Series and product representations of Jacobi theta functions

2ba423

\theta_{1}\!\left(z , \tau\right) = 2 {e}^{\pi i \tau / 4} \sum_{n=0}^{\infty} {\left(-1\right)}^{n} {q}^{n \left(n + 1\right)} \sin\!\left(\left(2 n + 1\right) \pi z\right)\; \text{ where } q = {e}^{\pi i \tau}

Assumptions:

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

TeX:

\theta_{1}\!\left(z , \tau\right) = 2 {e}^{\pi i \tau / 4} \sum_{n=0}^{\infty} {\left(-1\right)}^{n} {q}^{n \left(n + 1\right)} \sin\!\left(\left(2 n + 1\right) \pi z\right)\; \text{ where } q = {e}^{\pi i \tau}

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
`Sum`	$\sum_{n} f(n)$	Sum
`Pow`	${a}^{b}$	Power
`Sin`	$\sin(z)$	Sine
`Infinity`	$\infty$	Positive infinity
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("2ba423"),
    Formula(Equal(JacobiTheta(1, z, tau), Where(Mul(Mul(2, Exp(Div(Mul(Mul(Pi, ConstI), tau), 4))), Sum(Mul(Mul(Pow(-1, n), Pow(q, Mul(n, Add(n, 1)))), Sin(Mul(Mul(Add(Mul(2, n), 1), Pi), z))), For(n, 0, Infinity))), Equal(q, Exp(Mul(Mul(Pi, ConstI), tau)))))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH))))

06633e

\theta_{2}\!\left(z , \tau\right) = 2 {e}^{\pi i \tau / 4} \sum_{n=0}^{\infty} {q}^{n \left(n + 1\right)} \cos\!\left(\left(2 n + 1\right) \pi z\right)\; \text{ where } q = {e}^{\pi i \tau}

Assumptions:

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

TeX:

\theta_{2}\!\left(z , \tau\right) = 2 {e}^{\pi i \tau / 4} \sum_{n=0}^{\infty} {q}^{n \left(n + 1\right)} \cos\!\left(\left(2 n + 1\right) \pi z\right)\; \text{ where } q = {e}^{\pi i \tau}

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
`Sum`	$\sum_{n} f(n)$	Sum
`Pow`	${a}^{b}$	Power
`Cos`	$\cos(z)$	Cosine
`Infinity`	$\infty$	Positive infinity
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("06633e"),
    Formula(Equal(JacobiTheta(2, z, tau), Where(Mul(Mul(2, Exp(Div(Mul(Mul(Pi, ConstI), tau), 4))), Sum(Mul(Pow(q, Mul(n, Add(n, 1))), Cos(Mul(Mul(Add(Mul(2, n), 1), Pi), z))), For(n, 0, Infinity))), Equal(q, Exp(Mul(Mul(Pi, ConstI), tau)))))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH))))

f3e75c

\theta_{3}\!\left(z , \tau\right) = 1 + 2 \sum_{n=1}^{\infty} {q}^{{n}^{2}} \cos\!\left(2 n \pi z\right)\; \text{ where } q = {e}^{\pi i \tau}

Assumptions:

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

TeX:

\theta_{3}\!\left(z , \tau\right) = 1 + 2 \sum_{n=1}^{\infty} {q}^{{n}^{2}} \cos\!\left(2 n \pi z\right)\; \text{ where } q = {e}^{\pi i \tau}

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
`Sum`	$\sum_{n} f(n)$	Sum
`Pow`	${a}^{b}$	Power
`Cos`	$\cos(z)$	Cosine
`Pi`	$\pi$	The constant pi (3.14...)
`Infinity`	$\infty$	Positive infinity
`Exp`	${e}^{z}$	Exponential function
`ConstI`	$i$	Imaginary unit
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("f3e75c"),
    Formula(Equal(JacobiTheta(3, z, tau), Where(Add(1, Mul(2, Sum(Mul(Pow(q, Pow(n, 2)), Cos(Mul(Mul(Mul(2, n), Pi), z))), For(n, 1, Infinity)))), Equal(q, Exp(Mul(Mul(Pi, ConstI), tau)))))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH))))

8a34d1

\theta_{4}\!\left(z , \tau\right) = 1 + 2 \sum_{n=1}^{\infty} {\left(-1\right)}^{n} {q}^{{n}^{2}} \cos\!\left(2 n \pi z\right)\; \text{ where } q = {e}^{\pi i \tau}

Assumptions:

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

TeX:

\theta_{4}\!\left(z , \tau\right) = 1 + 2 \sum_{n=1}^{\infty} {\left(-1\right)}^{n} {q}^{{n}^{2}} \cos\!\left(2 n \pi z\right)\; \text{ where } q = {e}^{\pi i \tau}

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
`Sum`	$\sum_{n} f(n)$	Sum
`Pow`	${a}^{b}$	Power
`Cos`	$\cos(z)$	Cosine
`Pi`	$\pi$	The constant pi (3.14...)
`Infinity`	$\infty$	Positive infinity
`Exp`	${e}^{z}$	Exponential function
`ConstI`	$i$	Imaginary unit
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("8a34d1"),
    Formula(Equal(JacobiTheta(4, z, tau), Where(Add(1, Mul(2, Sum(Mul(Mul(Pow(-1, n), Pow(q, Pow(n, 2))), Cos(Mul(Mul(Mul(2, n), Pi), z))), For(n, 1, Infinity)))), Equal(q, Exp(Mul(Mul(Pi, ConstI), tau)))))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH))))

700d94

\theta_{1}\!\left(z , \tau\right) = -i {e}^{\pi i \tau / 4} \sum_{n=-\infty}^{\infty} {\left(-1\right)}^{n} {q}^{n \left(n + 1\right)} {w}^{2 n + 1}\; \text{ where } q = {e}^{\pi i \tau},\;w = {e}^{\pi i z}

Assumptions:

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

TeX:

\theta_{1}\!\left(z , \tau\right) = -i {e}^{\pi i \tau / 4} \sum_{n=-\infty}^{\infty} {\left(-1\right)}^{n} {q}^{n \left(n + 1\right)} {w}^{2 n + 1}\; \text{ where } q = {e}^{\pi i \tau},\;w = {e}^{\pi i z}

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...)
`Sum`	$\sum_{n} f(n)$	Sum
`Pow`	${a}^{b}$	Power
`Infinity`	$\infty$	Positive infinity
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("700d94"),
    Formula(Equal(JacobiTheta(1, z, tau), Where(Mul(Mul(Neg(ConstI), Exp(Div(Mul(Mul(Pi, ConstI), tau), 4))), Sum(Mul(Mul(Pow(-1, n), Pow(q, Mul(n, Add(n, 1)))), Pow(w, Add(Mul(2, n), 1))), For(n, Neg(Infinity), Infinity))), Equal(q, Exp(Mul(Mul(Pi, ConstI), tau))), Equal(w, Exp(Mul(Mul(Pi, ConstI), z)))))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH))))

495a98

\theta_{2}\!\left(z , \tau\right) = {e}^{\pi i \tau / 4} \sum_{n=-\infty}^{\infty} {q}^{n \left(n + 1\right)} {w}^{2 n + 1}\; \text{ where } q = {e}^{\pi i \tau},\;w = {e}^{\pi i z}

Assumptions:

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

TeX:

\theta_{2}\!\left(z , \tau\right) = {e}^{\pi i \tau / 4} \sum_{n=-\infty}^{\infty} {q}^{n \left(n + 1\right)} {w}^{2 n + 1}\; \text{ where } q = {e}^{\pi i \tau},\;w = {e}^{\pi i z}

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
`Sum`	$\sum_{n} f(n)$	Sum
`Pow`	${a}^{b}$	Power
`Infinity`	$\infty$	Positive infinity
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("495a98"),
    Formula(Equal(JacobiTheta(2, z, tau), Where(Mul(Exp(Div(Mul(Mul(Pi, ConstI), tau), 4)), Sum(Mul(Pow(q, Mul(n, Add(n, 1))), Pow(w, Add(Mul(2, n), 1))), For(n, Neg(Infinity), Infinity))), Equal(q, Exp(Mul(Mul(Pi, ConstI), tau))), Equal(w, Exp(Mul(Mul(Pi, ConstI), z)))))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH))))

2f97f5

\theta_{3}\!\left(z , \tau\right) = \sum_{n=-\infty}^{\infty} {q}^{{n}^{2}} {w}^{2 n}\; \text{ where } q = {e}^{\pi i \tau},\;w = {e}^{\pi i z}

Assumptions:

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

TeX:

\theta_{3}\!\left(z , \tau\right) = \sum_{n=-\infty}^{\infty} {q}^{{n}^{2}} {w}^{2 n}\; \text{ where } q = {e}^{\pi i \tau},\;w = {e}^{\pi i z}

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
`Sum`	$\sum_{n} f(n)$	Sum
`Pow`	${a}^{b}$	Power
`Infinity`	$\infty$	Positive infinity
`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("2f97f5"),
    Formula(Equal(JacobiTheta(3, z, tau), Where(Sum(Mul(Pow(q, Pow(n, 2)), Pow(w, Mul(2, n))), For(n, Neg(Infinity), Infinity)), Equal(q, Exp(Mul(Mul(Pi, ConstI), tau))), Equal(w, Exp(Mul(Mul(Pi, ConstI), z)))))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH))))

d923de

\theta_{4}\!\left(z , \tau\right) = \sum_{n=-\infty}^{\infty} {\left(-1\right)}^{n} {q}^{{n}^{2}} {w}^{2 n}\; \text{ where } q = {e}^{\pi i \tau},\;w = {e}^{\pi i z}

Assumptions:

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

TeX:

\theta_{4}\!\left(z , \tau\right) = \sum_{n=-\infty}^{\infty} {\left(-1\right)}^{n} {q}^{{n}^{2}} {w}^{2 n}\; \text{ where } q = {e}^{\pi i \tau},\;w = {e}^{\pi i z}

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
`Sum`	$\sum_{n} f(n)$	Sum
`Pow`	${a}^{b}$	Power
`Infinity`	$\infty$	Positive infinity
`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("d923de"),
    Formula(Equal(JacobiTheta(4, z, tau), Where(Sum(Mul(Mul(Pow(-1, n), Pow(q, Pow(n, 2))), Pow(w, Mul(2, n))), For(n, Neg(Infinity), Infinity)), Equal(q, Exp(Mul(Mul(Pi, ConstI), tau))), Equal(w, Exp(Mul(Mul(Pi, ConstI), z)))))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH))))

ed4ce5

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

Assumptions:

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

TeX:

\theta_{1}\!\left(z , \tau\right) = \sum_{n=-\infty}^{\infty} {e}^{\pi i \left({\left(n + 1 / 2\right)}^{2} \tau + \left(2 n + 1\right) z + n - 1 / 2\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
`Sum`	$\sum_{n} f(n)$	Sum
`Exp`	${e}^{z}$	Exponential function
`Pi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit
`Pow`	${a}^{b}$	Power
`Infinity`	$\infty$	Positive infinity
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("ed4ce5"),
    Formula(Equal(JacobiTheta(1, z, tau), Sum(Exp(Mul(Mul(Pi, ConstI), Sub(Add(Add(Mul(Pow(Add(n, Div(1, 2)), 2), tau), Mul(Add(Mul(2, n), 1), z)), n), Div(1, 2)))), For(n, Neg(Infinity), Infinity)))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH))))

7cb651

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

Assumptions:

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

TeX:

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

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

Definitions:

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

Source code for this entry:

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

580ba0

\theta_{3}\!\left(z , \tau\right) = \sum_{n=-\infty}^{\infty} {e}^{\pi i \left({n}^{2} \tau + 2 n z\right)}

Assumptions:

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

TeX:

\theta_{3}\!\left(z , \tau\right) = \sum_{n=-\infty}^{\infty} {e}^{\pi i \left({n}^{2} \tau + 2 n z\right)}

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

Definitions:

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

Source code for this entry:

Entry(ID("580ba0"),
    Formula(Equal(JacobiTheta(3, z, tau), Sum(Exp(Mul(Mul(Pi, ConstI), Add(Mul(Pow(n, 2), tau), Mul(Mul(2, n), z)))), For(n, Neg(Infinity), Infinity)))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH))))

27c319

\theta_{4}\!\left(z , \tau\right) = \sum_{n=-\infty}^{\infty} {e}^{\pi i \left({n}^{2} \tau + 2 n z + n\right)}

Assumptions:

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

TeX:

\theta_{4}\!\left(z , \tau\right) = \sum_{n=-\infty}^{\infty} {e}^{\pi i \left({n}^{2} \tau + 2 n z + n\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
`Sum`	$\sum_{n} f(n)$	Sum
`Exp`	${e}^{z}$	Exponential function
`Pi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit
`Pow`	${a}^{b}$	Power
`Infinity`	$\infty$	Positive infinity
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("27c319"),
    Formula(Equal(JacobiTheta(4, z, tau), Sum(Exp(Mul(Mul(Pi, ConstI), Add(Add(Mul(Pow(n, 2), tau), Mul(Mul(2, n), z)), n))), For(n, Neg(Infinity), Infinity)))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH))))

2ae142

\theta^{(r)}_{1}\!\left(z , \tau\right) = -i {\left(\pi i\right)}^{r} {e}^{\pi i \tau / 4} \sum_{n=-\infty}^{\infty} {\left(-1\right)}^{n} {\left(2 n + 1\right)}^{r} {q}^{n \left(n + 1\right)} {w}^{2 n + 1}\; \text{ where } q = {e}^{\pi i \tau},\;w = {e}^{\pi i z}

Assumptions:

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H} \;\mathbin{\operatorname{and}}\; r \in \mathbb{Z}_{\ge 0}

TeX:

\theta^{(r)}_{1}\!\left(z , \tau\right) = -i {\left(\pi i\right)}^{r} {e}^{\pi i \tau / 4} \sum_{n=-\infty}^{\infty} {\left(-1\right)}^{n} {\left(2 n + 1\right)}^{r} {q}^{n \left(n + 1\right)} {w}^{2 n + 1}\; \text{ where } q = {e}^{\pi i \tau},\;w = {e}^{\pi i z}

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H} \;\mathbin{\operatorname{and}}\; r \in \mathbb{Z}_{\ge 0}

Definitions:

Fungrim symbol	Notation	Short description
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	Jacobi theta function
`ConstI`	$i$	Imaginary unit
`Pow`	${a}^{b}$	Power
`Pi`	$\pi$	The constant pi (3.14...)
`Exp`	${e}^{z}$	Exponential function
`Sum`	$\sum_{n} f(n)$	Sum
`Infinity`	$\infty$	Positive infinity
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("2ae142"),
    Formula(Equal(JacobiTheta(1, z, tau, r), Where(Mul(Mul(Mul(Neg(ConstI), Pow(Mul(Pi, ConstI), r)), Exp(Div(Mul(Mul(Pi, ConstI), tau), 4))), Sum(Mul(Mul(Mul(Pow(-1, n), Pow(Add(Mul(2, n), 1), r)), Pow(q, Mul(n, Add(n, 1)))), Pow(w, Add(Mul(2, n), 1))), For(n, Neg(Infinity), Infinity))), Equal(q, Exp(Mul(Mul(Pi, ConstI), tau))), Equal(w, Exp(Mul(Mul(Pi, ConstI), z)))))),
    Variables(z, tau, r),
    Assumptions(And(Element(z, CC), Element(tau, HH), Element(r, ZZGreaterEqual(0)))))

42d832

\theta^{(r)}_{2}\!\left(z , \tau\right) = {\left(\pi i\right)}^{r} {e}^{\pi i \tau / 4} \sum_{n=-\infty}^{\infty} {\left(2 n + 1\right)}^{r} {q}^{n \left(n + 1\right)} {w}^{2 n + 1}\; \text{ where } q = {e}^{\pi i \tau},\;w = {e}^{\pi i z}

Assumptions:

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H} \;\mathbin{\operatorname{and}}\; r \in \mathbb{Z}_{\ge 0}

TeX:

\theta^{(r)}_{2}\!\left(z , \tau\right) = {\left(\pi i\right)}^{r} {e}^{\pi i \tau / 4} \sum_{n=-\infty}^{\infty} {\left(2 n + 1\right)}^{r} {q}^{n \left(n + 1\right)} {w}^{2 n + 1}\; \text{ where } q = {e}^{\pi i \tau},\;w = {e}^{\pi i z}

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H} \;\mathbin{\operatorname{and}}\; r \in \mathbb{Z}_{\ge 0}

Definitions:

Fungrim symbol	Notation	Short description
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	Jacobi theta function
`Pow`	${a}^{b}$	Power
`Pi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit
`Exp`	${e}^{z}$	Exponential function
`Sum`	$\sum_{n} f(n)$	Sum
`Infinity`	$\infty$	Positive infinity
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("42d832"),
    Formula(Equal(JacobiTheta(2, z, tau, r), Where(Mul(Mul(Pow(Mul(Pi, ConstI), r), Exp(Div(Mul(Mul(Pi, ConstI), tau), 4))), Sum(Mul(Mul(Pow(Add(Mul(2, n), 1), r), Pow(q, Mul(n, Add(n, 1)))), Pow(w, Add(Mul(2, n), 1))), For(n, Neg(Infinity), Infinity))), Equal(q, Exp(Mul(Mul(Pi, ConstI), tau))), Equal(w, Exp(Mul(Mul(Pi, ConstI), z)))))),
    Variables(z, tau, r),
    Assumptions(And(Element(z, CC), Element(tau, HH), Element(r, ZZGreaterEqual(0)))))

f551ca

\theta^{(r)}_{3}\!\left(z , \tau\right) = {\left(2 \pi i\right)}^{r} \sum_{n=-\infty}^{\infty} {n}^{r} {q}^{{n}^{2}} {w}^{2 n}\; \text{ where } q = {e}^{\pi i \tau},\;w = {e}^{\pi i z}

Assumptions:

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H} \;\mathbin{\operatorname{and}}\; r \in \mathbb{Z}_{\ge 0}

TeX:

\theta^{(r)}_{3}\!\left(z , \tau\right) = {\left(2 \pi i\right)}^{r} \sum_{n=-\infty}^{\infty} {n}^{r} {q}^{{n}^{2}} {w}^{2 n}\; \text{ where } q = {e}^{\pi i \tau},\;w = {e}^{\pi i z}

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H} \;\mathbin{\operatorname{and}}\; r \in \mathbb{Z}_{\ge 0}

Definitions:

Fungrim symbol	Notation	Short description
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	Jacobi theta function
`Pow`	${a}^{b}$	Power
`Pi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit
`Sum`	$\sum_{n} f(n)$	Sum
`Infinity`	$\infty$	Positive infinity
`Exp`	${e}^{z}$	Exponential function
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("f551ca"),
    Formula(Equal(JacobiTheta(3, z, tau, r), Where(Mul(Pow(Mul(Mul(2, Pi), ConstI), r), Sum(Mul(Mul(Pow(n, r), Pow(q, Pow(n, 2))), Pow(w, Mul(2, n))), For(n, Neg(Infinity), Infinity))), Equal(q, Exp(Mul(Mul(Pi, ConstI), tau))), Equal(w, Exp(Mul(Mul(Pi, ConstI), z)))))),
    Variables(z, tau, r),
    Assumptions(And(Element(z, CC), Element(tau, HH), Element(r, ZZGreaterEqual(0)))))

1842d9

\theta^{(r)}_{4}\!\left(z , \tau\right) = {\left(2 \pi i\right)}^{r} \sum_{n=-\infty}^{\infty} {\left(-1\right)}^{n} {n}^{r} {q}^{{n}^{2}} {w}^{2 n}\; \text{ where } q = {e}^{\pi i \tau},\;w = {e}^{\pi i z}

Assumptions:

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H} \;\mathbin{\operatorname{and}}\; r \in \mathbb{Z}_{\ge 0}

TeX:

\theta^{(r)}_{4}\!\left(z , \tau\right) = {\left(2 \pi i\right)}^{r} \sum_{n=-\infty}^{\infty} {\left(-1\right)}^{n} {n}^{r} {q}^{{n}^{2}} {w}^{2 n}\; \text{ where } q = {e}^{\pi i \tau},\;w = {e}^{\pi i z}

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H} \;\mathbin{\operatorname{and}}\; r \in \mathbb{Z}_{\ge 0}

Definitions:

Fungrim symbol	Notation	Short description
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	Jacobi theta function
`Pow`	${a}^{b}$	Power
`Pi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit
`Sum`	$\sum_{n} f(n)$	Sum
`Infinity`	$\infty$	Positive infinity
`Exp`	${e}^{z}$	Exponential function
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("1842d9"),
    Formula(Equal(JacobiTheta(4, z, tau, r), Where(Mul(Pow(Mul(Mul(2, Pi), ConstI), r), Sum(Mul(Mul(Mul(Pow(-1, n), Pow(n, r)), Pow(q, Pow(n, 2))), Pow(w, Mul(2, n))), For(n, Neg(Infinity), Infinity))), Equal(q, Exp(Mul(Mul(Pi, ConstI), tau))), Equal(w, Exp(Mul(Mul(Pi, ConstI), z)))))),
    Variables(z, tau, r),
    Assumptions(And(Element(z, CC), Element(tau, HH), Element(r, ZZGreaterEqual(0)))))

024a84

\theta_{1}\!\left(z , \tau\right) = 2 {e}^{\pi i \tau / 4} \sin\!\left(\pi z\right) \prod_{n=1}^{\infty} \left(1 - {q}^{2 n}\right) \left(1 - 2 {q}^{2 n} \cos\!\left(2 \pi z\right) + {q}^{4 n}\right)\; \text{ where } q = {e}^{\pi i \tau}

Assumptions:

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

TeX:

\theta_{1}\!\left(z , \tau\right) = 2 {e}^{\pi i \tau / 4} \sin\!\left(\pi z\right) \prod_{n=1}^{\infty} \left(1 - {q}^{2 n}\right) \left(1 - 2 {q}^{2 n} \cos\!\left(2 \pi z\right) + {q}^{4 n}\right)\; \text{ where } q = {e}^{\pi i \tau}

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
`Sin`	$\sin(z)$	Sine
`Product`	$\prod_{n} f(n)$	Product
`Pow`	${a}^{b}$	Power
`Cos`	$\cos(z)$	Cosine
`Infinity`	$\infty$	Positive infinity
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("024a84"),
    Formula(Equal(JacobiTheta(1, z, tau), Where(Mul(Mul(Mul(2, Exp(Div(Mul(Mul(Pi, ConstI), tau), 4))), Sin(Mul(Pi, z))), Product(Mul(Sub(1, Pow(q, Mul(2, n))), Add(Sub(1, Mul(Mul(2, Pow(q, Mul(2, n))), Cos(Mul(Mul(2, Pi), z)))), Pow(q, Mul(4, n)))), For(n, 1, Infinity))), Equal(q, Exp(Mul(Mul(Pi, ConstI), tau)))))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH))))

d6a799

\theta_{2}\!\left(z , \tau\right) = 2 {e}^{\pi i \tau / 4} \cos\!\left(\pi z\right) \prod_{n=1}^{\infty} \left(1 - {q}^{2 n}\right) \left(1 + 2 {q}^{2 n} \cos\!\left(2 \pi z\right) + {q}^{4 n}\right)\; \text{ where } q = {e}^{\pi i \tau}

Assumptions:

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

TeX:

\theta_{2}\!\left(z , \tau\right) = 2 {e}^{\pi i \tau / 4} \cos\!\left(\pi z\right) \prod_{n=1}^{\infty} \left(1 - {q}^{2 n}\right) \left(1 + 2 {q}^{2 n} \cos\!\left(2 \pi z\right) + {q}^{4 n}\right)\; \text{ where } q = {e}^{\pi i \tau}

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
`Cos`	$\cos(z)$	Cosine
`Product`	$\prod_{n} f(n)$	Product
`Pow`	${a}^{b}$	Power
`Infinity`	$\infty$	Positive infinity
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("d6a799"),
    Formula(Equal(JacobiTheta(2, z, tau), Where(Mul(Mul(Mul(2, Exp(Div(Mul(Mul(Pi, ConstI), tau), 4))), Cos(Mul(Pi, z))), Product(Mul(Sub(1, Pow(q, Mul(2, n))), Add(Add(1, Mul(Mul(2, Pow(q, Mul(2, n))), Cos(Mul(Mul(2, Pi), z)))), Pow(q, Mul(4, n)))), For(n, 1, Infinity))), Equal(q, Exp(Mul(Mul(Pi, ConstI), tau)))))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH))))

77aed2

\theta_{3}\!\left(z , \tau\right) = \prod_{n=1}^{\infty} \left(1 - {q}^{2 n}\right) \left(1 + 2 {q}^{2 n - 1} \cos\!\left(2 \pi z\right) + {q}^{4 n - 2}\right)\; \text{ where } q = {e}^{\pi i \tau}

Assumptions:

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

TeX:

\theta_{3}\!\left(z , \tau\right) = \prod_{n=1}^{\infty} \left(1 - {q}^{2 n}\right) \left(1 + 2 {q}^{2 n - 1} \cos\!\left(2 \pi z\right) + {q}^{4 n - 2}\right)\; \text{ where } q = {e}^{\pi i \tau}

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
`Product`	$\prod_{n} f(n)$	Product
`Pow`	${a}^{b}$	Power
`Cos`	$\cos(z)$	Cosine
`Pi`	$\pi$	The constant pi (3.14...)
`Infinity`	$\infty$	Positive infinity
`Exp`	${e}^{z}$	Exponential function
`ConstI`	$i$	Imaginary unit
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("77aed2"),
    Formula(Equal(JacobiTheta(3, z, tau), Where(Product(Mul(Sub(1, Pow(q, Mul(2, n))), Add(Add(1, Mul(Mul(2, Pow(q, Sub(Mul(2, n), 1))), Cos(Mul(Mul(2, Pi), z)))), Pow(q, Sub(Mul(4, n), 2)))), For(n, 1, Infinity)), Equal(q, Exp(Mul(Mul(Pi, ConstI), tau)))))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH))))

2a2a38

\theta_{4}\!\left(z , \tau\right) = \prod_{n=1}^{\infty} \left(1 - {q}^{2 n}\right) \left(1 - 2 {q}^{2 n - 1} \cos\!\left(2 \pi z\right) + {q}^{4 n - 2}\right)\; \text{ where } q = {e}^{\pi i \tau}

Assumptions:

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

TeX:

\theta_{4}\!\left(z , \tau\right) = \prod_{n=1}^{\infty} \left(1 - {q}^{2 n}\right) \left(1 - 2 {q}^{2 n - 1} \cos\!\left(2 \pi z\right) + {q}^{4 n - 2}\right)\; \text{ where } q = {e}^{\pi i \tau}

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
`Product`	$\prod_{n} f(n)$	Product
`Pow`	${a}^{b}$	Power
`Cos`	$\cos(z)$	Cosine
`Pi`	$\pi$	The constant pi (3.14...)
`Infinity`	$\infty$	Positive infinity
`Exp`	${e}^{z}$	Exponential function
`ConstI`	$i$	Imaginary unit
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("2a2a38"),
    Formula(Equal(JacobiTheta(4, z, tau), Where(Product(Mul(Sub(1, Pow(q, Mul(2, n))), Add(Sub(1, Mul(Mul(2, Pow(q, Sub(Mul(2, n), 1))), Cos(Mul(Mul(2, Pi), z)))), Pow(q, Sub(Mul(4, n), 2)))), For(n, 1, Infinity)), Equal(q, Exp(Mul(Mul(Pi, ConstI), tau)))))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH))))

39b699

\theta_{2}\!\left(z , \tau\right) = -i {e}^{\pi i \tau / 4} \left(w - {w}^{-1}\right) \prod_{n=1}^{\infty} \left(1 - {q}^{2 n}\right) \left(1 - {q}^{2 n} {w}^{2}\right) \left(1 - {q}^{2 n} {w}^{-2}\right)\; \text{ where } q = {e}^{\pi i \tau},\;w = {e}^{\pi i z}

Assumptions:

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

TeX:

\theta_{2}\!\left(z , \tau\right) = -i {e}^{\pi i \tau / 4} \left(w - {w}^{-1}\right) \prod_{n=1}^{\infty} \left(1 - {q}^{2 n}\right) \left(1 - {q}^{2 n} {w}^{2}\right) \left(1 - {q}^{2 n} {w}^{-2}\right)\; \text{ where } q = {e}^{\pi i \tau},\;w = {e}^{\pi i z}

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...)
`Pow`	${a}^{b}$	Power
`Product`	$\prod_{n} f(n)$	Product
`Infinity`	$\infty$	Positive infinity
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("39b699"),
    Formula(Where(Equal(JacobiTheta(2, z, tau), Neg(Mul(Mul(Mul(ConstI, Exp(Div(Mul(Mul(Pi, ConstI), tau), 4))), Sub(w, Pow(w, -1))), Product(Mul(Mul(Sub(1, Pow(q, Mul(2, n))), Sub(1, Mul(Pow(q, Mul(2, n)), Pow(w, 2)))), Sub(1, Mul(Pow(q, Mul(2, n)), Pow(w, -2)))), For(n, 1, Infinity))))), Equal(q, Exp(Mul(Mul(Pi, ConstI), tau))), Equal(w, Exp(Mul(Mul(Pi, ConstI), z))))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH))))

465810

\theta_{2}\!\left(z , \tau\right) = {e}^{\pi i \tau / 4} \left(w + {w}^{-1}\right) \prod_{n=1}^{\infty} \left(1 - {q}^{2 n}\right) \left(1 + {q}^{2 n} {w}^{2}\right) \left(1 + {q}^{2 n} {w}^{-2}\right)\; \text{ where } q = {e}^{\pi i \tau},\;w = {e}^{\pi i z}

Assumptions:

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

TeX:

\theta_{2}\!\left(z , \tau\right) = {e}^{\pi i \tau / 4} \left(w + {w}^{-1}\right) \prod_{n=1}^{\infty} \left(1 - {q}^{2 n}\right) \left(1 + {q}^{2 n} {w}^{2}\right) \left(1 + {q}^{2 n} {w}^{-2}\right)\; \text{ where } q = {e}^{\pi i \tau},\;w = {e}^{\pi i z}

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
`Pow`	${a}^{b}$	Power
`Product`	$\prod_{n} f(n)$	Product
`Infinity`	$\infty$	Positive infinity
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("465810"),
    Formula(Where(Equal(JacobiTheta(2, z, tau), Mul(Mul(Exp(Div(Mul(Mul(Pi, ConstI), tau), 4)), Add(w, Pow(w, -1))), Product(Mul(Mul(Sub(1, Pow(q, Mul(2, n))), Add(1, Mul(Pow(q, Mul(2, n)), Pow(w, 2)))), Add(1, Mul(Pow(q, Mul(2, n)), Pow(w, -2)))), For(n, 1, Infinity)))), Equal(q, Exp(Mul(Mul(Pi, ConstI), tau))), Equal(w, Exp(Mul(Mul(Pi, ConstI), z))))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH))))

21851b

\theta_{3}\!\left(z , \tau\right) = \prod_{n=1}^{\infty} \left(1 - {q}^{2 n}\right) \left(1 + {q}^{2 n - 1} {w}^{2}\right) \left(1 + {q}^{2 n - 1} {w}^{-2}\right)\; \text{ where } q = {e}^{\pi i \tau},\;w = {e}^{\pi i z}

Assumptions:

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

TeX:

\theta_{3}\!\left(z , \tau\right) = \prod_{n=1}^{\infty} \left(1 - {q}^{2 n}\right) \left(1 + {q}^{2 n - 1} {w}^{2}\right) \left(1 + {q}^{2 n - 1} {w}^{-2}\right)\; \text{ where } q = {e}^{\pi i \tau},\;w = {e}^{\pi i z}

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
`Product`	$\prod_{n} f(n)$	Product
`Pow`	${a}^{b}$	Power
`Infinity`	$\infty$	Positive infinity
`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("21851b"),
    Formula(Where(Equal(JacobiTheta(3, z, tau), Product(Mul(Mul(Sub(1, Pow(q, Mul(2, n))), Add(1, Mul(Pow(q, Sub(Mul(2, n), 1)), Pow(w, 2)))), Add(1, Mul(Pow(q, Sub(Mul(2, n), 1)), Pow(w, -2)))), For(n, 1, Infinity))), Equal(q, Exp(Mul(Mul(Pi, ConstI), tau))), Equal(w, Exp(Mul(Mul(Pi, ConstI), z))))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH))))

d45548

\theta_{4}\!\left(z , \tau\right) = \prod_{n=1}^{\infty} \left(1 - {q}^{2 n}\right) \left(1 - {q}^{2 n - 1} {w}^{2}\right) \left(1 - {q}^{2 n - 1} {w}^{-2}\right)\; \text{ where } q = {e}^{\pi i \tau},\;w = {e}^{\pi i z}

Assumptions:

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

TeX:

\theta_{4}\!\left(z , \tau\right) = \prod_{n=1}^{\infty} \left(1 - {q}^{2 n}\right) \left(1 - {q}^{2 n - 1} {w}^{2}\right) \left(1 - {q}^{2 n - 1} {w}^{-2}\right)\; \text{ where } q = {e}^{\pi i \tau},\;w = {e}^{\pi i z}

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
`Product`	$\prod_{n} f(n)$	Product
`Pow`	${a}^{b}$	Power
`Infinity`	$\infty$	Positive infinity
`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("d45548"),
    Formula(Where(Equal(JacobiTheta(4, z, tau), Product(Mul(Mul(Sub(1, Pow(q, Mul(2, n))), Sub(1, Mul(Pow(q, Sub(Mul(2, n), 1)), Pow(w, 2)))), Sub(1, Mul(Pow(q, Sub(Mul(2, n), 1)), Pow(w, -2)))), For(n, 1, Infinity))), Equal(q, Exp(Mul(Mul(Pi, ConstI), tau))), Equal(w, Exp(Mul(Mul(Pi, ConstI), z))))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH))))

13d2a1

\theta_{3}\!\left(z , \tau\right) = \sum_{n=-\infty}^{\infty} {q}^{{n}^{2}} {w}^{2 n} = \prod_{n=1}^{\infty} \left(1 - {q}^{2 n}\right) \left(1 + {q}^{2 n - 1} {w}^{2}\right) \left(1 + {q}^{2 n - 1} {w}^{-2}\right)\; \text{ where } q = {e}^{\pi i \tau},\;w = {e}^{\pi i z}

Assumptions:

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

TeX:

\theta_{3}\!\left(z , \tau\right) = \sum_{n=-\infty}^{\infty} {q}^{{n}^{2}} {w}^{2 n} = \prod_{n=1}^{\infty} \left(1 - {q}^{2 n}\right) \left(1 + {q}^{2 n - 1} {w}^{2}\right) \left(1 + {q}^{2 n - 1} {w}^{-2}\right)\; \text{ where } q = {e}^{\pi i \tau},\;w = {e}^{\pi i z}

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
`Sum`	$\sum_{n} f(n)$	Sum
`Pow`	${a}^{b}$	Power
`Infinity`	$\infty$	Positive infinity
`Product`	$\prod_{n} f(n)$	Product
`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("13d2a1"),
    Formula(Where(Equal(JacobiTheta(3, z, tau), Sum(Mul(Pow(q, Pow(n, 2)), Pow(w, Mul(2, n))), For(n, Neg(Infinity), Infinity)), Product(Mul(Mul(Sub(1, Pow(q, Mul(2, n))), Add(1, Mul(Pow(q, Sub(Mul(2, n), 1)), Pow(w, 2)))), Add(1, Mul(Pow(q, Sub(Mul(2, n), 1)), Pow(w, -2)))), For(n, 1, Infinity))), Equal(q, Exp(Mul(Mul(Pi, ConstI), tau))), Equal(w, Exp(Mul(Mul(Pi, ConstI), z))))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH))))

d2f183

\theta_{1}\!\left(z , \tau\right) = \frac{\theta'_{1}\!\left(0 , \tau\right)}{\pi} \sin\!\left(\pi z\right) \prod_{n=1}^{\infty} \frac{\sin\!\left(\pi \left(n \tau + z\right)\right) \sin\!\left(\pi \left(n \tau - z\right)\right)}{\sin^{2}\!\left(\pi n \tau\right)}

Assumptions:

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

TeX:

\theta_{1}\!\left(z , \tau\right) = \frac{\theta'_{1}\!\left(0 , \tau\right)}{\pi} \sin\!\left(\pi z\right) \prod_{n=1}^{\infty} \frac{\sin\!\left(\pi \left(n \tau + z\right)\right) \sin\!\left(\pi \left(n \tau - z\right)\right)}{\sin^{2}\!\left(\pi n \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
`Pi`	$\pi$	The constant pi (3.14...)
`Sin`	$\sin(z)$	Sine
`Product`	$\prod_{n} f(n)$	Product
`Pow`	${a}^{b}$	Power
`Infinity`	$\infty$	Positive infinity
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("d2f183"),
    Formula(Equal(JacobiTheta(1, z, tau), Mul(Mul(Div(JacobiTheta(1, 0, tau, 1), Pi), Sin(Mul(Pi, z))), Product(Div(Mul(Sin(Mul(Pi, Add(Mul(n, tau), z))), Sin(Mul(Pi, Sub(Mul(n, tau), z)))), Pow(Sin(Mul(Mul(Pi, n), tau)), 2)), For(n, 1, Infinity))))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH))))

64081c

\theta_{2}\!\left(z , \tau\right) = \theta_{2}\!\left(0 , \tau\right) \cos\!\left(\pi z\right) \prod_{n=1}^{\infty} \frac{\cos\!\left(\pi \left(n \tau + z\right)\right) \cos\!\left(\pi \left(n \tau - z\right)\right)}{\cos^{2}\!\left(\pi n \tau\right)}

Assumptions:

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

TeX:

\theta_{2}\!\left(z , \tau\right) = \theta_{2}\!\left(0 , \tau\right) \cos\!\left(\pi z\right) \prod_{n=1}^{\infty} \frac{\cos\!\left(\pi \left(n \tau + z\right)\right) \cos\!\left(\pi \left(n \tau - z\right)\right)}{\cos^{2}\!\left(\pi n \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
`Cos`	$\cos(z)$	Cosine
`Pi`	$\pi$	The constant pi (3.14...)
`Product`	$\prod_{n} f(n)$	Product
`Pow`	${a}^{b}$	Power
`Infinity`	$\infty$	Positive infinity
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("64081c"),
    Formula(Equal(JacobiTheta(2, z, tau), Mul(Mul(JacobiTheta(2, 0, tau), Cos(Mul(Pi, z))), Product(Div(Mul(Cos(Mul(Pi, Add(Mul(n, tau), z))), Cos(Mul(Pi, Sub(Mul(n, tau), z)))), Pow(Cos(Mul(Mul(Pi, n), tau)), 2)), For(n, 1, Infinity))))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH))))

816057

\theta_{3}\!\left(z , \tau\right) = \theta_{3}\!\left(0 , \tau\right) \prod_{n=1}^{\infty} \frac{\cos\!\left(\pi \left(\left(n - \frac{1}{2}\right) \tau + z\right)\right) \cos\!\left(\pi \left(\left(n - \frac{1}{2}\right) \tau - z\right)\right)}{\cos^{2}\!\left(\pi \left(n - \frac{1}{2}\right) \tau\right)}

Assumptions:

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

TeX:

\theta_{3}\!\left(z , \tau\right) = \theta_{3}\!\left(0 , \tau\right) \prod_{n=1}^{\infty} \frac{\cos\!\left(\pi \left(\left(n - \frac{1}{2}\right) \tau + z\right)\right) \cos\!\left(\pi \left(\left(n - \frac{1}{2}\right) \tau - z\right)\right)}{\cos^{2}\!\left(\pi \left(n - \frac{1}{2}\right) \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
`Product`	$\prod_{n} f(n)$	Product
`Cos`	$\cos(z)$	Cosine
`Pi`	$\pi$	The constant pi (3.14...)
`Pow`	${a}^{b}$	Power
`Infinity`	$\infty$	Positive infinity
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("816057"),
    Formula(Equal(JacobiTheta(3, z, tau), Mul(JacobiTheta(3, 0, tau), Product(Div(Mul(Cos(Mul(Pi, Add(Mul(Sub(n, Div(1, 2)), tau), z))), Cos(Mul(Pi, Sub(Mul(Sub(n, Div(1, 2)), tau), z)))), Pow(Cos(Mul(Pi, Mul(Sub(n, Div(1, 2)), tau))), 2)), For(n, 1, Infinity))))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH))))

3c88a7

\theta_{4}\!\left(z , \tau\right) = \theta_{4}\!\left(0 , \tau\right) \prod_{n=1}^{\infty} \frac{\sin\!\left(\pi \left(\left(n - \frac{1}{2}\right) \tau + z\right)\right) \sin\!\left(\pi \left(\left(n - \frac{1}{2}\right) \tau - z\right)\right)}{\sin^{2}\!\left(\pi \left(n - \frac{1}{2}\right) \tau\right)}

Assumptions:

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

TeX:

\theta_{4}\!\left(z , \tau\right) = \theta_{4}\!\left(0 , \tau\right) \prod_{n=1}^{\infty} \frac{\sin\!\left(\pi \left(\left(n - \frac{1}{2}\right) \tau + z\right)\right) \sin\!\left(\pi \left(\left(n - \frac{1}{2}\right) \tau - z\right)\right)}{\sin^{2}\!\left(\pi \left(n - \frac{1}{2}\right) \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
`Product`	$\prod_{n} f(n)$	Product
`Sin`	$\sin(z)$	Sine
`Pi`	$\pi$	The constant pi (3.14...)
`Pow`	${a}^{b}$	Power
`Infinity`	$\infty$	Positive infinity
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("3c88a7"),
    Formula(Equal(JacobiTheta(4, z, tau), Mul(JacobiTheta(4, 0, tau), Product(Div(Mul(Sin(Mul(Pi, Add(Mul(Sub(n, Div(1, 2)), tau), z))), Sin(Mul(Pi, Sub(Mul(Sub(n, Div(1, 2)), tau), z)))), Pow(Sin(Mul(Pi, Mul(Sub(n, Div(1, 2)), tau))), 2)), For(n, 1, Infinity))))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH))))

dfbddd

\frac{1}{\pi} \frac{\theta'_{1}\!\left(z , \tau\right)}{\theta_{1}\!\left(z , \tau\right)} = \cot\!\left(\pi z\right) + 4 \sum_{n=1}^{\infty} \frac{{q}^{2 n}}{1 - {q}^{2 n}} \sin\!\left(2 \pi n z\right)\; \text{ where } q = {e}^{\pi i \tau}

Assumptions:

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H} \;\mathbin{\operatorname{and}}\; \left|\operatorname{Im}(z)\right| < \left|\operatorname{Im}(\tau)\right| \;\mathbin{\operatorname{and}}\; \sin\!\left(\pi z\right) \ne 0

TeX:

\frac{1}{\pi} \frac{\theta'_{1}\!\left(z , \tau\right)}{\theta_{1}\!\left(z , \tau\right)} = \cot\!\left(\pi z\right) + 4 \sum_{n=1}^{\infty} \frac{{q}^{2 n}}{1 - {q}^{2 n}} \sin\!\left(2 \pi n z\right)\; \text{ where } q = {e}^{\pi i \tau}

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H} \;\mathbin{\operatorname{and}}\; \left|\operatorname{Im}(z)\right| < \left|\operatorname{Im}(\tau)\right| \;\mathbin{\operatorname{and}}\; \sin\!\left(\pi z\right) \ne 0

Definitions:

Fungrim symbol	Notation	Short description
`Pi`	$\pi$	The constant pi (3.14...)
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	Jacobi theta function
`Sum`	$\sum_{n} f(n)$	Sum
`Pow`	${a}^{b}$	Power
`Sin`	$\sin(z)$	Sine
`Infinity`	$\infty$	Positive infinity
`Exp`	${e}^{z}$	Exponential function
`ConstI`	$i$	Imaginary unit
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane
`Abs`	$\left\|z\right\|$	Absolute value
`Im`	$\operatorname{Im}(z)$	Imaginary part

Source code for this entry:

Entry(ID("dfbddd"),
    Formula(Equal(Mul(Div(1, Pi), Div(JacobiTheta(1, z, tau, 1), JacobiTheta(1, z, tau))), Where(Add(Cot(Mul(Pi, z)), Mul(4, Sum(Mul(Div(Pow(q, Mul(2, n)), Sub(1, Pow(q, Mul(2, n)))), Sin(Mul(Mul(Mul(2, Pi), n), z))), For(n, 1, Infinity)))), Equal(q, Exp(Mul(Mul(Pi, ConstI), tau)))))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH), Less(Abs(Im(z)), Abs(Im(tau))), NotEqual(Sin(Mul(Pi, z)), 0))))

c7f7a5

\frac{1}{\pi} \frac{\theta'_{2}\!\left(z , \tau\right)}{\theta_{2}\!\left(z , \tau\right)} = -\tan\!\left(\pi z\right) + 4 \sum_{n=1}^{\infty} {\left(-1\right)}^{n} \frac{{q}^{2 n}}{1 - {q}^{2 n}} \sin\!\left(2 \pi n z\right)\; \text{ where } q = {e}^{\pi i \tau}

Assumptions:

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H} \;\mathbin{\operatorname{and}}\; \left|\operatorname{Im}(z)\right| < \left|\operatorname{Im}(\tau)\right| \;\mathbin{\operatorname{and}}\; \cos\!\left(\pi z\right) \ne 0

TeX:

\frac{1}{\pi} \frac{\theta'_{2}\!\left(z , \tau\right)}{\theta_{2}\!\left(z , \tau\right)} = -\tan\!\left(\pi z\right) + 4 \sum_{n=1}^{\infty} {\left(-1\right)}^{n} \frac{{q}^{2 n}}{1 - {q}^{2 n}} \sin\!\left(2 \pi n z\right)\; \text{ where } q = {e}^{\pi i \tau}

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H} \;\mathbin{\operatorname{and}}\; \left|\operatorname{Im}(z)\right| < \left|\operatorname{Im}(\tau)\right| \;\mathbin{\operatorname{and}}\; \cos\!\left(\pi z\right) \ne 0

Definitions:

Fungrim symbol	Notation	Short description
`Pi`	$\pi$	The constant pi (3.14...)
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	Jacobi theta function
`Sum`	$\sum_{n} f(n)$	Sum
`Pow`	${a}^{b}$	Power
`Sin`	$\sin(z)$	Sine
`Infinity`	$\infty$	Positive infinity
`Exp`	${e}^{z}$	Exponential function
`ConstI`	$i$	Imaginary unit
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane
`Abs`	$\left\|z\right\|$	Absolute value
`Im`	$\operatorname{Im}(z)$	Imaginary part
`Cos`	$\cos(z)$	Cosine

Source code for this entry:

Entry(ID("c7f7a5"),
    Formula(Equal(Mul(Div(1, Pi), Div(JacobiTheta(2, z, tau, 1), JacobiTheta(2, z, tau))), Where(Add(Neg(Tan(Mul(Pi, z))), Mul(4, Sum(Mul(Mul(Pow(-1, n), Div(Pow(q, Mul(2, n)), Sub(1, Pow(q, Mul(2, n))))), Sin(Mul(Mul(Mul(2, Pi), n), z))), For(n, 1, Infinity)))), Equal(q, Exp(Mul(Mul(Pi, ConstI), tau)))))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH), Less(Abs(Im(z)), Abs(Im(tau))), NotEqual(Cos(Mul(Pi, z)), 0))))

44e8fb

\frac{1}{\pi} \frac{\theta'_{3}\!\left(z , \tau\right)}{\theta_{3}\!\left(z , \tau\right)} = 4 \sum_{n=1}^{\infty} {\left(-1\right)}^{n} \frac{{q}^{n}}{1 - {q}^{2 n}} \sin\!\left(2 \pi n z\right)\; \text{ where } q = {e}^{\pi i \tau}

Assumptions:

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H} \;\mathbin{\operatorname{and}}\; \left|\operatorname{Im}(z)\right| < \frac{1}{2} \left|\operatorname{Im}(\tau)\right|

TeX:

\frac{1}{\pi} \frac{\theta'_{3}\!\left(z , \tau\right)}{\theta_{3}\!\left(z , \tau\right)} = 4 \sum_{n=1}^{\infty} {\left(-1\right)}^{n} \frac{{q}^{n}}{1 - {q}^{2 n}} \sin\!\left(2 \pi n z\right)\; \text{ where } q = {e}^{\pi i \tau}

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H} \;\mathbin{\operatorname{and}}\; \left|\operatorname{Im}(z)\right| < \frac{1}{2} \left|\operatorname{Im}(\tau)\right|

Definitions:

Fungrim symbol	Notation	Short description
`Pi`	$\pi$	The constant pi (3.14...)
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	Jacobi theta function
`Sum`	$\sum_{n} f(n)$	Sum
`Pow`	${a}^{b}$	Power
`Sin`	$\sin(z)$	Sine
`Infinity`	$\infty$	Positive infinity
`Exp`	${e}^{z}$	Exponential function
`ConstI`	$i$	Imaginary unit
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane
`Abs`	$\left\|z\right\|$	Absolute value
`Im`	$\operatorname{Im}(z)$	Imaginary part

Source code for this entry:

Entry(ID("44e8fb"),
    Formula(Equal(Mul(Div(1, Pi), Div(JacobiTheta(3, z, tau, 1), JacobiTheta(3, z, tau))), Where(Mul(4, Sum(Mul(Mul(Pow(-1, n), Div(Pow(q, n), Sub(1, Pow(q, Mul(2, n))))), Sin(Mul(Mul(Mul(2, Pi), n), z))), For(n, 1, Infinity))), Equal(q, Exp(Mul(Mul(Pi, ConstI), tau)))))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH), Less(Abs(Im(z)), Mul(Div(1, 2), Abs(Im(tau)))))))

1848f1

\frac{1}{\pi} \frac{\theta'_{4}\!\left(z , \tau\right)}{\theta_{4}\!\left(z , \tau\right)} = 4 \sum_{n=1}^{\infty} \frac{{q}^{n}}{1 - {q}^{2 n}} \sin\!\left(2 \pi n z\right)\; \text{ where } q = {e}^{\pi i \tau}

Assumptions:

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H} \;\mathbin{\operatorname{and}}\; \left|\operatorname{Im}(z)\right| < \frac{1}{2} \left|\operatorname{Im}(\tau)\right|

TeX:

\frac{1}{\pi} \frac{\theta'_{4}\!\left(z , \tau\right)}{\theta_{4}\!\left(z , \tau\right)} = 4 \sum_{n=1}^{\infty} \frac{{q}^{n}}{1 - {q}^{2 n}} \sin\!\left(2 \pi n z\right)\; \text{ where } q = {e}^{\pi i \tau}

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H} \;\mathbin{\operatorname{and}}\; \left|\operatorname{Im}(z)\right| < \frac{1}{2} \left|\operatorname{Im}(\tau)\right|

Definitions:

Fungrim symbol	Notation	Short description
`Pi`	$\pi$	The constant pi (3.14...)
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	Jacobi theta function
`Sum`	$\sum_{n} f(n)$	Sum
`Pow`	${a}^{b}$	Power
`Sin`	$\sin(z)$	Sine
`Infinity`	$\infty$	Positive infinity
`Exp`	${e}^{z}$	Exponential function
`ConstI`	$i$	Imaginary unit
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane
`Abs`	$\left\|z\right\|$	Absolute value
`Im`	$\operatorname{Im}(z)$	Imaginary part

Source code for this entry:

Entry(ID("1848f1"),
    Formula(Equal(Mul(Div(1, Pi), Div(JacobiTheta(4, z, tau, 1), JacobiTheta(4, z, tau))), Where(Mul(4, Sum(Mul(Div(Pow(q, n), Sub(1, Pow(q, Mul(2, n)))), Sin(Mul(Mul(Mul(2, Pi), n), z))), For(n, 1, Infinity))), Equal(q, Exp(Mul(Mul(Pi, ConstI), tau)))))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH), Less(Abs(Im(z)), Mul(Div(1, 2), Abs(Im(tau)))))))

d81f05

\frac{d^{2}}{{d z}^{2}} \log\!\left(\theta_{1}\!\left(z , \tau\right)\right) = {\pi}^{2} \sum_{n=-\infty}^{\infty} \frac{1}{\sin^{2}\!\left(\pi \left(z + n \tau\right)\right)}

Assumptions:

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H} \;\mathbin{\operatorname{and}}\; \theta_{1}\!\left(z , \tau\right) \ne 0

TeX:

\frac{d^{2}}{{d z}^{2}} \log\!\left(\theta_{1}\!\left(z , \tau\right)\right) = {\pi}^{2} \sum_{n=-\infty}^{\infty} \frac{1}{\sin^{2}\!\left(\pi \left(z + n \tau\right)\right)}

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H} \;\mathbin{\operatorname{and}}\; \theta_{1}\!\left(z , \tau\right) \ne 0

Definitions:

Fungrim symbol	Notation	Short description
`ComplexBranchDerivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Complex derivative, allowing branch cuts
`Log`	$\log(z)$	Natural logarithm
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	Jacobi theta function
`Pow`	${a}^{b}$	Power
`Pi`	$\pi$	The constant pi (3.14...)
`Sum`	$\sum_{n} f(n)$	Sum
`Sin`	$\sin(z)$	Sine
`Infinity`	$\infty$	Positive infinity
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("d81f05"),
    Formula(Equal(ComplexBranchDerivative(Log(JacobiTheta(1, z, tau)), For(z, z, 2)), Mul(Pow(Pi, 2), Sum(Div(1, Pow(Sin(Mul(Pi, Add(z, Mul(n, tau)))), 2)), For(n, Neg(Infinity), Infinity))))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH), NotEqual(JacobiTheta(1, z, tau), 0))))

561d75

\frac{d^{2}}{{d z}^{2}} \log\!\left(\theta_{2}\!\left(z , \tau\right)\right) = {\pi}^{2} \sum_{n=-\infty}^{\infty} \frac{1}{\cos^{2}\!\left(\pi \left(z + n \tau\right)\right)}

Assumptions:

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H} \;\mathbin{\operatorname{and}}\; \theta_{2}\!\left(z , \tau\right) \ne 0

TeX:

\frac{d^{2}}{{d z}^{2}} \log\!\left(\theta_{2}\!\left(z , \tau\right)\right) = {\pi}^{2} \sum_{n=-\infty}^{\infty} \frac{1}{\cos^{2}\!\left(\pi \left(z + n \tau\right)\right)}

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H} \;\mathbin{\operatorname{and}}\; \theta_{2}\!\left(z , \tau\right) \ne 0

Definitions:

Fungrim symbol	Notation	Short description
`ComplexBranchDerivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Complex derivative, allowing branch cuts
`Log`	$\log(z)$	Natural logarithm
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	Jacobi theta function
`Pow`	${a}^{b}$	Power
`Pi`	$\pi$	The constant pi (3.14...)
`Sum`	$\sum_{n} f(n)$	Sum
`Cos`	$\cos(z)$	Cosine
`Infinity`	$\infty$	Positive infinity
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("561d75"),
    Formula(Equal(ComplexBranchDerivative(Log(JacobiTheta(2, z, tau)), For(z, z, 2)), Mul(Pow(Pi, 2), Sum(Div(1, Pow(Cos(Mul(Pi, Add(z, Mul(n, tau)))), 2)), For(n, Neg(Infinity), Infinity))))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH), NotEqual(JacobiTheta(2, z, tau), 0))))

1cdd7b

\theta_{j}\!\left(z + x , \tau\right) = \sum_{n=0}^{\infty} \frac{\theta^{(n)}_{j}\!\left(z , \tau\right)}{n !} {x}^{n}

Assumptions:

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

TeX:

\theta_{j}\!\left(z + x , \tau\right) = \sum_{n=0}^{\infty} \frac{\theta^{(n)}_{j}\!\left(z , \tau\right)}{n !} {x}^{n}

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

Definitions:

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

Source code for this entry:

Entry(ID("1cdd7b"),
    Formula(Equal(JacobiTheta(j, Add(z, x), tau), Sum(Mul(Div(JacobiTheta(j, z, tau, n), Factorial(n)), Pow(x, n)), For(n, 0, Infinity)))),
    Variables(j, z, tau, x),
    Assumptions(And(Element(j, Set(1, 2, 3, 4)), Element(z, CC), Element(tau, HH), Element(x, CC))))

d637c5

\theta_{j}\!\left(z , \tau + x\right) = \sum_{n=0}^{\infty} \frac{1}{{\left(4 \pi i\right)}^{n}} \frac{\theta^{(2 n)}_{j}\!\left(z , \tau\right)}{n !} {x}^{n}

Assumptions:

j \in \left\{1, 2, 3, 4\right\} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left|x\right| < \operatorname{Im}(\tau)

TeX:

\theta_{j}\!\left(z , \tau + x\right) = \sum_{n=0}^{\infty} \frac{1}{{\left(4 \pi i\right)}^{n}} \frac{\theta^{(2 n)}_{j}\!\left(z , \tau\right)}{n !} {x}^{n}

j \in \left\{1, 2, 3, 4\right\} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left|x\right| < \operatorname{Im}(\tau)

Definitions:

Fungrim symbol	Notation	Short description
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	Jacobi theta function
`Sum`	$\sum_{n} f(n)$	Sum
`Pow`	${a}^{b}$	Power
`Pi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit
`Factorial`	$n !$	Factorial
`Infinity`	$\infty$	Positive infinity
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane
`Abs`	$\left\|z\right\|$	Absolute value
`Im`	$\operatorname{Im}(z)$	Imaginary part

Source code for this entry:

Entry(ID("d637c5"),
    Formula(Equal(JacobiTheta(j, z, Add(tau, x)), Sum(Mul(Mul(Div(1, Pow(Mul(Mul(4, Pi), ConstI), n)), Div(JacobiTheta(j, z, tau, Mul(2, n)), Factorial(n))), Pow(x, n)), For(n, 0, Infinity)))),
    Variables(j, z, tau, x),
    Assumptions(And(Element(j, Set(1, 2, 3, 4)), Element(z, CC), Element(tau, HH), Element(x, CC), Less(Abs(x), Im(tau)))))

a5e568

\theta_{3}^{k}\!\left(0, \tau\right) = \sum_{n=0}^{\infty} r_{k}\!\left(n\right) {q}^{n}\; \text{ where } q = {e}^{\pi i \tau}

Assumptions:

k \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H}

TeX:

\theta_{3}^{k}\!\left(0, \tau\right) = \sum_{n=0}^{\infty} r_{k}\!\left(n\right) {q}^{n}\; \text{ where } q = {e}^{\pi i \tau}

k \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H}

Definitions:

Fungrim symbol	Notation	Short description
`Pow`	${a}^{b}$	Power
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	Jacobi theta function
`Sum`	$\sum_{n} f(n)$	Sum
`SquaresR`	$r_{k}\!\left(n\right)$	Sum of squares function
`Infinity`	$\infty$	Positive infinity
`Exp`	${e}^{z}$	Exponential function
`Pi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("a5e568"),
    Formula(Equal(Pow(JacobiTheta(3, 0, tau), k), Where(Sum(Mul(SquaresR(k, n), Pow(q, n)), For(n, 0, Infinity)), Equal(q, Exp(Mul(Mul(Pi, ConstI), tau)))))),
    Variables(k, tau),
    Assumptions(And(Element(k, ZZGreaterEqual(0)), Element(tau, HH))))

7c90eb

\theta_{4}^{k}\!\left(0, \tau\right) = \sum_{n=0}^{\infty} {\left(-1\right)}^{n} r_{k}\!\left(n\right) {q}^{n}\; \text{ where } q = {e}^{\pi i \tau}

Assumptions:

k \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H}

TeX:

\theta_{4}^{k}\!\left(0, \tau\right) = \sum_{n=0}^{\infty} {\left(-1\right)}^{n} r_{k}\!\left(n\right) {q}^{n}\; \text{ where } q = {e}^{\pi i \tau}

k \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H}

Definitions:

Fungrim symbol	Notation	Short description
`Pow`	${a}^{b}$	Power
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	Jacobi theta function
`Sum`	$\sum_{n} f(n)$	Sum
`SquaresR`	$r_{k}\!\left(n\right)$	Sum of squares function
`Infinity`	$\infty$	Positive infinity
`Exp`	${e}^{z}$	Exponential function
`Pi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("7c90eb"),
    Formula(Equal(Pow(JacobiTheta(4, 0, tau), k), Where(Sum(Mul(Mul(Pow(-1, n), SquaresR(k, n)), Pow(q, n)), For(n, 0, Infinity)), Equal(q, Exp(Mul(Mul(Pi, ConstI), tau)))))),
    Variables(k, tau),
    Assumptions(And(Element(k, ZZGreaterEqual(0)), Element(tau, HH))))

df88a0

\theta_{3}^{2}\!\left(0, \tau\right) + \theta_{4}^{2}\!\left(0, \tau\right) = 2 \sum_{n=0}^{\infty} r_{2}\!\left(2 n\right) {q}^{2 n}\; \text{ where } q = {e}^{\pi i \tau}

Assumptions:

\tau \in \mathbb{H}

TeX:

\theta_{3}^{2}\!\left(0, \tau\right) + \theta_{4}^{2}\!\left(0, \tau\right) = 2 \sum_{n=0}^{\infty} r_{2}\!\left(2 n\right) {q}^{2 n}\; \text{ where } q = {e}^{\pi i \tau}

\tau \in \mathbb{H}

Definitions:

Fungrim symbol	Notation	Short description
`Pow`	${a}^{b}$	Power
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	Jacobi theta function
`Sum`	$\sum_{n} f(n)$	Sum
`SquaresR`	$r_{k}\!\left(n\right)$	Sum of squares function
`Infinity`	$\infty$	Positive infinity
`Exp`	${e}^{z}$	Exponential function
`Pi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("df88a0"),
    Formula(Equal(Add(Pow(JacobiTheta(3, 0, tau), 2), Pow(JacobiTheta(4, 0, tau), 2)), Mul(2, Where(Sum(Mul(SquaresR(2, Mul(2, n)), Pow(q, Mul(2, n))), For(n, 0, Infinity)), Equal(q, Exp(Mul(Mul(Pi, ConstI), tau))))))),
    Variables(tau),
    Assumptions(Element(tau, HH)))

290f36

\theta_{3}^{2}\!\left(0, \tau\right) - \theta_{3}^{2}\!\left(0, 2 \tau\right) = \sum_{n=0}^{\infty} r_{2}\!\left(2 n + 1\right) {q}^{2 n + 1}\; \text{ where } q = {e}^{\pi i \tau}

Assumptions:

\tau \in \mathbb{H}

TeX:

\theta_{3}^{2}\!\left(0, \tau\right) - \theta_{3}^{2}\!\left(0, 2 \tau\right) = \sum_{n=0}^{\infty} r_{2}\!\left(2 n + 1\right) {q}^{2 n + 1}\; \text{ where } q = {e}^{\pi i \tau}

\tau \in \mathbb{H}

Definitions:

Fungrim symbol	Notation	Short description
`Pow`	${a}^{b}$	Power
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	Jacobi theta function
`Sum`	$\sum_{n} f(n)$	Sum
`SquaresR`	$r_{k}\!\left(n\right)$	Sum of squares function
`Infinity`	$\infty$	Positive infinity
`Exp`	${e}^{z}$	Exponential function
`Pi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("290f36"),
    Formula(Equal(Sub(Pow(JacobiTheta(3, 0, tau), 2), Pow(JacobiTheta(3, 0, Mul(2, tau)), 2)), Where(Sum(Mul(SquaresR(2, Add(Mul(2, n), 1)), Pow(q, Add(Mul(2, n), 1))), For(n, 0, Infinity)), Equal(q, Exp(Mul(Mul(Pi, ConstI), tau)))))),
    Variables(tau),
    Assumptions(Element(tau, HH)))

a0ba58

\frac{\theta_{3}\!\left(0 , \tau\right)}{\theta_{4}\!\left(0 , \tau\right)} = \prod_{n=1}^{\infty} {\left(\frac{1 + {q}^{2 n - 1}}{1 - {q}^{2 n - 1}}\right)}^{2}\; \text{ where } q = {e}^{\pi i \tau}

Assumptions:

\tau \in \mathbb{H}

TeX:

\frac{\theta_{3}\!\left(0 , \tau\right)}{\theta_{4}\!\left(0 , \tau\right)} = \prod_{n=1}^{\infty} {\left(\frac{1 + {q}^{2 n - 1}}{1 - {q}^{2 n - 1}}\right)}^{2}\; \text{ where } q = {e}^{\pi i \tau}

\tau \in \mathbb{H}

Definitions:

Fungrim symbol	Notation	Short description
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	Jacobi theta function
`Product`	$\prod_{n} f(n)$	Product
`Pow`	${a}^{b}$	Power
`Infinity`	$\infty$	Positive infinity
`Exp`	${e}^{z}$	Exponential function
`Pi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("a0ba58"),
    Formula(Equal(Div(JacobiTheta(3, 0, tau), JacobiTheta(4, 0, tau)), Where(Product(Pow(Div(Add(1, Pow(q, Sub(Mul(2, n), 1))), Sub(1, Pow(q, Sub(Mul(2, n), 1)))), 2), For(n, 1, Infinity)), Equal(q, Exp(Mul(Mul(Pi, ConstI), tau)))))),
    Variables(tau),
    Assumptions(Element(tau, HH)))

f1f42f

\frac{\theta_{2}\!\left(0 , \tau\right)}{\theta_{3}\!\left(0 , \tau\right)} = 2 {e}^{\pi i \tau / 4} \prod_{n=1}^{\infty} {\left(\frac{1 + {q}^{2 n}}{1 + {q}^{2 n - 1}}\right)}^{2}\; \text{ where } q = {e}^{\pi i \tau}

Assumptions:

\tau \in \mathbb{H}

TeX:

\frac{\theta_{2}\!\left(0 , \tau\right)}{\theta_{3}\!\left(0 , \tau\right)} = 2 {e}^{\pi i \tau / 4} \prod_{n=1}^{\infty} {\left(\frac{1 + {q}^{2 n}}{1 + {q}^{2 n - 1}}\right)}^{2}\; \text{ where } q = {e}^{\pi i \tau}

\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
`Product`	$\prod_{n} f(n)$	Product
`Pow`	${a}^{b}$	Power
`Infinity`	$\infty$	Positive infinity
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("f1f42f"),
    Formula(Equal(Div(JacobiTheta(2, 0, tau), JacobiTheta(3, 0, tau)), Where(Mul(Mul(2, Exp(Div(Mul(Mul(Pi, ConstI), tau), 4))), Product(Pow(Div(Add(1, Pow(q, Mul(2, n))), Add(1, Pow(q, Sub(Mul(2, n), 1)))), 2), For(n, 1, Infinity))), Equal(q, Exp(Mul(Mul(Pi, ConstI), tau)))))),
    Variables(tau),
    Assumptions(Element(tau, HH)))

e4e707

\theta_{3}\!\left(0 , \tau\right) = 1 + 2 \sum_{n=1}^{\infty} \frac{\lambda(n) {q}^{n}}{1 - {q}^{n}}\; \text{ where } q = {e}^{\pi i \tau}

Assumptions:

\tau \in \mathbb{H}

TeX:

\theta_{3}\!\left(0 , \tau\right) = 1 + 2 \sum_{n=1}^{\infty} \frac{\lambda(n) {q}^{n}}{1 - {q}^{n}}\; \text{ where } q = {e}^{\pi i \tau}

\tau \in \mathbb{H}

Definitions:

Fungrim symbol	Notation	Short description
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	Jacobi theta function
`Sum`	$\sum_{n} f(n)$	Sum
`LiouvilleLambda`	$\lambda(n)$	Liouville function
`Pow`	${a}^{b}$	Power
`Infinity`	$\infty$	Positive infinity
`Exp`	${e}^{z}$	Exponential function
`Pi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("e4e707"),
    Formula(Equal(JacobiTheta(3, 0, tau), Where(Add(1, Mul(2, Sum(Div(Mul(LiouvilleLambda(n), Pow(q, n)), Sub(1, Pow(q, n))), For(n, 1, Infinity)))), Equal(q, Exp(Mul(Mul(Pi, ConstI), tau)))))),
    Variables(tau),
    Assumptions(Element(tau, HH)))

0650f8

\theta_{3}^{2}\!\left(0, \tau\right) = 1 + 4 \sum_{n=1}^{\infty} \frac{{q}^{n}}{1 + {q}^{2 n}}\; \text{ where } q = {e}^{\pi i \tau}

Assumptions:

\tau \in \mathbb{H}

TeX:

\theta_{3}^{2}\!\left(0, \tau\right) = 1 + 4 \sum_{n=1}^{\infty} \frac{{q}^{n}}{1 + {q}^{2 n}}\; \text{ where } q = {e}^{\pi i \tau}

\tau \in \mathbb{H}

Definitions:

Fungrim symbol	Notation	Short description
`Pow`	${a}^{b}$	Power
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	Jacobi theta function
`Sum`	$\sum_{n} f(n)$	Sum
`Infinity`	$\infty$	Positive infinity
`Exp`	${e}^{z}$	Exponential function
`Pi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("0650f8"),
    Formula(Equal(Pow(JacobiTheta(3, 0, tau), 2), Where(Add(1, Mul(4, Sum(Div(Pow(q, n), Add(1, Pow(q, Mul(2, n)))), For(n, 1, Infinity)))), Equal(q, Exp(Mul(Mul(Pi, ConstI), tau)))))),
    Variables(tau),
    Assumptions(Element(tau, HH)))

c743eb

\theta_{2}^{4}\!\left(0, \tau\right) = 8 \sum_{n=0}^{\infty} \frac{\left(2 n + 1\right) {q}^{2 n + 1}}{1 + {q}^{2 n + 1}} + 8 \sum_{n=0}^{\infty} \frac{\left(2 n + 1\right) {q}^{2 n + 1}}{1 - {q}^{2 n + 1}}\; \text{ where } q = {e}^{\pi i \tau}

Assumptions:

\tau \in \mathbb{H}

TeX:

\theta_{2}^{4}\!\left(0, \tau\right) = 8 \sum_{n=0}^{\infty} \frac{\left(2 n + 1\right) {q}^{2 n + 1}}{1 + {q}^{2 n + 1}} + 8 \sum_{n=0}^{\infty} \frac{\left(2 n + 1\right) {q}^{2 n + 1}}{1 - {q}^{2 n + 1}}\; \text{ where } q = {e}^{\pi i \tau}

\tau \in \mathbb{H}

Definitions:

Fungrim symbol	Notation	Short description
`Pow`	${a}^{b}$	Power
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	Jacobi theta function
`Sum`	$\sum_{n} f(n)$	Sum
`Infinity`	$\infty$	Positive infinity
`Exp`	${e}^{z}$	Exponential function
`Pi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("c743eb"),
    Formula(Equal(Pow(JacobiTheta(2, 0, tau), 4), Where(Add(Mul(8, Sum(Div(Mul(Add(Mul(2, n), 1), Pow(q, Add(Mul(2, n), 1))), Add(1, Pow(q, Add(Mul(2, n), 1)))), For(n, 0, Infinity))), Mul(8, Sum(Div(Mul(Add(Mul(2, n), 1), Pow(q, Add(Mul(2, n), 1))), Sub(1, Pow(q, Add(Mul(2, n), 1)))), For(n, 0, Infinity)))), Equal(q, Exp(Mul(Mul(Pi, ConstI), tau)))))),
    Variables(tau),
    Assumptions(Element(tau, HH)))

8a316c

\theta_{3}^{4}\!\left(0, \tau\right) = 1 + 8 \sum_{n=0}^{\infty} \frac{2 n {q}^{2 n}}{1 + {q}^{2 n}} + 8 \sum_{n=0}^{\infty} \frac{\left(2 n + 1\right) {q}^{2 n + 1}}{1 - {q}^{2 n + 1}}\; \text{ where } q = {e}^{\pi i \tau}

Assumptions:

\tau \in \mathbb{H}

TeX:

\theta_{3}^{4}\!\left(0, \tau\right) = 1 + 8 \sum_{n=0}^{\infty} \frac{2 n {q}^{2 n}}{1 + {q}^{2 n}} + 8 \sum_{n=0}^{\infty} \frac{\left(2 n + 1\right) {q}^{2 n + 1}}{1 - {q}^{2 n + 1}}\; \text{ where } q = {e}^{\pi i \tau}

\tau \in \mathbb{H}

Definitions:

Fungrim symbol	Notation	Short description
`Pow`	${a}^{b}$	Power
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	Jacobi theta function
`Sum`	$\sum_{n} f(n)$	Sum
`Infinity`	$\infty$	Positive infinity
`Exp`	${e}^{z}$	Exponential function
`Pi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("8a316c"),
    Formula(Equal(Pow(JacobiTheta(3, 0, tau), 4), Where(Add(Add(1, Mul(8, Sum(Div(Mul(Mul(2, n), Pow(q, Mul(2, n))), Add(1, Pow(q, Mul(2, n)))), For(n, 0, Infinity)))), Mul(8, Sum(Div(Mul(Add(Mul(2, n), 1), Pow(q, Add(Mul(2, n), 1))), Sub(1, Pow(q, Add(Mul(2, n), 1)))), For(n, 0, Infinity)))), Equal(q, Exp(Mul(Mul(Pi, ConstI), tau)))))),
    Variables(tau),
    Assumptions(Element(tau, HH)))

dc7c83

\theta_{4}^{4}\!\left(0, \tau\right) = 1 + 8 \sum_{n=0}^{\infty} \frac{2 n {q}^{2 n}}{1 + {q}^{2 n}} - 8 \sum_{n=0}^{\infty} \frac{\left(2 n + 1\right) {q}^{2 n + 1}}{1 + {q}^{2 n + 1}}\; \text{ where } q = {e}^{\pi i \tau}

Assumptions:

\tau \in \mathbb{H}

TeX:

\theta_{4}^{4}\!\left(0, \tau\right) = 1 + 8 \sum_{n=0}^{\infty} \frac{2 n {q}^{2 n}}{1 + {q}^{2 n}} - 8 \sum_{n=0}^{\infty} \frac{\left(2 n + 1\right) {q}^{2 n + 1}}{1 + {q}^{2 n + 1}}\; \text{ where } q = {e}^{\pi i \tau}

\tau \in \mathbb{H}

Definitions:

Fungrim symbol	Notation	Short description
`Pow`	${a}^{b}$	Power
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	Jacobi theta function
`Sum`	$\sum_{n} f(n)$	Sum
`Infinity`	$\infty$	Positive infinity
`Exp`	${e}^{z}$	Exponential function
`Pi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("dc7c83"),
    Formula(Equal(Pow(JacobiTheta(4, 0, tau), 4), Where(Sub(Add(1, Mul(8, Sum(Div(Mul(Mul(2, n), Pow(q, Mul(2, n))), Add(1, Pow(q, Mul(2, n)))), For(n, 0, Infinity)))), Mul(8, Sum(Div(Mul(Add(Mul(2, n), 1), Pow(q, Add(Mul(2, n), 1))), Add(1, Pow(q, Add(Mul(2, n), 1)))), For(n, 0, Infinity)))), Equal(q, Exp(Mul(Mul(Pi, ConstI), tau)))))),
    Variables(tau),
    Assumptions(Element(tau, HH)))

1cec67

\theta_{4}^{4}\!\left(0, \tau\right) - \theta_{2}^{4}\!\left(0, \tau\right) = 1 - 24 \sum_{n=0}^{\infty} \frac{\left(2 n + 1\right) {q}^{2 n + 1}}{1 + {q}^{2 n + 1}}\; \text{ where } q = {e}^{\pi i \tau}

Assumptions:

\tau \in \mathbb{H}

TeX:

\theta_{4}^{4}\!\left(0, \tau\right) - \theta_{2}^{4}\!\left(0, \tau\right) = 1 - 24 \sum_{n=0}^{\infty} \frac{\left(2 n + 1\right) {q}^{2 n + 1}}{1 + {q}^{2 n + 1}}\; \text{ where } q = {e}^{\pi i \tau}

\tau \in \mathbb{H}

Definitions:

Fungrim symbol	Notation	Short description
`Pow`	${a}^{b}$	Power
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	Jacobi theta function
`Sum`	$\sum_{n} f(n)$	Sum
`Infinity`	$\infty$	Positive infinity
`Exp`	${e}^{z}$	Exponential function
`Pi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("1cec67"),
    Formula(Equal(Sub(Pow(JacobiTheta(4, 0, tau), 4), Pow(JacobiTheta(2, 0, tau), 4)), Where(Sub(1, Mul(24, Sum(Div(Mul(Add(Mul(2, n), 1), Pow(q, Add(Mul(2, n), 1))), Add(1, Pow(q, Add(Mul(2, n), 1)))), For(n, 0, Infinity)))), Equal(q, Exp(Mul(Mul(Pi, ConstI), tau)))))),
    Variables(tau),
    Assumptions(Element(tau, HH)))

4d26ec

\theta_{4}^{8}\!\left(0, \tau\right) = 1 + 16 \sum_{n=1}^{\infty} \frac{{\left(-1\right)}^{n} {n}^{3} {q}^{n}}{1 - {q}^{n}}\; \text{ where } q = {e}^{\pi i \tau}

Assumptions:

\tau \in \mathbb{H}

TeX:

\theta_{4}^{8}\!\left(0, \tau\right) = 1 + 16 \sum_{n=1}^{\infty} \frac{{\left(-1\right)}^{n} {n}^{3} {q}^{n}}{1 - {q}^{n}}\; \text{ where } q = {e}^{\pi i \tau}

\tau \in \mathbb{H}

Definitions:

Fungrim symbol	Notation	Short description
`Pow`	${a}^{b}$	Power
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	Jacobi theta function
`Sum`	$\sum_{n} f(n)$	Sum
`Infinity`	$\infty$	Positive infinity
`Exp`	${e}^{z}$	Exponential function
`Pi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("4d26ec"),
    Formula(Equal(Pow(JacobiTheta(4, 0, tau), 8), Where(Add(1, Mul(16, Sum(Div(Mul(Mul(Pow(-1, n), Pow(n, 3)), Pow(q, n)), Sub(1, Pow(q, n))), For(n, 1, Infinity)))), Equal(q, Exp(Mul(Mul(Pi, ConstI), tau)))))),
    Variables(tau),
    Assumptions(Element(tau, HH)))

9b7d8c

\theta_{2}^{2}\!\left(0, \tau\right) = \sum_{n=-\infty}^{\infty} \frac{1}{\cos\!\left(\pi \tau \left(n + \frac{1}{2}\right)\right)}

Assumptions:

\tau \in \mathbb{H}

TeX:

\theta_{2}^{2}\!\left(0, \tau\right) = \sum_{n=-\infty}^{\infty} \frac{1}{\cos\!\left(\pi \tau \left(n + \frac{1}{2}\right)\right)}

\tau \in \mathbb{H}

Definitions:

Fungrim symbol	Notation	Short description
`Pow`	${a}^{b}$	Power
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	Jacobi theta function
`Sum`	$\sum_{n} f(n)$	Sum
`Cos`	$\cos(z)$	Cosine
`Pi`	$\pi$	The constant pi (3.14...)
`Infinity`	$\infty$	Positive infinity
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("9b7d8c"),
    Formula(Equal(Pow(JacobiTheta(2, 0, tau), 2), Sum(Div(1, Cos(Mul(Mul(Pi, tau), Add(n, Div(1, 2))))), For(n, Neg(Infinity), Infinity)))),
    Variables(tau),
    Assumptions(Element(tau, HH)))

f8cd8f

\theta_{3}^{2}\!\left(0, \tau\right) = 1 + 2 \sum_{n=1}^{\infty} \frac{1}{\cos\!\left(\pi \tau n\right)}

Assumptions:

\tau \in \mathbb{H}

TeX:

\theta_{3}^{2}\!\left(0, \tau\right) = 1 + 2 \sum_{n=1}^{\infty} \frac{1}{\cos\!\left(\pi \tau n\right)}

\tau \in \mathbb{H}

Definitions:

Fungrim symbol	Notation	Short description
`Pow`	${a}^{b}$	Power
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	Jacobi theta function
`Sum`	$\sum_{n} f(n)$	Sum
`Cos`	$\cos(z)$	Cosine
`Pi`	$\pi$	The constant pi (3.14...)
`Infinity`	$\infty$	Positive infinity
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("f8cd8f"),
    Formula(Equal(Pow(JacobiTheta(3, 0, tau), 2), Add(1, Mul(2, Sum(Div(1, Cos(Mul(Mul(Pi, tau), n))), For(n, 1, Infinity)))))),
    Variables(tau),
    Assumptions(Element(tau, HH)))

7b3ac4

\theta_{3}^{2}\!\left(0, \tau\right) = \sum_{n=-\infty}^{\infty} \frac{1}{\cos\!\left(\pi \tau n\right)}

Assumptions:

\tau \in \mathbb{H}

TeX:

\theta_{3}^{2}\!\left(0, \tau\right) = \sum_{n=-\infty}^{\infty} \frac{1}{\cos\!\left(\pi \tau n\right)}

\tau \in \mathbb{H}

Definitions:

Fungrim symbol	Notation	Short description
`Pow`	${a}^{b}$	Power
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	Jacobi theta function
`Sum`	$\sum_{n} f(n)$	Sum
`Cos`	$\cos(z)$	Cosine
`Pi`	$\pi$	The constant pi (3.14...)
`Infinity`	$\infty$	Positive infinity
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("7b3ac4"),
    Formula(Equal(Pow(JacobiTheta(3, 0, tau), 2), Sum(Div(1, Cos(Mul(Mul(Pi, tau), n))), For(n, Neg(Infinity), Infinity)))),
    Variables(tau),
    Assumptions(Element(tau, HH)))

ab1c77

\theta_{4}^{2}\!\left(0, \tau\right) = \sum_{n=-\infty}^{\infty} \frac{1}{\cos\!\left(\pi \left(\tau + 1\right) n\right)}

Assumptions:

\tau \in \mathbb{H}

TeX:

\theta_{4}^{2}\!\left(0, \tau\right) = \sum_{n=-\infty}^{\infty} \frac{1}{\cos\!\left(\pi \left(\tau + 1\right) n\right)}

\tau \in \mathbb{H}

Definitions:

Fungrim symbol	Notation	Short description
`Pow`	${a}^{b}$	Power
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	Jacobi theta function
`Sum`	$\sum_{n} f(n)$	Sum
`Cos`	$\cos(z)$	Cosine
`Pi`	$\pi$	The constant pi (3.14...)
`Infinity`	$\infty$	Positive infinity
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("ab1c77"),
    Formula(Equal(Pow(JacobiTheta(4, 0, tau), 2), Sum(Div(1, Cos(Mul(Mul(Pi, Add(tau, 1)), n))), For(n, Neg(Infinity), Infinity)))),
    Variables(tau),
    Assumptions(Element(tau, HH)))

Fourier series

Trigonometric Fourier series

Exponential Fourier series

Pure exponential series

Fourier series for derivatives

Exponential Fourier series

Infinite products

Infinite q-products with trigonometric factors

Infinite q-products with exponential factors

Jacobi triple product

Trigonometric infinite products

Series for logarithmic derivatives

Lambert series with trigonometric factors

Reciprocal trigonometric series

Taylor series

Theta constants

Fourier series (q-series) with linear exponents

Infinite products for quotients

Lambert series

Reciprocal trigonometric series