \tau\right) = {\left(-1\right)}^{n} {e}^{-\pi i \left(\tau {n}^{2} + 2 n z\right)} \theta_4\!\left(z, \tau\right)

Assumptions:

z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \tau \in \mathbb{H} \,\mathbin{\operatorname{and}}\, m \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, n \in \mathbb{Z}

TeX:

\theta_4\!\left(z + m + n \tau, \tau\right) = {\left(-1\right)}^{n} {e}^{-\pi i \left(\tau {n}^{2} + 2 n z\right)} \theta_4\!\left(z, \tau\right)

z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \tau \in \mathbb{H} \,\mathbin{\operatorname{and}}\, m \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, n \in \mathbb{Z}

Definitions:

Fungrim symbol	Notation	Short description
`JacobiTheta4`	$\theta_4\!\left(z, \tau\right)$	Jacobi theta function
`Pow`	${a}^{b}$	Power
`Exp`	${e}^{z}$	Exponential function
`ConstPi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane
`ZZ`	$\mathbb{Z}$	Integers

Source code for this entry:

Entry(ID("8d6a1d"),
    Formula(Equal(JacobiTheta4(Add(z, Add(m, Mul(n, tau))), tau), Mul(Mul(Pow(-1, n), Exp(Neg(Mul(Mul(ConstPi, ConstI), Add(Mul(tau, Pow(n, 2)), Mul(Mul(2, n), z)))))), JacobiTheta4(z, tau)))),
    Variables(z, tau, m, n),
    Assumptions(And(Element(z, CC), Element(tau, HH), Element(m, ZZ), Element(n, ZZ))))

Exponential Fourier series

Trigonometric Fourier series

Zeros

Symmetry

Periodicity

Quasi-periodicity