Argument transformations for Jacobi theta functions

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
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	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(JacobiTheta(1, Neg(z), tau), Neg(JacobiTheta(1, 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
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	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(JacobiTheta(2, Neg(z), tau), JacobiTheta(2, 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
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	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(JacobiTheta(3, Neg(z), tau), JacobiTheta(3, 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
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	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(JacobiTheta(4, Neg(z), tau), JacobiTheta(4, z, tau))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH))))

a891da

\theta_{j}\!\left(\overline{z} , \tau\right) = \overline{\theta_{j}\!\left(z , -\overline{\tau}\right)}

Assumptions:

j \in \left\{1, 2, 3, 4\right\} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H}

TeX:

\theta_{j}\!\left(\overline{z} , \tau\right) = \overline{\theta_{j}\!\left(z , -\overline{\tau}\right)}

j \in \left\{1, 2, 3, 4\right\} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H}

Definitions:

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

Source code for this entry:

Entry(ID("a891da"),
    Formula(Equal(JacobiTheta(j, Conjugate(z), tau), Conjugate(JacobiTheta(j, z, Neg(Conjugate(tau)))))),
    Variables(j, z, tau),
    Assumptions(And(Element(j, Set(1, 2, 3, 4)), 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
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	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(JacobiTheta(1, Add(z, Mul(2, n)), tau), JacobiTheta(1, 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
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	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(JacobiTheta(2, Add(z, Mul(2, n)), tau), JacobiTheta(2, z, tau))),
    Variables(z, tau, n),
    Assumptions(And(Element(z, CC), Element(tau, HH), Element(n, ZZ))))

5cdae6

\theta_{1}\!\left(z + n , \tau\right) = {\left(-1\right)}^{n} \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 + n , \tau\right) = {\left(-1\right)}^{n} \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
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	Jacobi theta function
`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("5cdae6"),
    Formula(Equal(JacobiTheta(1, Add(z, n), tau), Mul(Pow(-1, n), JacobiTheta(1, z, tau)))),
    Variables(z, tau, n),
    Assumptions(And(Element(z, CC), Element(tau, HH), Element(n, ZZ))))

f697d5

\theta_{2}\!\left(z + n , \tau\right) = {\left(-1\right)}^{n} \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 + n , \tau\right) = {\left(-1\right)}^{n} \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
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	Jacobi theta function
`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("f697d5"),
    Formula(Equal(JacobiTheta(2, Add(z, n), tau), Mul(Pow(-1, n), JacobiTheta(2, z, tau)))),
    Variables(z, tau, n),
    Assumptions(And(Element(z, CC), Element(tau, HH), Element(n, ZZ))))

e56f77

\theta_{3}\!\left(z + 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 + 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
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	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(JacobiTheta(3, Add(z, n), tau), JacobiTheta(3, z, tau))),
    Variables(z, tau, n),
    Assumptions(And(Element(z, CC), Element(tau, HH), Element(n, ZZ))))

4448f1

\theta_{4}\!\left(z + n , \tau\right) = \theta_{4}\!\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 + n , \tau\right) = \theta_{4}\!\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
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	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(JacobiTheta(4, Add(z, n), tau), JacobiTheta(4, z, tau))),
    Variables(z, tau, n),
    Assumptions(And(Element(z, CC), Element(tau, HH), Element(n, ZZ))))

d989cd

\theta_{1}\!\left(z + \tau , \tau\right) = -{e}^{-\pi i \left(2 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 , \tau\right) = -{e}^{-\pi i \left(2 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
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	Jacobi theta function
`Exp`	${e}^{z}$	Exponential function
`Pi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("d989cd"),
    Formula(Equal(JacobiTheta(1, Add(z, tau), tau), Mul(Neg(Exp(Neg(Mul(Mul(Pi, ConstI), Add(Mul(2, z), tau))))), JacobiTheta(1, z, tau)))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH))))

cd5f45

\theta_{2}\!\left(z + \tau , \tau\right) = {e}^{-\pi i \left(2 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 , \tau\right) = {e}^{-\pi i \left(2 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
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	Jacobi theta function
`Exp`	${e}^{z}$	Exponential function
`Pi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("cd5f45"),
    Formula(Equal(JacobiTheta(2, Add(z, tau), tau), Mul(Exp(Neg(Mul(Mul(Pi, ConstI), Add(Mul(2, z), tau)))), JacobiTheta(2, z, tau)))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH))))

103bfb

\theta_{3}\!\left(z + \tau , \tau\right) = {e}^{-\pi i \left(2 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 , \tau\right) = {e}^{-\pi i \left(2 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
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	Jacobi theta function
`Exp`	${e}^{z}$	Exponential function
`Pi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("103bfb"),
    Formula(Equal(JacobiTheta(3, Add(z, tau), tau), Mul(Exp(Neg(Mul(Mul(Pi, ConstI), Add(Mul(2, z), tau)))), JacobiTheta(3, z, tau)))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH))))

b83f63

\theta_{4}\!\left(z + \tau , \tau\right) = -{e}^{-\pi i \left(2 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 , \tau\right) = -{e}^{-\pi i \left(2 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
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	Jacobi theta function
`Exp`	${e}^{z}$	Exponential function
`Pi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("b83f63"),
    Formula(Equal(JacobiTheta(4, Add(z, tau), tau), Mul(Neg(Exp(Neg(Mul(Mul(Pi, ConstI), Add(Mul(2, z), tau))))), JacobiTheta(4, z, tau)))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH))))

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
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	Jacobi theta function
`Pow`	${a}^{b}$	Power
`Exp`	${e}^{z}$	Exponential function
`Pi`	$\pi$	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(JacobiTheta(1, Add(z, Add(m, Mul(n, tau))), tau), Mul(Mul(Pow(-1, Add(m, n)), Exp(Neg(Mul(Mul(Pi, ConstI), Add(Mul(tau, Pow(n, 2)), Mul(Mul(2, n), z)))))), JacobiTheta(1, 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
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	Jacobi theta function
`Pow`	${a}^{b}$	Power
`Exp`	${e}^{z}$	Exponential function
`Pi`	$\pi$	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(JacobiTheta(2, Add(z, Add(m, Mul(n, tau))), tau), Mul(Mul(Pow(-1, m), Exp(Neg(Mul(Mul(Pi, ConstI), Add(Mul(tau, Pow(n, 2)), Mul(Mul(2, n), z)))))), JacobiTheta(2, 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
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	Jacobi theta function
`Exp`	${e}^{z}$	Exponential function
`Pi`	$\pi$	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(JacobiTheta(3, Add(z, Add(m, Mul(n, tau))), tau), Mul(Exp(Neg(Mul(Mul(Pi, ConstI), Add(Mul(tau, Pow(n, 2)), Mul(Mul(2, n), z))))), JacobiTheta(3, 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
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	Jacobi theta function
`Pow`	${a}^{b}$	Power
`Exp`	${e}^{z}$	Exponential function
`Pi`	$\pi$	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(JacobiTheta(4, Add(z, Add(m, Mul(n, tau))), tau), Mul(Mul(Pow(-1, n), Exp(Neg(Mul(Mul(Pi, ConstI), Add(Mul(tau, Pow(n, 2)), Mul(Mul(2, n), z)))))), JacobiTheta(4, z, tau)))),
    Variables(z, tau, m, n),
    Assumptions(And(Element(z, CC), Element(tau, HH), Element(m, ZZ), Element(n, ZZ))))

563d18

\theta_{1}\!\left(z + \frac{1}{2} , \tau\right) = \theta_{2}\!\left(z , \tau\right)

Assumptions:

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

TeX:

\theta_{1}\!\left(z + \frac{1}{2} , \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
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	Jacobi theta function
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("563d18"),
    Formula(Equal(JacobiTheta(1, Add(z, Div(1, 2)), tau), JacobiTheta(2, z, tau))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH))))

47f6dd

\theta_{2}\!\left(z + \frac{1}{2} , \tau\right) = -\theta_{1}\!\left(z , \tau\right)

Assumptions:

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

TeX:

\theta_{2}\!\left(z + \frac{1}{2} , \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
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	Jacobi theta function
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("47f6dd"),
    Formula(Equal(JacobiTheta(2, Add(z, Div(1, 2)), tau), Neg(JacobiTheta(1, z, tau)))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH))))

7d559c

\theta_{3}\!\left(z + \frac{1}{2} , \tau\right) = \theta_{4}\!\left(z , \tau\right)

Assumptions:

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

TeX:

\theta_{3}\!\left(z + \frac{1}{2} , \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
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	Jacobi theta function
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("7d559c"),
    Formula(Equal(JacobiTheta(3, Add(z, Div(1, 2)), tau), JacobiTheta(4, z, tau))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH))))

bb2d01

\theta_{4}\!\left(z + \frac{1}{2} , \tau\right) = \theta_{3}\!\left(z , \tau\right)

Assumptions:

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

TeX:

\theta_{4}\!\left(z + \frac{1}{2} , \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
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	Jacobi theta function
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("bb2d01"),
    Formula(Equal(JacobiTheta(4, Add(z, Div(1, 2)), tau), JacobiTheta(3, z, tau))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH))))

d5a29e

\theta_{1}\!\left(z + \frac{1}{2} \tau , \tau\right) = {e}^{-\pi i \left(z + \tau / 4\right)} i \theta_{4}\!\left(z , \tau\right)

Assumptions:

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

TeX:

\theta_{1}\!\left(z + \frac{1}{2} \tau , \tau\right) = {e}^{-\pi i \left(z + \tau / 4\right)} i \theta_{4}\!\left(z , \tau\right)

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

Definitions:

Fungrim symbol	Notation	Short description
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	Jacobi theta function
`Exp`	${e}^{z}$	Exponential function
`Pi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("d5a29e"),
    Formula(Equal(JacobiTheta(1, Add(z, Mul(Div(1, 2), tau)), tau), Mul(Mul(Exp(Neg(Mul(Mul(Pi, ConstI), Add(z, Div(tau, 4))))), ConstI), JacobiTheta(4, z, tau)))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH))))

cc6d21

\theta_{2}\!\left(z + \frac{1}{2} \tau , \tau\right) = {e}^{-\pi i \left(z + \tau / 4\right)} \theta_{3}\!\left(z , \tau\right)

Assumptions:

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

TeX:

\theta_{2}\!\left(z + \frac{1}{2} \tau , \tau\right) = {e}^{-\pi i \left(z + \tau / 4\right)} \theta_{3}\!\left(z , \tau\right)

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

Definitions:

Fungrim symbol	Notation	Short description
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	Jacobi theta function
`Exp`	${e}^{z}$	Exponential function
`Pi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("cc6d21"),
    Formula(Equal(JacobiTheta(2, Add(z, Mul(Div(1, 2), tau)), tau), Mul(Exp(Neg(Mul(Mul(Pi, ConstI), Add(z, Div(tau, 4))))), JacobiTheta(3, z, tau)))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH))))

2d2dde

\theta_{3}\!\left(z + \frac{1}{2} \tau , \tau\right) = {e}^{-\pi i \left(z + \tau / 4\right)} \theta_{2}\!\left(z , \tau\right)

Assumptions:

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

TeX:

\theta_{3}\!\left(z + \frac{1}{2} \tau , \tau\right) = {e}^{-\pi i \left(z + \tau / 4\right)} \theta_{2}\!\left(z , \tau\right)

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

Definitions:

Fungrim symbol	Notation	Short description
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	Jacobi theta function
`Exp`	${e}^{z}$	Exponential function
`Pi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("2d2dde"),
    Formula(Equal(JacobiTheta(3, Add(z, Mul(Div(1, 2), tau)), tau), Mul(Exp(Neg(Mul(Mul(Pi, ConstI), Add(z, Div(tau, 4))))), JacobiTheta(2, z, tau)))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH))))

429093

\theta_{4}\!\left(z + \frac{1}{2} \tau , \tau\right) = {e}^{-\pi i \left(z + \tau / 4\right)} i \theta_{1}\!\left(z , \tau\right)

Assumptions:

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

TeX:

\theta_{4}\!\left(z + \frac{1}{2} \tau , \tau\right) = {e}^{-\pi i \left(z + \tau / 4\right)} i \theta_{1}\!\left(z , \tau\right)

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

Definitions:

Fungrim symbol	Notation	Short description
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	Jacobi theta function
`Exp`	${e}^{z}$	Exponential function
`Pi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("429093"),
    Formula(Equal(JacobiTheta(4, Add(z, Mul(Div(1, 2), tau)), tau), Mul(Mul(Exp(Neg(Mul(Mul(Pi, ConstI), Add(z, Div(tau, 4))))), ConstI), JacobiTheta(1, z, tau)))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH))))

95988c

\theta_{1}\!\left(z , \tau\right) = -\theta_{2}\!\left(z + \frac{1}{2} , \tau\right)

Assumptions:

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

TeX:

\theta_{1}\!\left(z , \tau\right) = -\theta_{2}\!\left(z + \frac{1}{2} , \tau\right)

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

Definitions:

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

Source code for this entry:

Entry(ID("95988c"),
    Formula(Equal(JacobiTheta(1, z, tau), Neg(JacobiTheta(2, Add(z, Div(1, 2)), tau)))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH))))

4c462b

\theta_{1}\!\left(z , \tau\right) = -i {e}^{\pi i \left(z + \tau / 4\right)} \theta_{4}\!\left(z + \frac{1}{2} \tau , \tau\right)

Assumptions:

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

TeX:

\theta_{1}\!\left(z , \tau\right) = -i {e}^{\pi i \left(z + \tau / 4\right)} \theta_{4}\!\left(z + \frac{1}{2} \tau , \tau\right)

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

Definitions:

Fungrim symbol	Notation	Short description
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	Jacobi theta function
`ConstI`	$i$	Imaginary unit
`Exp`	${e}^{z}$	Exponential function
`Pi`	$\pi$	The constant pi (3.14...)
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("4c462b"),
    Formula(Equal(JacobiTheta(1, z, tau), Mul(Mul(Neg(ConstI), Exp(Mul(Mul(Pi, ConstI), Add(z, Div(tau, 4))))), JacobiTheta(4, Add(z, Mul(Div(1, 2), tau)), tau)))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH))))

ed0756

\theta_{1}\!\left(z , \tau\right) = -i {e}^{\pi i \left(z + \tau / 4\right)} \theta_{3}\!\left(z + \frac{1}{2} + \frac{1}{2} \tau , \tau\right)

Assumptions:

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

TeX:

\theta_{1}\!\left(z , \tau\right) = -i {e}^{\pi i \left(z + \tau / 4\right)} \theta_{3}\!\left(z + \frac{1}{2} + \frac{1}{2} \tau , \tau\right)

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

Definitions:

Fungrim symbol	Notation	Short description
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	Jacobi theta function
`ConstI`	$i$	Imaginary unit
`Exp`	${e}^{z}$	Exponential function
`Pi`	$\pi$	The constant pi (3.14...)
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("ed0756"),
    Formula(Equal(JacobiTheta(1, z, tau), Mul(Mul(Neg(ConstI), Exp(Mul(Mul(Pi, ConstI), Add(z, Div(tau, 4))))), JacobiTheta(3, Add(Add(z, Div(1, 2)), Mul(Div(1, 2), tau)), tau)))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH))))

785668

\theta_{2}\!\left(z , \tau\right) = \theta_{1}\!\left(z + \frac{1}{2} , \tau\right)

Assumptions:

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

TeX:

\theta_{2}\!\left(z , \tau\right) = \theta_{1}\!\left(z + \frac{1}{2} , \tau\right)

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

Definitions:

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

Source code for this entry:

Entry(ID("785668"),
    Formula(Equal(JacobiTheta(2, z, tau), JacobiTheta(1, Add(z, Div(1, 2)), tau))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH))))

0878a4

\theta_{2}\!\left(z , \tau\right) = {e}^{\pi i \left(z + \tau / 4\right)} \theta_{3}\!\left(z + \frac{1}{2} \tau , \tau\right)

Assumptions:

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

TeX:

\theta_{2}\!\left(z , \tau\right) = {e}^{\pi i \left(z + \tau / 4\right)} \theta_{3}\!\left(z + \frac{1}{2} \tau , \tau\right)

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

Definitions:

Fungrim symbol	Notation	Short description
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	Jacobi theta function
`Exp`	${e}^{z}$	Exponential function
`Pi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("0878a4"),
    Formula(Equal(JacobiTheta(2, z, tau), Mul(Exp(Mul(Mul(Pi, ConstI), Add(z, Div(tau, 4)))), JacobiTheta(3, Add(z, Mul(Div(1, 2), tau)), tau)))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH))))

6a7704

\theta_{2}\!\left(z , \tau\right) = {e}^{\pi i \left(z + \tau / 4\right)} \theta_{4}\!\left(z + \frac{1}{2} + \frac{1}{2} \tau , \tau\right)

Assumptions:

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

TeX:

\theta_{2}\!\left(z , \tau\right) = {e}^{\pi i \left(z + \tau / 4\right)} \theta_{4}\!\left(z + \frac{1}{2} + \frac{1}{2} \tau , \tau\right)

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

Definitions:

Fungrim symbol	Notation	Short description
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	Jacobi theta function
`Exp`	${e}^{z}$	Exponential function
`Pi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("6a7704"),
    Formula(Equal(JacobiTheta(2, z, tau), Mul(Exp(Mul(Mul(Pi, ConstI), Add(z, Div(tau, 4)))), JacobiTheta(4, Add(Add(z, Div(1, 2)), Mul(Div(1, 2), tau)), tau)))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH))))

b3fc6d

\theta_{3}\!\left(z , \tau\right) = \theta_{4}\!\left(z + \frac{1}{2} , \tau\right)

Assumptions:

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

TeX:

\theta_{3}\!\left(z , \tau\right) = \theta_{4}\!\left(z + \frac{1}{2} , \tau\right)

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

Definitions:

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

Source code for this entry:

Entry(ID("b3fc6d"),
    Formula(Equal(JacobiTheta(3, z, tau), JacobiTheta(4, Add(z, Div(1, 2)), tau))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH))))

71d5ee

\theta_{3}\!\left(z , \tau\right) = {e}^{\pi i \left(z + \tau / 4\right)} \theta_{2}\!\left(z + \frac{1}{2} \tau , \tau\right)

Assumptions:

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

TeX:

\theta_{3}\!\left(z , \tau\right) = {e}^{\pi i \left(z + \tau / 4\right)} \theta_{2}\!\left(z + \frac{1}{2} \tau , \tau\right)

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

Definitions:

Fungrim symbol	Notation	Short description
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	Jacobi theta function
`Exp`	${e}^{z}$	Exponential function
`Pi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("71d5ee"),
    Formula(Equal(JacobiTheta(3, z, tau), Mul(Exp(Mul(Mul(Pi, ConstI), Add(z, Div(tau, 4)))), JacobiTheta(2, Add(z, Mul(Div(1, 2), tau)), tau)))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH))))

235d0d

\theta_{3}\!\left(z , \tau\right) = {e}^{\pi i \left(z + \tau / 4\right)} \theta_{1}\!\left(z + \frac{1}{2} + \frac{1}{2} \tau , \tau\right)

Assumptions:

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

TeX:

\theta_{3}\!\left(z , \tau\right) = {e}^{\pi i \left(z + \tau / 4\right)} \theta_{1}\!\left(z + \frac{1}{2} + \frac{1}{2} \tau , \tau\right)

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

Definitions:

Fungrim symbol	Notation	Short description
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	Jacobi theta function
`Exp`	${e}^{z}$	Exponential function
`Pi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("235d0d"),
    Formula(Equal(JacobiTheta(3, z, tau), Mul(Exp(Mul(Mul(Pi, ConstI), Add(z, Div(tau, 4)))), JacobiTheta(1, Add(Add(z, Div(1, 2)), Mul(Div(1, 2), tau)), tau)))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH))))

5d41b1

\theta_{4}\!\left(z , \tau\right) = \theta_{3}\!\left(z + \frac{1}{2} , \tau\right)

Assumptions:

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

TeX:

\theta_{4}\!\left(z , \tau\right) = \theta_{3}\!\left(z + \frac{1}{2} , \tau\right)

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

Definitions:

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

Source code for this entry:

Entry(ID("5d41b1"),
    Formula(Equal(JacobiTheta(4, z, tau), JacobiTheta(3, Add(z, Div(1, 2)), tau))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH))))

6d918c

\theta_{4}\!\left(z , \tau\right) = -i {e}^{\pi i \left(z + \tau / 4\right)} \theta_{1}\!\left(z + \frac{1}{2} \tau , \tau\right)

Assumptions:

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

TeX:

\theta_{4}\!\left(z , \tau\right) = -i {e}^{\pi i \left(z + \tau / 4\right)} \theta_{1}\!\left(z + \frac{1}{2} \tau , \tau\right)

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

Definitions:

Fungrim symbol	Notation	Short description
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	Jacobi theta function
`ConstI`	$i$	Imaginary unit
`Exp`	${e}^{z}$	Exponential function
`Pi`	$\pi$	The constant pi (3.14...)
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("6d918c"),
    Formula(Equal(JacobiTheta(4, z, tau), Mul(Mul(Neg(ConstI), Exp(Mul(Mul(Pi, ConstI), Add(z, Div(tau, 4))))), JacobiTheta(1, Add(z, Mul(Div(1, 2), tau)), tau)))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH))))

10ca40

\theta_{4}\!\left(z , \tau\right) = i {e}^{\pi i \left(z + \tau / 4\right)} \theta_{2}\!\left(z + \frac{1}{2} + \frac{1}{2} \tau , \tau\right)

Assumptions:

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

TeX:

\theta_{4}\!\left(z , \tau\right) = i {e}^{\pi i \left(z + \tau / 4\right)} \theta_{2}\!\left(z + \frac{1}{2} + \frac{1}{2} \tau , \tau\right)

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

Definitions:

Fungrim symbol	Notation	Short description
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	Jacobi theta function
`ConstI`	$i$	Imaginary unit
`Exp`	${e}^{z}$	Exponential function
`Pi`	$\pi$	The constant pi (3.14...)
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("10ca40"),
    Formula(Equal(JacobiTheta(4, z, tau), Mul(Mul(ConstI, Exp(Mul(Mul(Pi, ConstI), Add(z, Div(tau, 4))))), JacobiTheta(2, Add(Add(z, Div(1, 2)), Mul(Div(1, 2), tau)), tau)))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH))))

5fe58d

\theta_{1}\!\left(2 z , \tau\right) = \frac{2 \theta_{1}\!\left(z , \tau\right) \theta_{2}\!\left(z , \tau\right) \theta_{3}\!\left(z , \tau\right) \theta_{4}\!\left(z , \tau\right)}{\theta_{2}\!\left(0 , \tau\right) \theta_{3}\!\left(0 , \tau\right) \theta_{4}\!\left(0 , \tau\right)}

Assumptions:

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

TeX:

\theta_{1}\!\left(2 z , \tau\right) = \frac{2 \theta_{1}\!\left(z , \tau\right) \theta_{2}\!\left(z , \tau\right) \theta_{3}\!\left(z , \tau\right) \theta_{4}\!\left(z , \tau\right)}{\theta_{2}\!\left(0 , \tau\right) \theta_{3}\!\left(0 , \tau\right) \theta_{4}\!\left(0 , \tau\right)}

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

Definitions:

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

Source code for this entry:

Entry(ID("5fe58d"),
    Formula(Equal(JacobiTheta(1, Mul(2, z), tau), Div(Mul(Mul(Mul(Mul(2, JacobiTheta(1, z, tau)), JacobiTheta(2, z, tau)), JacobiTheta(3, z, tau)), JacobiTheta(4, z, tau)), Mul(Mul(JacobiTheta(2, 0, tau), JacobiTheta(3, 0, tau)), JacobiTheta(4, 0, tau))))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH))))

3a77e0

\theta_{2}\!\left(2 z , \tau\right) = \frac{\theta_{2}^{4}\!\left(z, \tau\right) - \theta_{1}^{4}\!\left(z, \tau\right)}{\theta_{2}^{3}\!\left(0, \tau\right)}

Assumptions:

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

TeX:

\theta_{2}\!\left(2 z , \tau\right) = \frac{\theta_{2}^{4}\!\left(z, \tau\right) - \theta_{1}^{4}\!\left(z, \tau\right)}{\theta_{2}^{3}\!\left(0, \tau\right)}

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

Definitions:

Fungrim symbol	Notation	Short description
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	Jacobi theta function
`Pow`	${a}^{b}$	Power
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("3a77e0"),
    Formula(Equal(JacobiTheta(2, Mul(2, z), tau), Div(Sub(Pow(JacobiTheta(2, z, tau), 4), Pow(JacobiTheta(1, z, tau), 4)), Pow(JacobiTheta(2, 0, tau), 3)))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH))))

e6dc09

\theta_{2}\!\left(2 z , \tau\right) = \frac{\theta_{3}^{4}\!\left(z, \tau\right) - \theta_{4}^{4}\!\left(z, \tau\right)}{\theta_{2}^{3}\!\left(0, \tau\right)}

Assumptions:

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

TeX:

\theta_{2}\!\left(2 z , \tau\right) = \frac{\theta_{3}^{4}\!\left(z, \tau\right) - \theta_{4}^{4}\!\left(z, \tau\right)}{\theta_{2}^{3}\!\left(0, \tau\right)}

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

Definitions:

Fungrim symbol	Notation	Short description
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	Jacobi theta function
`Pow`	${a}^{b}$	Power
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("e6dc09"),
    Formula(Equal(JacobiTheta(2, Mul(2, z), tau), Div(Sub(Pow(JacobiTheta(3, z, tau), 4), Pow(JacobiTheta(4, z, tau), 4)), Pow(JacobiTheta(2, 0, tau), 3)))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH))))

aaa582

\theta_{2}\!\left(2 z , \tau\right) = \frac{\theta_{2}^{2}\!\left(z, \tau\right) \theta_{3}^{2}\!\left(z, \tau\right) - \theta_{1}^{2}\!\left(z, \tau\right) \theta_{4}^{2}\!\left(z, \tau\right)}{\theta_{2}\!\left(0 , \tau\right) \theta_{3}^{2}\!\left(0, \tau\right)}

Assumptions:

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

TeX:

\theta_{2}\!\left(2 z , \tau\right) = \frac{\theta_{2}^{2}\!\left(z, \tau\right) \theta_{3}^{2}\!\left(z, \tau\right) - \theta_{1}^{2}\!\left(z, \tau\right) \theta_{4}^{2}\!\left(z, \tau\right)}{\theta_{2}\!\left(0 , \tau\right) \theta_{3}^{2}\!\left(0, \tau\right)}

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

Definitions:

Fungrim symbol	Notation	Short description
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	Jacobi theta function
`Pow`	${a}^{b}$	Power
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("aaa582"),
    Formula(Equal(JacobiTheta(2, Mul(2, z), tau), Div(Sub(Mul(Pow(JacobiTheta(2, z, tau), 2), Pow(JacobiTheta(3, z, tau), 2)), Mul(Pow(JacobiTheta(1, z, tau), 2), Pow(JacobiTheta(4, z, tau), 2))), Mul(JacobiTheta(2, 0, tau), Pow(JacobiTheta(3, 0, tau), 2))))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH))))

b1d07b

\theta_{2}\!\left(2 z , \tau\right) = \frac{\theta_{2}^{2}\!\left(z, \tau\right) \theta_{4}^{2}\!\left(z, \tau\right) - \theta_{1}^{2}\!\left(z, \tau\right) \theta_{3}^{2}\!\left(z, \tau\right)}{\theta_{2}\!\left(0 , \tau\right) \theta_{4}^{2}\!\left(0, \tau\right)}

Assumptions:

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

TeX:

\theta_{2}\!\left(2 z , \tau\right) = \frac{\theta_{2}^{2}\!\left(z, \tau\right) \theta_{4}^{2}\!\left(z, \tau\right) - \theta_{1}^{2}\!\left(z, \tau\right) \theta_{3}^{2}\!\left(z, \tau\right)}{\theta_{2}\!\left(0 , \tau\right) \theta_{4}^{2}\!\left(0, \tau\right)}

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

Definitions:

Fungrim symbol	Notation	Short description
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	Jacobi theta function
`Pow`	${a}^{b}$	Power
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("b1d07b"),
    Formula(Equal(JacobiTheta(2, Mul(2, z), tau), Div(Sub(Mul(Pow(JacobiTheta(2, z, tau), 2), Pow(JacobiTheta(4, z, tau), 2)), Mul(Pow(JacobiTheta(1, z, tau), 2), Pow(JacobiTheta(3, z, tau), 2))), Mul(JacobiTheta(2, 0, tau), Pow(JacobiTheta(4, 0, tau), 2))))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH))))

20d581

\theta_{3}\!\left(2 z , \tau\right) = \frac{\theta_{1}^{4}\!\left(z, \tau\right) + \theta_{3}^{4}\!\left(z, \tau\right)}{\theta_{3}^{3}\!\left(0, \tau\right)}

Assumptions:

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

TeX:

\theta_{3}\!\left(2 z , \tau\right) = \frac{\theta_{1}^{4}\!\left(z, \tau\right) + \theta_{3}^{4}\!\left(z, \tau\right)}{\theta_{3}^{3}\!\left(0, \tau\right)}

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

Definitions:

Fungrim symbol	Notation	Short description
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	Jacobi theta function
`Pow`	${a}^{b}$	Power
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("20d581"),
    Formula(Equal(JacobiTheta(3, Mul(2, z), tau), Div(Add(Pow(JacobiTheta(1, z, tau), 4), Pow(JacobiTheta(3, z, tau), 4)), Pow(JacobiTheta(3, 0, tau), 3)))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH))))

ed3ff9

\theta_{3}\!\left(2 z , \tau\right) = \frac{\theta_{2}^{4}\!\left(z, \tau\right) + \theta_{4}^{4}\!\left(z, \tau\right)}{\theta_{3}^{3}\!\left(0, \tau\right)}

Assumptions:

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

TeX:

\theta_{3}\!\left(2 z , \tau\right) = \frac{\theta_{2}^{4}\!\left(z, \tau\right) + \theta_{4}^{4}\!\left(z, \tau\right)}{\theta_{3}^{3}\!\left(0, \tau\right)}

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

Definitions:

Fungrim symbol	Notation	Short description
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	Jacobi theta function
`Pow`	${a}^{b}$	Power
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("ed3ff9"),
    Formula(Equal(JacobiTheta(3, Mul(2, z), tau), Div(Add(Pow(JacobiTheta(2, z, tau), 4), Pow(JacobiTheta(4, z, tau), 4)), Pow(JacobiTheta(3, 0, tau), 3)))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH))))

794106

\theta_{3}\!\left(2 z , \tau\right) = \frac{\theta_{2}^{2}\!\left(z, \tau\right) \theta_{3}^{2}\!\left(z, \tau\right) + \theta_{1}^{2}\!\left(z, \tau\right) \theta_{4}^{2}\!\left(z, \tau\right)}{\theta_{3}\!\left(0 , \tau\right) \theta_{2}^{2}\!\left(0, \tau\right)}

Assumptions:

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

TeX:

\theta_{3}\!\left(2 z , \tau\right) = \frac{\theta_{2}^{2}\!\left(z, \tau\right) \theta_{3}^{2}\!\left(z, \tau\right) + \theta_{1}^{2}\!\left(z, \tau\right) \theta_{4}^{2}\!\left(z, \tau\right)}{\theta_{3}\!\left(0 , \tau\right) \theta_{2}^{2}\!\left(0, \tau\right)}

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

Definitions:

Fungrim symbol	Notation	Short description
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	Jacobi theta function
`Pow`	${a}^{b}$	Power
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("794106"),
    Formula(Equal(JacobiTheta(3, Mul(2, z), tau), Div(Add(Mul(Pow(JacobiTheta(2, z, tau), 2), Pow(JacobiTheta(3, z, tau), 2)), Mul(Pow(JacobiTheta(1, z, tau), 2), Pow(JacobiTheta(4, z, tau), 2))), Mul(JacobiTheta(3, 0, tau), Pow(JacobiTheta(2, 0, tau), 2))))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH))))

a94b43

\theta_{3}\!\left(2 z , \tau\right) = \frac{\theta_{3}^{2}\!\left(z, \tau\right) \theta_{4}^{2}\!\left(z, \tau\right) - \theta_{1}^{2}\!\left(z, \tau\right) \theta_{2}^{2}\!\left(z, \tau\right)}{\theta_{3}\!\left(0 , \tau\right) \theta_{4}^{2}\!\left(0, \tau\right)}

Assumptions:

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

TeX:

\theta_{3}\!\left(2 z , \tau\right) = \frac{\theta_{3}^{2}\!\left(z, \tau\right) \theta_{4}^{2}\!\left(z, \tau\right) - \theta_{1}^{2}\!\left(z, \tau\right) \theta_{2}^{2}\!\left(z, \tau\right)}{\theta_{3}\!\left(0 , \tau\right) \theta_{4}^{2}\!\left(0, \tau\right)}

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

Definitions:

Fungrim symbol	Notation	Short description
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	Jacobi theta function
`Pow`	${a}^{b}$	Power
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("a94b43"),
    Formula(Equal(JacobiTheta(3, Mul(2, z), tau), Div(Sub(Mul(Pow(JacobiTheta(3, z, tau), 2), Pow(JacobiTheta(4, z, tau), 2)), Mul(Pow(JacobiTheta(1, z, tau), 2), Pow(JacobiTheta(2, z, tau), 2))), Mul(JacobiTheta(3, 0, tau), Pow(JacobiTheta(4, 0, tau), 2))))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH))))

8b825c

\theta_{4}\!\left(2 z , \tau\right) = \frac{\theta_{4}^{4}\!\left(z, \tau\right) - \theta_{1}^{4}\!\left(z, \tau\right)}{\theta_{4}^{3}\!\left(0, \tau\right)}

Assumptions:

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

TeX:

\theta_{4}\!\left(2 z , \tau\right) = \frac{\theta_{4}^{4}\!\left(z, \tau\right) - \theta_{1}^{4}\!\left(z, \tau\right)}{\theta_{4}^{3}\!\left(0, \tau\right)}

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

Definitions:

Fungrim symbol	Notation	Short description
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	Jacobi theta function
`Pow`	${a}^{b}$	Power
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("8b825c"),
    Formula(Equal(JacobiTheta(4, Mul(2, z), tau), Div(Sub(Pow(JacobiTheta(4, z, tau), 4), Pow(JacobiTheta(1, z, tau), 4)), Pow(JacobiTheta(4, 0, tau), 3)))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH))))

7131cd

\theta_{4}\!\left(2 z , \tau\right) = \frac{\theta_{3}^{4}\!\left(z, \tau\right) - \theta_{2}^{4}\!\left(z, \tau\right)}{\theta_{4}^{3}\!\left(0, \tau\right)}

Assumptions:

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

TeX:

\theta_{4}\!\left(2 z , \tau\right) = \frac{\theta_{3}^{4}\!\left(z, \tau\right) - \theta_{2}^{4}\!\left(z, \tau\right)}{\theta_{4}^{3}\!\left(0, \tau\right)}

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

Definitions:

Fungrim symbol	Notation	Short description
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	Jacobi theta function
`Pow`	${a}^{b}$	Power
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("7131cd"),
    Formula(Equal(JacobiTheta(4, Mul(2, z), tau), Div(Sub(Pow(JacobiTheta(3, z, tau), 4), Pow(JacobiTheta(2, z, tau), 4)), Pow(JacobiTheta(4, 0, tau), 3)))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH))))

931201

\theta_{4}\!\left(2 z , \tau\right) = \frac{\theta_{1}^{2}\!\left(z, \tau\right) \theta_{3}^{2}\!\left(z, \tau\right) + \theta_{2}^{2}\!\left(z, \tau\right) \theta_{4}^{2}\!\left(z, \tau\right)}{\theta_{4}\!\left(0 , \tau\right) \theta_{2}^{2}\!\left(0, \tau\right)}

Assumptions:

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

TeX:

\theta_{4}\!\left(2 z , \tau\right) = \frac{\theta_{1}^{2}\!\left(z, \tau\right) \theta_{3}^{2}\!\left(z, \tau\right) + \theta_{2}^{2}\!\left(z, \tau\right) \theta_{4}^{2}\!\left(z, \tau\right)}{\theta_{4}\!\left(0 , \tau\right) \theta_{2}^{2}\!\left(0, \tau\right)}

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

Definitions:

Fungrim symbol	Notation	Short description
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	Jacobi theta function
`Pow`	${a}^{b}$	Power
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("931201"),
    Formula(Equal(JacobiTheta(4, Mul(2, z), tau), Div(Add(Mul(Pow(JacobiTheta(1, z, tau), 2), Pow(JacobiTheta(3, z, tau), 2)), Mul(Pow(JacobiTheta(2, z, tau), 2), Pow(JacobiTheta(4, z, tau), 2))), Mul(JacobiTheta(4, 0, tau), Pow(JacobiTheta(2, 0, tau), 2))))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH))))

21dc98

\theta_{4}\!\left(2 z , \tau\right) = \frac{\theta_{1}^{2}\!\left(z, \tau\right) \theta_{2}^{2}\!\left(z, \tau\right) + \theta_{3}^{2}\!\left(z, \tau\right) \theta_{4}^{2}\!\left(z, \tau\right)}{\theta_{4}\!\left(0 , \tau\right) \theta_{3}^{2}\!\left(0, \tau\right)}

Assumptions:

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

TeX:

\theta_{4}\!\left(2 z , \tau\right) = \frac{\theta_{1}^{2}\!\left(z, \tau\right) \theta_{2}^{2}\!\left(z, \tau\right) + \theta_{3}^{2}\!\left(z, \tau\right) \theta_{4}^{2}\!\left(z, \tau\right)}{\theta_{4}\!\left(0 , \tau\right) \theta_{3}^{2}\!\left(0, \tau\right)}

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

Definitions:

Fungrim symbol	Notation	Short description
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	Jacobi theta function
`Pow`	${a}^{b}$	Power
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("21dc98"),
    Formula(Equal(JacobiTheta(4, Mul(2, z), tau), Div(Add(Mul(Pow(JacobiTheta(1, z, tau), 2), Pow(JacobiTheta(2, z, tau), 2)), Mul(Pow(JacobiTheta(3, z, tau), 2), Pow(JacobiTheta(4, z, tau), 2))), Mul(JacobiTheta(4, 0, tau), Pow(JacobiTheta(3, 0, tau), 2))))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH))))

1792a9

\theta_{1}\!\left(z , \tau\right) \theta_{1}\!\left(w , \tau\right) = \theta_{3}\!\left(z + w , 2 \tau\right) \theta_{2}\!\left(z - w , 2 \tau\right) - \theta_{2}\!\left(z + w , 2 \tau\right) \theta_{3}\!\left(z - w , 2 \tau\right)

Assumptions:

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

TeX:

\theta_{1}\!\left(z , \tau\right) \theta_{1}\!\left(w , \tau\right) = \theta_{3}\!\left(z + w , 2 \tau\right) \theta_{2}\!\left(z - w , 2 \tau\right) - \theta_{2}\!\left(z + w , 2 \tau\right) \theta_{3}\!\left(z - w , 2 \tau\right)

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

Definitions:

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

Source code for this entry:

Entry(ID("1792a9"),
    Formula(Equal(Mul(JacobiTheta(1, z, tau), JacobiTheta(1, w, tau)), Sub(Mul(JacobiTheta(3, Add(z, w), Mul(2, tau)), JacobiTheta(2, Sub(z, w), Mul(2, tau))), Mul(JacobiTheta(2, Add(z, w), Mul(2, tau)), JacobiTheta(3, Sub(z, w), Mul(2, tau)))))),
    Variables(z, w, tau),
    Assumptions(And(Element(z, CC), Element(w, tau), Element(tau, HH))))

5f9e54

\theta_{1}\!\left(z , \tau\right) \theta_{2}\!\left(w , \tau\right) = \theta_{1}\!\left(z + w , 2 \tau\right) \theta_{4}\!\left(z - w , 2 \tau\right) + \theta_{4}\!\left(z + w , 2 \tau\right) \theta_{1}\!\left(z - w , 2 \tau\right)

Assumptions:

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

TeX:

\theta_{1}\!\left(z , \tau\right) \theta_{2}\!\left(w , \tau\right) = \theta_{1}\!\left(z + w , 2 \tau\right) \theta_{4}\!\left(z - w , 2 \tau\right) + \theta_{4}\!\left(z + w , 2 \tau\right) \theta_{1}\!\left(z - w , 2 \tau\right)

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

Definitions:

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

Source code for this entry:

Entry(ID("5f9e54"),
    Formula(Equal(Mul(JacobiTheta(1, z, tau), JacobiTheta(2, w, tau)), Add(Mul(JacobiTheta(1, Add(z, w), Mul(2, tau)), JacobiTheta(4, Sub(z, w), Mul(2, tau))), Mul(JacobiTheta(4, Add(z, w), Mul(2, tau)), JacobiTheta(1, Sub(z, w), Mul(2, tau)))))),
    Variables(z, w, tau),
    Assumptions(And(Element(z, CC), Element(w, tau), Element(tau, HH))))

9a9487

\theta_{2}\!\left(z , \tau\right) \theta_{2}\!\left(w , \tau\right) = \theta_{2}\!\left(z + w , 2 \tau\right) \theta_{3}\!\left(z - w , 2 \tau\right) + \theta_{3}\!\left(z + w , 2 \tau\right) \theta_{2}\!\left(z - w , 2 \tau\right)

Assumptions:

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

TeX:

\theta_{2}\!\left(z , \tau\right) \theta_{2}\!\left(w , \tau\right) = \theta_{2}\!\left(z + w , 2 \tau\right) \theta_{3}\!\left(z - w , 2 \tau\right) + \theta_{3}\!\left(z + w , 2 \tau\right) \theta_{2}\!\left(z - w , 2 \tau\right)

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

Definitions:

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

Source code for this entry:

Entry(ID("9a9487"),
    Formula(Equal(Mul(JacobiTheta(2, z, tau), JacobiTheta(2, w, tau)), Add(Mul(JacobiTheta(2, Add(z, w), Mul(2, tau)), JacobiTheta(3, Sub(z, w), Mul(2, tau))), Mul(JacobiTheta(3, Add(z, w), Mul(2, tau)), JacobiTheta(2, Sub(z, w), Mul(2, tau)))))),
    Variables(z, w, tau),
    Assumptions(And(Element(z, CC), Element(w, tau), Element(tau, HH))))

f4554f

\theta_{3}\!\left(z , \tau\right) \theta_{3}\!\left(w , \tau\right) = \theta_{3}\!\left(z + w , 2 \tau\right) \theta_{3}\!\left(z - w , 2 \tau\right) + \theta_{2}\!\left(z + w , 2 \tau\right) \theta_{2}\!\left(z - w , 2 \tau\right)

Assumptions:

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

TeX:

\theta_{3}\!\left(z , \tau\right) \theta_{3}\!\left(w , \tau\right) = \theta_{3}\!\left(z + w , 2 \tau\right) \theta_{3}\!\left(z - w , 2 \tau\right) + \theta_{2}\!\left(z + w , 2 \tau\right) \theta_{2}\!\left(z - w , 2 \tau\right)

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

Definitions:

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

Source code for this entry:

Entry(ID("f4554f"),
    Formula(Equal(Mul(JacobiTheta(3, z, tau), JacobiTheta(3, w, tau)), Add(Mul(JacobiTheta(3, Add(z, w), Mul(2, tau)), JacobiTheta(3, Sub(z, w), Mul(2, tau))), Mul(JacobiTheta(2, Add(z, w), Mul(2, tau)), JacobiTheta(2, Sub(z, w), Mul(2, tau)))))),
    Variables(z, w, tau),
    Assumptions(And(Element(z, CC), Element(w, tau), Element(tau, HH))))

d36e97

\theta_{3}\!\left(z , \tau\right) \theta_{4}\!\left(w , \tau\right) = \theta_{4}\!\left(z + w , 2 \tau\right) \theta_{4}\!\left(z - w , 2 \tau\right) - \theta_{1}\!\left(z + w , 2 \tau\right) \theta_{1}\!\left(z - w , 2 \tau\right)

Assumptions:

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

TeX:

\theta_{3}\!\left(z , \tau\right) \theta_{4}\!\left(w , \tau\right) = \theta_{4}\!\left(z + w , 2 \tau\right) \theta_{4}\!\left(z - w , 2 \tau\right) - \theta_{1}\!\left(z + w , 2 \tau\right) \theta_{1}\!\left(z - w , 2 \tau\right)

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

Definitions:

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

Source code for this entry:

Entry(ID("d36e97"),
    Formula(Equal(Mul(JacobiTheta(3, z, tau), JacobiTheta(4, w, tau)), Sub(Mul(JacobiTheta(4, Add(z, w), Mul(2, tau)), JacobiTheta(4, Sub(z, w), Mul(2, tau))), Mul(JacobiTheta(1, Add(z, w), Mul(2, tau)), JacobiTheta(1, Sub(z, w), Mul(2, tau)))))),
    Variables(z, w, tau),
    Assumptions(And(Element(z, CC), Element(w, tau), Element(tau, HH))))

73eb5d

\theta_{4}\!\left(z , \tau\right) \theta_{4}\!\left(w , \tau\right) = \theta_{3}\!\left(z + w , 2 \tau\right) \theta_{3}\!\left(z - w , 2 \tau\right) - \theta_{2}\!\left(z + w , 2 \tau\right) \theta_{2}\!\left(z - w , 2 \tau\right)

Assumptions:

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

TeX:

\theta_{4}\!\left(z , \tau\right) \theta_{4}\!\left(w , \tau\right) = \theta_{3}\!\left(z + w , 2 \tau\right) \theta_{3}\!\left(z - w , 2 \tau\right) - \theta_{2}\!\left(z + w , 2 \tau\right) \theta_{2}\!\left(z - w , 2 \tau\right)

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

Definitions:

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

Source code for this entry:

Entry(ID("73eb5d"),
    Formula(Equal(Mul(JacobiTheta(4, z, tau), JacobiTheta(4, w, tau)), Sub(Mul(JacobiTheta(3, Add(z, w), Mul(2, tau)), JacobiTheta(3, Sub(z, w), Mul(2, tau))), Mul(JacobiTheta(2, Add(z, w), Mul(2, tau)), JacobiTheta(2, Sub(z, w), Mul(2, tau)))))),
    Variables(z, w, tau),
    Assumptions(And(Element(z, CC), Element(w, tau), Element(tau, HH))))

34d1c6

\theta_{3}\!\left(0 , \tau\right) \theta_{4}\!\left(0 , \tau\right) \theta_{1}\!\left(z + w , \tau\right) \theta_{2}\!\left(z - w , \tau\right) = \theta_{1}\!\left(z , \tau\right) \theta_{2}\!\left(z , \tau\right) \theta_{3}\!\left(w , \tau\right) \theta_{4}\!\left(w , \tau\right) + \theta_{3}\!\left(z , \tau\right) \theta_{4}\!\left(z , \tau\right) \theta_{1}\!\left(w , \tau\right) \theta_{2}\!\left(w , \tau\right)

Assumptions:

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

TeX:

\theta_{3}\!\left(0 , \tau\right) \theta_{4}\!\left(0 , \tau\right) \theta_{1}\!\left(z + w , \tau\right) \theta_{2}\!\left(z - w , \tau\right) = \theta_{1}\!\left(z , \tau\right) \theta_{2}\!\left(z , \tau\right) \theta_{3}\!\left(w , \tau\right) \theta_{4}\!\left(w , \tau\right) + \theta_{3}\!\left(z , \tau\right) \theta_{4}\!\left(z , \tau\right) \theta_{1}\!\left(w , \tau\right) \theta_{2}\!\left(w , \tau\right)

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

Definitions:

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

Source code for this entry:

Entry(ID("34d1c6"),
    Formula(Equal(Mul(Mul(Mul(JacobiTheta(3, 0, tau), JacobiTheta(4, 0, tau)), JacobiTheta(1, Add(z, w), tau)), JacobiTheta(2, Sub(z, w), tau)), Add(Mul(Mul(Mul(JacobiTheta(1, z, tau), JacobiTheta(2, z, tau)), JacobiTheta(3, w, tau)), JacobiTheta(4, w, tau)), Mul(Mul(Mul(JacobiTheta(3, z, tau), JacobiTheta(4, z, tau)), JacobiTheta(1, w, tau)), JacobiTheta(2, w, tau))))),
    Variables(z, w, tau),
    Assumptions(And(Element(z, CC), Element(w, tau), Element(tau, HH))))

47e587

\theta_{2}\!\left(0 , \tau\right) \theta_{4}\!\left(0 , \tau\right) \theta_{1}\!\left(z + w , \tau\right) \theta_{3}\!\left(z - w , \tau\right) = \theta_{1}\!\left(z , \tau\right) \theta_{3}\!\left(z , \tau\right) \theta_{2}\!\left(w , \tau\right) \theta_{4}\!\left(w , \tau\right) + \theta_{2}\!\left(z , \tau\right) \theta_{4}\!\left(z , \tau\right) \theta_{1}\!\left(w , \tau\right) \theta_{3}\!\left(w , \tau\right)

Assumptions:

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

TeX:

\theta_{2}\!\left(0 , \tau\right) \theta_{4}\!\left(0 , \tau\right) \theta_{1}\!\left(z + w , \tau\right) \theta_{3}\!\left(z - w , \tau\right) = \theta_{1}\!\left(z , \tau\right) \theta_{3}\!\left(z , \tau\right) \theta_{2}\!\left(w , \tau\right) \theta_{4}\!\left(w , \tau\right) + \theta_{2}\!\left(z , \tau\right) \theta_{4}\!\left(z , \tau\right) \theta_{1}\!\left(w , \tau\right) \theta_{3}\!\left(w , \tau\right)

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

Definitions:

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

Source code for this entry:

Entry(ID("47e587"),
    Formula(Equal(Mul(Mul(Mul(JacobiTheta(2, 0, tau), JacobiTheta(4, 0, tau)), JacobiTheta(1, Add(z, w), tau)), JacobiTheta(3, Sub(z, w), tau)), Add(Mul(Mul(Mul(JacobiTheta(1, z, tau), JacobiTheta(3, z, tau)), JacobiTheta(2, w, tau)), JacobiTheta(4, w, tau)), Mul(Mul(Mul(JacobiTheta(2, z, tau), JacobiTheta(4, z, tau)), JacobiTheta(1, w, tau)), JacobiTheta(3, w, tau))))),
    Variables(z, w, tau),
    Assumptions(And(Element(z, CC), Element(w, tau), Element(tau, HH))))

ee8617

\theta_{2}\!\left(0 , \tau\right) \theta_{3}\!\left(0 , \tau\right) \theta_{1}\!\left(z + w , \tau\right) \theta_{4}\!\left(z - w , \tau\right) = \theta_{1}\!\left(z , \tau\right) \theta_{4}\!\left(z , \tau\right) \theta_{2}\!\left(w , \tau\right) \theta_{3}\!\left(w , \tau\right) + \theta_{2}\!\left(z , \tau\right) \theta_{3}\!\left(z , \tau\right) \theta_{1}\!\left(w , \tau\right) \theta_{4}\!\left(w , \tau\right)

Assumptions:

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

TeX:

\theta_{2}\!\left(0 , \tau\right) \theta_{3}\!\left(0 , \tau\right) \theta_{1}\!\left(z + w , \tau\right) \theta_{4}\!\left(z - w , \tau\right) = \theta_{1}\!\left(z , \tau\right) \theta_{4}\!\left(z , \tau\right) \theta_{2}\!\left(w , \tau\right) \theta_{3}\!\left(w , \tau\right) + \theta_{2}\!\left(z , \tau\right) \theta_{3}\!\left(z , \tau\right) \theta_{1}\!\left(w , \tau\right) \theta_{4}\!\left(w , \tau\right)

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

Definitions:

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

Source code for this entry:

Entry(ID("ee8617"),
    Formula(Equal(Mul(Mul(Mul(JacobiTheta(2, 0, tau), JacobiTheta(3, 0, tau)), JacobiTheta(1, Add(z, w), tau)), JacobiTheta(4, Sub(z, w), tau)), Add(Mul(Mul(Mul(JacobiTheta(1, z, tau), JacobiTheta(4, z, tau)), JacobiTheta(2, w, tau)), JacobiTheta(3, w, tau)), Mul(Mul(Mul(JacobiTheta(2, z, tau), JacobiTheta(3, z, tau)), JacobiTheta(1, w, tau)), JacobiTheta(4, w, tau))))),
    Variables(z, w, tau),
    Assumptions(And(Element(z, CC), Element(w, tau), Element(tau, HH))))

dfea7d

\theta_{2}\!\left(0 , \tau\right) \theta_{3}\!\left(0 , \tau\right) \theta_{2}\!\left(z + w , \tau\right) \theta_{3}\!\left(z - w , \tau\right) = \theta_{2}\!\left(z , \tau\right) \theta_{3}\!\left(z , \tau\right) \theta_{2}\!\left(w , \tau\right) \theta_{3}\!\left(w , \tau\right) - \theta_{1}\!\left(z , \tau\right) \theta_{4}\!\left(z , \tau\right) \theta_{1}\!\left(w , \tau\right) \theta_{4}\!\left(w , \tau\right)

Assumptions:

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

TeX:

\theta_{2}\!\left(0 , \tau\right) \theta_{3}\!\left(0 , \tau\right) \theta_{2}\!\left(z + w , \tau\right) \theta_{3}\!\left(z - w , \tau\right) = \theta_{2}\!\left(z , \tau\right) \theta_{3}\!\left(z , \tau\right) \theta_{2}\!\left(w , \tau\right) \theta_{3}\!\left(w , \tau\right) - \theta_{1}\!\left(z , \tau\right) \theta_{4}\!\left(z , \tau\right) \theta_{1}\!\left(w , \tau\right) \theta_{4}\!\left(w , \tau\right)

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

Definitions:

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

Source code for this entry:

Entry(ID("dfea7d"),
    Formula(Equal(Mul(Mul(Mul(JacobiTheta(2, 0, tau), JacobiTheta(3, 0, tau)), JacobiTheta(2, Add(z, w), tau)), JacobiTheta(3, Sub(z, w), tau)), Sub(Mul(Mul(Mul(JacobiTheta(2, z, tau), JacobiTheta(3, z, tau)), JacobiTheta(2, w, tau)), JacobiTheta(3, w, tau)), Mul(Mul(Mul(JacobiTheta(1, z, tau), JacobiTheta(4, z, tau)), JacobiTheta(1, w, tau)), JacobiTheta(4, w, tau))))),
    Variables(z, w, tau),
    Assumptions(And(Element(z, CC), Element(w, tau), Element(tau, HH))))

9973ef

\theta_{2}\!\left(0 , \tau\right) \theta_{4}\!\left(0 , \tau\right) \theta_{2}\!\left(z + w , \tau\right) \theta_{4}\!\left(z - w , \tau\right) = \theta_{2}\!\left(z , \tau\right) \theta_{4}\!\left(z , \tau\right) \theta_{2}\!\left(w , \tau\right) \theta_{4}\!\left(w , \tau\right) - \theta_{1}\!\left(z , \tau\right) \theta_{3}\!\left(z , \tau\right) \theta_{1}\!\left(w , \tau\right) \theta_{3}\!\left(w , \tau\right)

Assumptions:

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

TeX:

\theta_{2}\!\left(0 , \tau\right) \theta_{4}\!\left(0 , \tau\right) \theta_{2}\!\left(z + w , \tau\right) \theta_{4}\!\left(z - w , \tau\right) = \theta_{2}\!\left(z , \tau\right) \theta_{4}\!\left(z , \tau\right) \theta_{2}\!\left(w , \tau\right) \theta_{4}\!\left(w , \tau\right) - \theta_{1}\!\left(z , \tau\right) \theta_{3}\!\left(z , \tau\right) \theta_{1}\!\left(w , \tau\right) \theta_{3}\!\left(w , \tau\right)

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

Definitions:

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

Source code for this entry:

Entry(ID("9973ef"),
    Formula(Equal(Mul(Mul(Mul(JacobiTheta(2, 0, tau), JacobiTheta(4, 0, tau)), JacobiTheta(2, Add(z, w), tau)), JacobiTheta(4, Sub(z, w), tau)), Sub(Mul(Mul(Mul(JacobiTheta(2, z, tau), JacobiTheta(4, z, tau)), JacobiTheta(2, w, tau)), JacobiTheta(4, w, tau)), Mul(Mul(Mul(JacobiTheta(1, z, tau), JacobiTheta(3, z, tau)), JacobiTheta(1, w, tau)), JacobiTheta(3, w, tau))))),
    Variables(z, w, tau),
    Assumptions(And(Element(z, CC), Element(w, tau), Element(tau, HH))))

077394

\theta_{3}\!\left(0 , \tau\right) \theta_{4}\!\left(0 , \tau\right) \theta_{3}\!\left(z + w , \tau\right) \theta_{4}\!\left(z - w , \tau\right) = \theta_{3}\!\left(z , \tau\right) \theta_{4}\!\left(z , \tau\right) \theta_{3}\!\left(w , \tau\right) \theta_{4}\!\left(w , \tau\right) - \theta_{1}\!\left(z , \tau\right) \theta_{2}\!\left(z , \tau\right) \theta_{1}\!\left(w , \tau\right) \theta_{2}\!\left(w , \tau\right)

Assumptions:

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

TeX:

\theta_{3}\!\left(0 , \tau\right) \theta_{4}\!\left(0 , \tau\right) \theta_{3}\!\left(z + w , \tau\right) \theta_{4}\!\left(z - w , \tau\right) = \theta_{3}\!\left(z , \tau\right) \theta_{4}\!\left(z , \tau\right) \theta_{3}\!\left(w , \tau\right) \theta_{4}\!\left(w , \tau\right) - \theta_{1}\!\left(z , \tau\right) \theta_{2}\!\left(z , \tau\right) \theta_{1}\!\left(w , \tau\right) \theta_{2}\!\left(w , \tau\right)

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

Definitions:

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

Source code for this entry:

Entry(ID("077394"),
    Formula(Equal(Mul(Mul(Mul(JacobiTheta(3, 0, tau), JacobiTheta(4, 0, tau)), JacobiTheta(3, Add(z, w), tau)), JacobiTheta(4, Sub(z, w), tau)), Sub(Mul(Mul(Mul(JacobiTheta(3, z, tau), JacobiTheta(4, z, tau)), JacobiTheta(3, w, tau)), JacobiTheta(4, w, tau)), Mul(Mul(Mul(JacobiTheta(1, z, tau), JacobiTheta(2, z, tau)), JacobiTheta(1, w, tau)), JacobiTheta(2, w, tau))))),
    Variables(z, w, tau),
    Assumptions(And(Element(z, CC), Element(w, tau), Element(tau, HH))))

45165c

\theta_{1}\!\left(z + w , \tau\right) \theta_{1}\!\left(z - w , \tau\right) \theta_{2}^{2}\!\left(0, \tau\right) = \theta_{1}^{2}\!\left(z, \tau\right) \theta_{2}^{2}\!\left(w, \tau\right) - \theta_{2}^{2}\!\left(z, \tau\right) \theta_{1}^{2}\!\left(w, \tau\right) = \theta_{4}^{2}\!\left(z, \tau\right) \theta_{3}^{2}\!\left(w, \tau\right) - \theta_{3}^{2}\!\left(z, \tau\right) \theta_{4}^{2}\!\left(w, \tau\right)

Assumptions:

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

TeX:

\theta_{1}\!\left(z + w , \tau\right) \theta_{1}\!\left(z - w , \tau\right) \theta_{2}^{2}\!\left(0, \tau\right) = \theta_{1}^{2}\!\left(z, \tau\right) \theta_{2}^{2}\!\left(w, \tau\right) - \theta_{2}^{2}\!\left(z, \tau\right) \theta_{1}^{2}\!\left(w, \tau\right) = \theta_{4}^{2}\!\left(z, \tau\right) \theta_{3}^{2}\!\left(w, \tau\right) - \theta_{3}^{2}\!\left(z, \tau\right) \theta_{4}^{2}\!\left(w, \tau\right)

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

Definitions:

Fungrim symbol	Notation	Short description
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	Jacobi theta function
`Pow`	${a}^{b}$	Power
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("45165c"),
    Formula(Equal(Mul(Mul(JacobiTheta(1, Add(z, w), tau), JacobiTheta(1, Sub(z, w), tau)), Pow(JacobiTheta(2, 0, tau), 2)), Sub(Mul(Pow(JacobiTheta(1, z, tau), 2), Pow(JacobiTheta(2, w, tau), 2)), Mul(Pow(JacobiTheta(2, z, tau), 2), Pow(JacobiTheta(1, w, tau), 2))), Sub(Mul(Pow(JacobiTheta(4, z, tau), 2), Pow(JacobiTheta(3, w, tau), 2)), Mul(Pow(JacobiTheta(3, z, tau), 2), Pow(JacobiTheta(4, w, tau), 2))))),
    Variables(z, w, tau),
    Assumptions(And(Element(z, CC), Element(w, tau), Element(tau, HH))))

75cb8c

\theta_{1}\!\left(z + w , \tau\right) \theta_{1}\!\left(z - w , \tau\right) \theta_{3}^{2}\!\left(0, \tau\right) = \theta_{1}^{2}\!\left(z, \tau\right) \theta_{3}^{2}\!\left(w, \tau\right) - \theta_{3}^{2}\!\left(z, \tau\right) \theta_{1}^{2}\!\left(w, \tau\right) = \theta_{4}^{2}\!\left(z, \tau\right) \theta_{2}^{2}\!\left(w, \tau\right) - \theta_{2}^{2}\!\left(z, \tau\right) \theta_{4}^{2}\!\left(w, \tau\right)

Assumptions:

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

TeX:

\theta_{1}\!\left(z + w , \tau\right) \theta_{1}\!\left(z - w , \tau\right) \theta_{3}^{2}\!\left(0, \tau\right) = \theta_{1}^{2}\!\left(z, \tau\right) \theta_{3}^{2}\!\left(w, \tau\right) - \theta_{3}^{2}\!\left(z, \tau\right) \theta_{1}^{2}\!\left(w, \tau\right) = \theta_{4}^{2}\!\left(z, \tau\right) \theta_{2}^{2}\!\left(w, \tau\right) - \theta_{2}^{2}\!\left(z, \tau\right) \theta_{4}^{2}\!\left(w, \tau\right)

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

Definitions:

Fungrim symbol	Notation	Short description
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	Jacobi theta function
`Pow`	${a}^{b}$	Power
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("75cb8c"),
    Formula(Equal(Mul(Mul(JacobiTheta(1, Add(z, w), tau), JacobiTheta(1, Sub(z, w), tau)), Pow(JacobiTheta(3, 0, tau), 2)), Sub(Mul(Pow(JacobiTheta(1, z, tau), 2), Pow(JacobiTheta(3, w, tau), 2)), Mul(Pow(JacobiTheta(3, z, tau), 2), Pow(JacobiTheta(1, w, tau), 2))), Sub(Mul(Pow(JacobiTheta(4, z, tau), 2), Pow(JacobiTheta(2, w, tau), 2)), Mul(Pow(JacobiTheta(2, z, tau), 2), Pow(JacobiTheta(4, w, tau), 2))))),
    Variables(z, w, tau),
    Assumptions(And(Element(z, CC), Element(w, tau), Element(tau, HH))))

663a02

\theta_{1}\!\left(z + w , \tau\right) \theta_{1}\!\left(z - w , \tau\right) \theta_{4}^{2}\!\left(0, \tau\right) = \theta_{3}^{2}\!\left(z, \tau\right) \theta_{2}^{2}\!\left(w, \tau\right) - \theta_{2}^{2}\!\left(z, \tau\right) \theta_{3}^{2}\!\left(w, \tau\right) = \theta_{1}^{2}\!\left(z, \tau\right) \theta_{4}^{2}\!\left(w, \tau\right) - \theta_{4}^{2}\!\left(z, \tau\right) \theta_{1}^{2}\!\left(w, \tau\right)

Assumptions:

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

TeX:

\theta_{1}\!\left(z + w , \tau\right) \theta_{1}\!\left(z - w , \tau\right) \theta_{4}^{2}\!\left(0, \tau\right) = \theta_{3}^{2}\!\left(z, \tau\right) \theta_{2}^{2}\!\left(w, \tau\right) - \theta_{2}^{2}\!\left(z, \tau\right) \theta_{3}^{2}\!\left(w, \tau\right) = \theta_{1}^{2}\!\left(z, \tau\right) \theta_{4}^{2}\!\left(w, \tau\right) - \theta_{4}^{2}\!\left(z, \tau\right) \theta_{1}^{2}\!\left(w, \tau\right)

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

Definitions:

Fungrim symbol	Notation	Short description
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	Jacobi theta function
`Pow`	${a}^{b}$	Power
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("663a02"),
    Formula(Equal(Mul(Mul(JacobiTheta(1, Add(z, w), tau), JacobiTheta(1, Sub(z, w), tau)), Pow(JacobiTheta(4, 0, tau), 2)), Sub(Mul(Pow(JacobiTheta(3, z, tau), 2), Pow(JacobiTheta(2, w, tau), 2)), Mul(Pow(JacobiTheta(2, z, tau), 2), Pow(JacobiTheta(3, w, tau), 2))), Sub(Mul(Pow(JacobiTheta(1, z, tau), 2), Pow(JacobiTheta(4, w, tau), 2)), Mul(Pow(JacobiTheta(4, z, tau), 2), Pow(JacobiTheta(1, w, tau), 2))))),
    Variables(z, w, tau),
    Assumptions(And(Element(z, CC), Element(w, tau), Element(tau, HH))))

1feda6

\theta_{2}\!\left(z + w , \tau\right) \theta_{2}\!\left(z - w , \tau\right) \theta_{2}^{2}\!\left(0, \tau\right) = \theta_{2}^{2}\!\left(z, \tau\right) \theta_{2}^{2}\!\left(w, \tau\right) - \theta_{1}^{2}\!\left(z, \tau\right) \theta_{1}^{2}\!\left(w, \tau\right) = \theta_{3}^{2}\!\left(z, \tau\right) \theta_{3}^{2}\!\left(w, \tau\right) - \theta_{4}^{2}\!\left(z, \tau\right) \theta_{4}^{2}\!\left(w, \tau\right)

Assumptions:

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

TeX:

\theta_{2}\!\left(z + w , \tau\right) \theta_{2}\!\left(z - w , \tau\right) \theta_{2}^{2}\!\left(0, \tau\right) = \theta_{2}^{2}\!\left(z, \tau\right) \theta_{2}^{2}\!\left(w, \tau\right) - \theta_{1}^{2}\!\left(z, \tau\right) \theta_{1}^{2}\!\left(w, \tau\right) = \theta_{3}^{2}\!\left(z, \tau\right) \theta_{3}^{2}\!\left(w, \tau\right) - \theta_{4}^{2}\!\left(z, \tau\right) \theta_{4}^{2}\!\left(w, \tau\right)

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

Definitions:

Fungrim symbol	Notation	Short description
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	Jacobi theta function
`Pow`	${a}^{b}$	Power
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("1feda6"),
    Formula(Equal(Mul(Mul(JacobiTheta(2, Add(z, w), tau), JacobiTheta(2, Sub(z, w), tau)), Pow(JacobiTheta(2, 0, tau), 2)), Sub(Mul(Pow(JacobiTheta(2, z, tau), 2), Pow(JacobiTheta(2, w, tau), 2)), Mul(Pow(JacobiTheta(1, z, tau), 2), Pow(JacobiTheta(1, w, tau), 2))), Sub(Mul(Pow(JacobiTheta(3, z, tau), 2), Pow(JacobiTheta(3, w, tau), 2)), Mul(Pow(JacobiTheta(4, z, tau), 2), Pow(JacobiTheta(4, w, tau), 2))))),
    Variables(z, w, tau),
    Assumptions(And(Element(z, CC), Element(w, tau), Element(tau, HH))))

89c9e4

\theta_{2}\!\left(z + w , \tau\right) \theta_{2}\!\left(z - w , \tau\right) \theta_{3}^{2}\!\left(0, \tau\right) = \theta_{2}^{2}\!\left(z, \tau\right) \theta_{3}^{2}\!\left(w, \tau\right) - \theta_{4}^{2}\!\left(z, \tau\right) \theta_{1}^{2}\!\left(w, \tau\right) = \theta_{3}^{2}\!\left(z, \tau\right) \theta_{2}^{2}\!\left(w, \tau\right) - \theta_{1}^{2}\!\left(z, \tau\right) \theta_{4}^{2}\!\left(w, \tau\right)

Assumptions:

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

TeX:

\theta_{2}\!\left(z + w , \tau\right) \theta_{2}\!\left(z - w , \tau\right) \theta_{3}^{2}\!\left(0, \tau\right) = \theta_{2}^{2}\!\left(z, \tau\right) \theta_{3}^{2}\!\left(w, \tau\right) - \theta_{4}^{2}\!\left(z, \tau\right) \theta_{1}^{2}\!\left(w, \tau\right) = \theta_{3}^{2}\!\left(z, \tau\right) \theta_{2}^{2}\!\left(w, \tau\right) - \theta_{1}^{2}\!\left(z, \tau\right) \theta_{4}^{2}\!\left(w, \tau\right)

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

Definitions:

Fungrim symbol	Notation	Short description
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	Jacobi theta function
`Pow`	${a}^{b}$	Power
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("89c9e4"),
    Formula(Equal(Mul(Mul(JacobiTheta(2, Add(z, w), tau), JacobiTheta(2, Sub(z, w), tau)), Pow(JacobiTheta(3, 0, tau), 2)), Sub(Mul(Pow(JacobiTheta(2, z, tau), 2), Pow(JacobiTheta(3, w, tau), 2)), Mul(Pow(JacobiTheta(4, z, tau), 2), Pow(JacobiTheta(1, w, tau), 2))), Sub(Mul(Pow(JacobiTheta(3, z, tau), 2), Pow(JacobiTheta(2, w, tau), 2)), Mul(Pow(JacobiTheta(1, z, tau), 2), Pow(JacobiTheta(4, w, tau), 2))))),
    Variables(z, w, tau),
    Assumptions(And(Element(z, CC), Element(w, tau), Element(tau, HH))))

48a1c6

\theta_{2}\!\left(z + w , \tau\right) \theta_{2}\!\left(z - w , \tau\right) \theta_{4}^{2}\!\left(0, \tau\right) = \theta_{4}^{2}\!\left(z, \tau\right) \theta_{2}^{2}\!\left(w, \tau\right) - \theta_{1}^{2}\!\left(z, \tau\right) \theta_{3}^{2}\!\left(w, \tau\right) = \theta_{2}^{2}\!\left(z, \tau\right) \theta_{4}^{2}\!\left(w, \tau\right) - \theta_{3}^{2}\!\left(z, \tau\right) \theta_{1}^{2}\!\left(w, \tau\right)

Assumptions:

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

TeX:

\theta_{2}\!\left(z + w , \tau\right) \theta_{2}\!\left(z - w , \tau\right) \theta_{4}^{2}\!\left(0, \tau\right) = \theta_{4}^{2}\!\left(z, \tau\right) \theta_{2}^{2}\!\left(w, \tau\right) - \theta_{1}^{2}\!\left(z, \tau\right) \theta_{3}^{2}\!\left(w, \tau\right) = \theta_{2}^{2}\!\left(z, \tau\right) \theta_{4}^{2}\!\left(w, \tau\right) - \theta_{3}^{2}\!\left(z, \tau\right) \theta_{1}^{2}\!\left(w, \tau\right)

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

Definitions:

Fungrim symbol	Notation	Short description
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	Jacobi theta function
`Pow`	${a}^{b}$	Power
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("48a1c6"),
    Formula(Equal(Mul(Mul(JacobiTheta(2, Add(z, w), tau), JacobiTheta(2, Sub(z, w), tau)), Pow(JacobiTheta(4, 0, tau), 2)), Sub(Mul(Pow(JacobiTheta(4, z, tau), 2), Pow(JacobiTheta(2, w, tau), 2)), Mul(Pow(JacobiTheta(1, z, tau), 2), Pow(JacobiTheta(3, w, tau), 2))), Sub(Mul(Pow(JacobiTheta(2, z, tau), 2), Pow(JacobiTheta(4, w, tau), 2)), Mul(Pow(JacobiTheta(3, z, tau), 2), Pow(JacobiTheta(1, w, tau), 2))))),
    Variables(z, w, tau),
    Assumptions(And(Element(z, CC), Element(w, tau), Element(tau, HH))))

66efb8

\theta_{3}\!\left(z + w , \tau\right) \theta_{3}\!\left(z - w , \tau\right) \theta_{2}^{2}\!\left(0, \tau\right) = \theta_{3}^{2}\!\left(z, \tau\right) \theta_{2}^{2}\!\left(w, \tau\right) + \theta_{4}^{2}\!\left(z, \tau\right) \theta_{1}^{2}\!\left(w, \tau\right) = \theta_{2}^{2}\!\left(z, \tau\right) \theta_{3}^{2}\!\left(w, \tau\right) + \theta_{1}^{2}\!\left(z, \tau\right) \theta_{4}^{2}\!\left(w, \tau\right)

Assumptions:

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

TeX:

\theta_{3}\!\left(z + w , \tau\right) \theta_{3}\!\left(z - w , \tau\right) \theta_{2}^{2}\!\left(0, \tau\right) = \theta_{3}^{2}\!\left(z, \tau\right) \theta_{2}^{2}\!\left(w, \tau\right) + \theta_{4}^{2}\!\left(z, \tau\right) \theta_{1}^{2}\!\left(w, \tau\right) = \theta_{2}^{2}\!\left(z, \tau\right) \theta_{3}^{2}\!\left(w, \tau\right) + \theta_{1}^{2}\!\left(z, \tau\right) \theta_{4}^{2}\!\left(w, \tau\right)

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

Definitions:

Fungrim symbol	Notation	Short description
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	Jacobi theta function
`Pow`	${a}^{b}$	Power
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("66efb8"),
    Formula(Equal(Mul(Mul(JacobiTheta(3, Add(z, w), tau), JacobiTheta(3, Sub(z, w), tau)), Pow(JacobiTheta(2, 0, tau), 2)), Add(Mul(Pow(JacobiTheta(3, z, tau), 2), Pow(JacobiTheta(2, w, tau), 2)), Mul(Pow(JacobiTheta(4, z, tau), 2), Pow(JacobiTheta(1, w, tau), 2))), Add(Mul(Pow(JacobiTheta(2, z, tau), 2), Pow(JacobiTheta(3, w, tau), 2)), Mul(Pow(JacobiTheta(1, z, tau), 2), Pow(JacobiTheta(4, w, tau), 2))))),
    Variables(z, w, tau),
    Assumptions(And(Element(z, CC), Element(w, tau), Element(tau, HH))))

9aa437

\theta_{3}\!\left(z + w , \tau\right) \theta_{3}\!\left(z - w , \tau\right) \theta_{3}^{2}\!\left(0, \tau\right) = \theta_{1}^{2}\!\left(z, \tau\right) \theta_{1}^{2}\!\left(w, \tau\right) + \theta_{3}^{2}\!\left(z, \tau\right) \theta_{3}^{2}\!\left(w, \tau\right) = \theta_{2}^{2}\!\left(z, \tau\right) \theta_{2}^{2}\!\left(w, \tau\right) + \theta_{4}^{2}\!\left(z, \tau\right) \theta_{4}^{2}\!\left(w, \tau\right)

Assumptions:

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

TeX:

\theta_{3}\!\left(z + w , \tau\right) \theta_{3}\!\left(z - w , \tau\right) \theta_{3}^{2}\!\left(0, \tau\right) = \theta_{1}^{2}\!\left(z, \tau\right) \theta_{1}^{2}\!\left(w, \tau\right) + \theta_{3}^{2}\!\left(z, \tau\right) \theta_{3}^{2}\!\left(w, \tau\right) = \theta_{2}^{2}\!\left(z, \tau\right) \theta_{2}^{2}\!\left(w, \tau\right) + \theta_{4}^{2}\!\left(z, \tau\right) \theta_{4}^{2}\!\left(w, \tau\right)

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

Definitions:

Fungrim symbol	Notation	Short description
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	Jacobi theta function
`Pow`	${a}^{b}$	Power
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("9aa437"),
    Formula(Equal(Mul(Mul(JacobiTheta(3, Add(z, w), tau), JacobiTheta(3, Sub(z, w), tau)), Pow(JacobiTheta(3, 0, tau), 2)), Add(Mul(Pow(JacobiTheta(1, z, tau), 2), Pow(JacobiTheta(1, w, tau), 2)), Mul(Pow(JacobiTheta(3, z, tau), 2), Pow(JacobiTheta(3, w, tau), 2))), Add(Mul(Pow(JacobiTheta(2, z, tau), 2), Pow(JacobiTheta(2, w, tau), 2)), Mul(Pow(JacobiTheta(4, z, tau), 2), Pow(JacobiTheta(4, w, tau), 2))))),
    Variables(z, w, tau),
    Assumptions(And(Element(z, CC), Element(w, tau), Element(tau, HH))))

5752b8

\theta_{3}\!\left(z + w , \tau\right) \theta_{3}\!\left(z - w , \tau\right) \theta_{4}^{2}\!\left(0, \tau\right) = \theta_{4}^{2}\!\left(z, \tau\right) \theta_{3}^{2}\!\left(w, \tau\right) - \theta_{1}^{2}\!\left(z, \tau\right) \theta_{2}^{2}\!\left(w, \tau\right) = \theta_{3}^{2}\!\left(z, \tau\right) \theta_{4}^{2}\!\left(w, \tau\right) - \theta_{2}^{2}\!\left(z, \tau\right) \theta_{1}^{2}\!\left(w, \tau\right)

Assumptions:

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

TeX:

\theta_{3}\!\left(z + w , \tau\right) \theta_{3}\!\left(z - w , \tau\right) \theta_{4}^{2}\!\left(0, \tau\right) = \theta_{4}^{2}\!\left(z, \tau\right) \theta_{3}^{2}\!\left(w, \tau\right) - \theta_{1}^{2}\!\left(z, \tau\right) \theta_{2}^{2}\!\left(w, \tau\right) = \theta_{3}^{2}\!\left(z, \tau\right) \theta_{4}^{2}\!\left(w, \tau\right) - \theta_{2}^{2}\!\left(z, \tau\right) \theta_{1}^{2}\!\left(w, \tau\right)

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

Definitions:

Fungrim symbol	Notation	Short description
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	Jacobi theta function
`Pow`	${a}^{b}$	Power
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("5752b8"),
    Formula(Equal(Mul(Mul(JacobiTheta(3, Add(z, w), tau), JacobiTheta(3, Sub(z, w), tau)), Pow(JacobiTheta(4, 0, tau), 2)), Sub(Mul(Pow(JacobiTheta(4, z, tau), 2), Pow(JacobiTheta(3, w, tau), 2)), Mul(Pow(JacobiTheta(1, z, tau), 2), Pow(JacobiTheta(2, w, tau), 2))), Sub(Mul(Pow(JacobiTheta(3, z, tau), 2), Pow(JacobiTheta(4, w, tau), 2)), Mul(Pow(JacobiTheta(2, z, tau), 2), Pow(JacobiTheta(1, w, tau), 2))))),
    Variables(z, w, tau),
    Assumptions(And(Element(z, CC), Element(w, tau), Element(tau, HH))))

c891a1

\theta_{4}\!\left(z + w , \tau\right) \theta_{4}\!\left(z - w , \tau\right) \theta_{2}^{2}\!\left(0, \tau\right) = \theta_{4}^{2}\!\left(z, \tau\right) \theta_{2}^{2}\!\left(w, \tau\right) + \theta_{3}^{2}\!\left(z, \tau\right) \theta_{1}^{2}\!\left(w, \tau\right) = \theta_{1}^{2}\!\left(z, \tau\right) \theta_{3}^{2}\!\left(w, \tau\right) + \theta_{2}^{2}\!\left(z, \tau\right) \theta_{4}^{2}\!\left(w, \tau\right)

Assumptions:

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

TeX:

\theta_{4}\!\left(z + w , \tau\right) \theta_{4}\!\left(z - w , \tau\right) \theta_{2}^{2}\!\left(0, \tau\right) = \theta_{4}^{2}\!\left(z, \tau\right) \theta_{2}^{2}\!\left(w, \tau\right) + \theta_{3}^{2}\!\left(z, \tau\right) \theta_{1}^{2}\!\left(w, \tau\right) = \theta_{1}^{2}\!\left(z, \tau\right) \theta_{3}^{2}\!\left(w, \tau\right) + \theta_{2}^{2}\!\left(z, \tau\right) \theta_{4}^{2}\!\left(w, \tau\right)

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

Definitions:

Fungrim symbol	Notation	Short description
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	Jacobi theta function
`Pow`	${a}^{b}$	Power
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("c891a1"),
    Formula(Equal(Mul(Mul(JacobiTheta(4, Add(z, w), tau), JacobiTheta(4, Sub(z, w), tau)), Pow(JacobiTheta(2, 0, tau), 2)), Add(Mul(Pow(JacobiTheta(4, z, tau), 2), Pow(JacobiTheta(2, w, tau), 2)), Mul(Pow(JacobiTheta(3, z, tau), 2), Pow(JacobiTheta(1, w, tau), 2))), Add(Mul(Pow(JacobiTheta(1, z, tau), 2), Pow(JacobiTheta(3, w, tau), 2)), Mul(Pow(JacobiTheta(2, z, tau), 2), Pow(JacobiTheta(4, w, tau), 2))))),
    Variables(z, w, tau),
    Assumptions(And(Element(z, CC), Element(w, tau), Element(tau, HH))))

3cac28

\theta_{4}\!\left(z + w , \tau\right) \theta_{4}\!\left(z - w , \tau\right) \theta_{3}^{2}\!\left(0, \tau\right) = \theta_{1}^{2}\!\left(z, \tau\right) \theta_{2}^{2}\!\left(w, \tau\right) + \theta_{3}^{2}\!\left(z, \tau\right) \theta_{4}^{2}\!\left(w, \tau\right) = \theta_{2}^{2}\!\left(z, \tau\right) \theta_{1}^{2}\!\left(w, \tau\right) + \theta_{4}^{2}\!\left(z, \tau\right) \theta_{3}^{2}\!\left(w, \tau\right)

Assumptions:

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

TeX:

\theta_{4}\!\left(z + w , \tau\right) \theta_{4}\!\left(z - w , \tau\right) \theta_{3}^{2}\!\left(0, \tau\right) = \theta_{1}^{2}\!\left(z, \tau\right) \theta_{2}^{2}\!\left(w, \tau\right) + \theta_{3}^{2}\!\left(z, \tau\right) \theta_{4}^{2}\!\left(w, \tau\right) = \theta_{2}^{2}\!\left(z, \tau\right) \theta_{1}^{2}\!\left(w, \tau\right) + \theta_{4}^{2}\!\left(z, \tau\right) \theta_{3}^{2}\!\left(w, \tau\right)

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

Definitions:

Fungrim symbol	Notation	Short description
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	Jacobi theta function
`Pow`	${a}^{b}$	Power
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("3cac28"),
    Formula(Equal(Mul(Mul(JacobiTheta(4, Add(z, w), tau), JacobiTheta(4, Sub(z, w), tau)), Pow(JacobiTheta(3, 0, tau), 2)), Add(Mul(Pow(JacobiTheta(1, z, tau), 2), Pow(JacobiTheta(2, w, tau), 2)), Mul(Pow(JacobiTheta(3, z, tau), 2), Pow(JacobiTheta(4, w, tau), 2))), Add(Mul(Pow(JacobiTheta(2, z, tau), 2), Pow(JacobiTheta(1, w, tau), 2)), Mul(Pow(JacobiTheta(4, z, tau), 2), Pow(JacobiTheta(3, w, tau), 2))))),
    Variables(z, w, tau),
    Assumptions(And(Element(z, CC), Element(w, tau), Element(tau, HH))))

45a130

\theta_{4}\!\left(z + w , \tau\right) \theta_{4}\!\left(z - w , \tau\right) \theta_{4}^{2}\!\left(0, \tau\right) = \theta_{3}^{2}\!\left(z, \tau\right) \theta_{3}^{2}\!\left(w, \tau\right) - \theta_{2}^{2}\!\left(z, \tau\right) \theta_{2}^{2}\!\left(w, \tau\right) = \theta_{4}^{2}\!\left(z, \tau\right) \theta_{4}^{2}\!\left(w, \tau\right) - \theta_{1}^{2}\!\left(z, \tau\right) \theta_{1}^{2}\!\left(w, \tau\right)

Assumptions:

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

TeX:

\theta_{4}\!\left(z + w , \tau\right) \theta_{4}\!\left(z - w , \tau\right) \theta_{4}^{2}\!\left(0, \tau\right) = \theta_{3}^{2}\!\left(z, \tau\right) \theta_{3}^{2}\!\left(w, \tau\right) - \theta_{2}^{2}\!\left(z, \tau\right) \theta_{2}^{2}\!\left(w, \tau\right) = \theta_{4}^{2}\!\left(z, \tau\right) \theta_{4}^{2}\!\left(w, \tau\right) - \theta_{1}^{2}\!\left(z, \tau\right) \theta_{1}^{2}\!\left(w, \tau\right)

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

Definitions:

Fungrim symbol	Notation	Short description
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	Jacobi theta function
`Pow`	${a}^{b}$	Power
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("45a130"),
    Formula(Equal(Mul(Mul(JacobiTheta(4, Add(z, w), tau), JacobiTheta(4, Sub(z, w), tau)), Pow(JacobiTheta(4, 0, tau), 2)), Sub(Mul(Pow(JacobiTheta(3, z, tau), 2), Pow(JacobiTheta(3, w, tau), 2)), Mul(Pow(JacobiTheta(2, z, tau), 2), Pow(JacobiTheta(2, w, tau), 2))), Sub(Mul(Pow(JacobiTheta(4, z, tau), 2), Pow(JacobiTheta(4, w, tau), 2)), Mul(Pow(JacobiTheta(1, z, tau), 2), Pow(JacobiTheta(1, w, tau), 2))))),
    Variables(z, w, tau),
    Assumptions(And(Element(z, CC), Element(w, tau), Element(tau, HH))))

Reflection symmetry

Even-odd symmetry

Conjugate symmetry

Periodicity

Quasi-periodicity

Single shifts

General shifts

Half-period or quarter-period shifts

Theta functions represented in terms of each other

Theta 1

Theta 2

Theta 3

Theta 4

Double argument

Theta 1

Theta 2

Theta 3

Theta 4

Relations involving sums and differences of arguments

Cross-products with two factors and double lattice parameter

Cross-products with four factors

Cross-products of squares