Differential equations for Jacobi theta functions

a222ed

\frac{d^{r}}{{d z}^{r}} \theta_{j}\!\left(z , \tau\right) = \theta^{(r)}_{j}\!\left(z , \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} \;\mathbin{\operatorname{and}}\; r \in \mathbb{Z}_{\ge 0}

TeX:

\frac{d^{r}}{{d z}^{r}} \theta_{j}\!\left(z , \tau\right) = \theta^{(r)}_{j}\!\left(z , \tau\right)

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}}\; r \in \mathbb{Z}_{\ge 0}

Definitions:

Fungrim symbol	Notation	Short description
`ComplexDerivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Complex derivative
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	Jacobi theta 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("a222ed"),
    Formula(Equal(ComplexDerivative(JacobiTheta(j, z, tau), For(z, z, r)), JacobiTheta(j, z, tau, r))),
    Variables(j, z, tau, r),
    Assumptions(And(Element(j, Set(1, 2, 3, 4)), Element(z, CC), Element(tau, HH), Element(r, ZZGreaterEqual(0)))))

37e644

\frac{d^{r}}{{d \tau}^{r}} \theta^{(s)}_{j}\!\left(z , \tau\right) = \frac{1}{{\left(4 \pi i\right)}^{r}} \theta^{(2 r + s)}_{j}\!\left(z , \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} \;\mathbin{\operatorname{and}}\; r \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; s \in \mathbb{Z}_{\ge 0}

TeX:

\frac{d^{r}}{{d \tau}^{r}} \theta^{(s)}_{j}\!\left(z , \tau\right) = \frac{1}{{\left(4 \pi i\right)}^{r}} \theta^{(2 r + s)}_{j}\!\left(z , \tau\right)

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}}\; r \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; s \in \mathbb{Z}_{\ge 0}

Definitions:

Fungrim symbol	Notation	Short description
`ComplexDerivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Complex derivative
`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
`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("37e644"),
    Formula(Equal(ComplexDerivative(JacobiTheta(j, z, tau, s), For(tau, tau, r)), Mul(Div(1, Pow(Mul(Mul(4, Pi), ConstI), r)), JacobiTheta(j, z, tau, Add(Mul(2, r), s))))),
    Variables(j, z, tau, r, s),
    Assumptions(And(Element(j, Set(1, 2, 3, 4)), Element(z, CC), Element(tau, HH), Element(r, ZZGreaterEqual(0)), Element(s, ZZGreaterEqual(0)))))

ebc673

\theta''_{j}\!\left(z , \tau\right) - 4 \pi i \frac{d}{d \tau}\, \theta_{j}\!\left(z , \tau\right) = 0

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(z , \tau\right) - 4 \pi i \frac{d}{d \tau}\, \theta_{j}\!\left(z , \tau\right) = 0

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
`Pi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit
`ComplexDerivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Complex derivative
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("ebc673"),
    Formula(Equal(Sub(JacobiTheta(j, z, tau, 2), Mul(Mul(Mul(4, Pi), ConstI), ComplexDerivative(JacobiTheta(j, z, tau), For(tau, tau)))), 0)),
    Variables(j, z, tau),
    Assumptions(And(Element(j, Set(1, 2, 3, 4)), Element(z, CC), Element(tau, HH))))

936694

{\left(30 D_{1}^{3} - 15 D_{0} D_{1} D_{2} + D_{0}^{2} D_{3}\right)}^{2} + 32 {\left(D_{0} D_{2} - 3 D_{1}^{2}\right)}^{3} + {\pi}^{2} {\left(D_{0} D_{2} - 3 D_{1}^{2}\right)}^{2} D_{0}^{10} = 0\; \text{ where } D_{r} = \frac{d^{r}}{{d \tau}^{r}} \theta_{j}\!\left(0 , \tau\right)

Assumptions:

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

TeX:

{\left(30 D_{1}^{3} - 15 D_{0} D_{1} D_{2} + D_{0}^{2} D_{3}\right)}^{2} + 32 {\left(D_{0} D_{2} - 3 D_{1}^{2}\right)}^{3} + {\pi}^{2} {\left(D_{0} D_{2} - 3 D_{1}^{2}\right)}^{2} D_{0}^{10} = 0\; \text{ where } D_{r} = \frac{d^{r}}{{d \tau}^{r}} \theta_{j}\!\left(0 , \tau\right)

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

Definitions:

Fungrim symbol	Notation	Short description
`Pow`	${a}^{b}$	Power
`Pi`	$\pi$	The constant pi (3.14...)
`ComplexDerivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Complex derivative
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	Jacobi theta function
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("936694"),
    Formula(Where(Equal(Add(Add(Pow(Add(Sub(Mul(30, Pow(D_(1), 3)), Mul(Mul(Mul(15, D_(0)), D_(1)), D_(2))), Mul(Pow(D_(0), 2), D_(3))), 2), Mul(32, Pow(Sub(Mul(D_(0), D_(2)), Mul(3, Pow(D_(1), 2))), 3))), Mul(Mul(Pow(Pi, 2), Pow(Sub(Mul(D_(0), D_(2)), Mul(3, Pow(D_(1), 2))), 2)), Pow(D_(0), 10))), 0), Def(D_(r), ComplexDerivative(JacobiTheta(j, 0, tau), For(tau, tau, r))))),
    Variables(j, tau),
    Assumptions(And(Element(j, Set(1, 2, 3, 4)), Element(tau, HH))))

d967af

\theta^{(2 r)}_{1}\!\left(0 , \tau\right) = 0

Assumptions:

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

TeX:

\theta^{(2 r)}_{1}\!\left(0 , \tau\right) = 0

\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
`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("d967af"),
    Formula(Equal(JacobiTheta(1, 0, tau, Mul(2, r)), 0)),
    Variables(tau, r),
    Assumptions(And(Element(tau, HH), Element(r, ZZGreaterEqual(0)))))

cdbdc7

\theta^{(2 r + 1)}_{2}\!\left(0 , \tau\right) = 0

Assumptions:

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

TeX:

\theta^{(2 r + 1)}_{2}\!\left(0 , \tau\right) = 0

\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
`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("cdbdc7"),
    Formula(Equal(JacobiTheta(2, 0, tau, Add(Mul(2, r), 1)), 0)),
    Variables(tau, r),
    Assumptions(And(Element(tau, HH), Element(r, ZZGreaterEqual(0)))))

a19141

\theta^{(2 r + 1)}_{3}\!\left(0 , \tau\right) = 0

Assumptions:

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

TeX:

\theta^{(2 r + 1)}_{3}\!\left(0 , \tau\right) = 0

\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
`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("a19141"),
    Formula(Equal(JacobiTheta(3, 0, tau, Add(Mul(2, r), 1)), 0)),
    Variables(tau, r),
    Assumptions(And(Element(tau, HH), Element(r, ZZGreaterEqual(0)))))

901934

\theta^{(2 r + 1)}_{4}\!\left(0 , \tau\right) = 0

Assumptions:

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

TeX:

\theta^{(2 r + 1)}_{4}\!\left(0 , \tau\right) = 0

\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
`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("901934"),
    Formula(Equal(JacobiTheta(4, 0, tau, Add(Mul(2, r), 1)), 0)),
    Variables(tau, r),
    Assumptions(And(Element(tau, HH), Element(r, ZZGreaterEqual(0)))))

f2e28a

\theta'_{1}\!\left(0 , \tau\right) = \pi \theta_{2}\!\left(0 , \tau\right) \theta_{3}\!\left(0 , \tau\right) \theta_{4}\!\left(0 , \tau\right)

Assumptions:

\tau \in \mathbb{H}

TeX:

\theta'_{1}\!\left(0 , \tau\right) = \pi \theta_{2}\!\left(0 , \tau\right) \theta_{3}\!\left(0 , \tau\right) \theta_{4}\!\left(0 , \tau\right)

\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...)
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("f2e28a"),
    Formula(Equal(JacobiTheta(1, 0, tau, 1), Mul(Mul(Mul(Pi, JacobiTheta(2, 0, tau)), JacobiTheta(3, 0, tau)), JacobiTheta(4, 0, tau)))),
    Variables(tau),
    Assumptions(And(Element(tau, HH))))

278274

\frac{\theta'''_{1}\!\left(0 , \tau\right)}{\theta'_{1}\!\left(0 , \tau\right)} = \frac{\theta''_{2}\!\left(0 , \tau\right)}{\theta_{2}\!\left(0 , \tau\right)} + \frac{\theta''_{3}\!\left(0 , \tau\right)}{\theta_{3}\!\left(0 , \tau\right)} + \frac{\theta''_{4}\!\left(0 , \tau\right)}{\theta_{4}\!\left(0 , \tau\right)}

Assumptions:

\tau \in \mathbb{H}

TeX:

\frac{\theta'''_{1}\!\left(0 , \tau\right)}{\theta'_{1}\!\left(0 , \tau\right)} = \frac{\theta''_{2}\!\left(0 , \tau\right)}{\theta_{2}\!\left(0 , \tau\right)} + \frac{\theta''_{3}\!\left(0 , \tau\right)}{\theta_{3}\!\left(0 , \tau\right)} + \frac{\theta''_{4}\!\left(0 , \tau\right)}{\theta_{4}\!\left(0 , \tau\right)}

\tau \in \mathbb{H}

Definitions:

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

Source code for this entry:

Entry(ID("278274"),
    Formula(Equal(Div(JacobiTheta(1, 0, tau, 3), JacobiTheta(1, 0, tau, 1)), Add(Add(Div(JacobiTheta(2, 0, tau, 2), JacobiTheta(2, 0, tau)), Div(JacobiTheta(3, 0, tau, 2), JacobiTheta(3, 0, tau))), Div(JacobiTheta(4, 0, tau, 2), JacobiTheta(4, 0, tau))))),
    Variables(tau),
    Assumptions(And(Element(tau, HH))))

59184e

\theta'_{1}\!\left(0 , \frac{\tau}{2}\right) \theta_{2}\!\left(0 , \frac{\tau}{2}\right) = 2 \theta'_{1}\!\left(0 , \tau\right) \theta_{4}\!\left(0 , \tau\right)

Assumptions:

\tau \in \mathbb{H}

TeX:

\theta'_{1}\!\left(0 , \frac{\tau}{2}\right) \theta_{2}\!\left(0 , \frac{\tau}{2}\right) = 2 \theta'_{1}\!\left(0 , \tau\right) \theta_{4}\!\left(0 , \tau\right)

\tau \in \mathbb{H}

Definitions:

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

Source code for this entry:

Entry(ID("59184e"),
    Formula(Equal(Mul(JacobiTheta(1, 0, Div(tau, 2), 1), JacobiTheta(2, 0, Div(tau, 2))), Mul(Mul(2, JacobiTheta(1, 0, tau, 1)), JacobiTheta(4, 0, tau)))),
    Variables(tau),
    Assumptions(Element(tau, HH)))

46f244

2 \theta'_{1}\!\left(0 , 2 \tau\right) \theta_{4}\!\left(0 , 2 \tau\right) = \theta'_{1}\!\left(0 , \tau\right) \theta_{2}\!\left(0 , \tau\right)

Assumptions:

\tau \in \mathbb{H}

TeX:

2 \theta'_{1}\!\left(0 , 2 \tau\right) \theta_{4}\!\left(0 , 2 \tau\right) = \theta'_{1}\!\left(0 , \tau\right) \theta_{2}\!\left(0 , \tau\right)

\tau \in \mathbb{H}

Definitions:

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

Source code for this entry:

Entry(ID("46f244"),
    Formula(Equal(Mul(Mul(2, JacobiTheta(1, 0, Mul(2, tau), 1)), JacobiTheta(4, 0, Mul(2, tau))), Mul(JacobiTheta(1, 0, tau, 1), JacobiTheta(2, 0, tau)))),
    Variables(tau),
    Assumptions(Element(tau, HH)))

cb493d

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

Assumptions:

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

TeX:

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

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

Definitions:

Fungrim symbol	Notation	Short description
`MeromorphicDerivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Complex derivative, allowing poles
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	Jacobi theta function
`Pi`	$\pi$	The constant pi (3.14...)
`Pow`	${a}^{b}$	Power
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("cb493d"),
    Formula(Equal(MeromorphicDerivative(Div(JacobiTheta(1, z, tau), JacobiTheta(2, z, tau)), For(z, z)), Mul(Mul(Pi, Pow(JacobiTheta(2, 0, tau), 2)), Div(Mul(JacobiTheta(3, z, tau), JacobiTheta(4, z, tau)), Pow(JacobiTheta(2, z, tau), 2))))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH))))

d41a95

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

Assumptions:

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

TeX:

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

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

Definitions:

Fungrim symbol	Notation	Short description
`MeromorphicDerivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Complex derivative, allowing poles
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	Jacobi theta function
`Pi`	$\pi$	The constant pi (3.14...)
`Pow`	${a}^{b}$	Power
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("d41a95"),
    Formula(Equal(MeromorphicDerivative(Div(JacobiTheta(1, z, tau), JacobiTheta(3, z, tau)), For(z, z)), Mul(Mul(Pi, Pow(JacobiTheta(3, 0, tau), 2)), Div(Mul(JacobiTheta(2, z, tau), JacobiTheta(4, z, tau)), Pow(JacobiTheta(3, z, tau), 2))))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH))))

a4eecf

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

Assumptions:

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

TeX:

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

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

Definitions:

Fungrim symbol	Notation	Short description
`MeromorphicDerivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Complex derivative, allowing poles
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	Jacobi theta function
`Pi`	$\pi$	The constant pi (3.14...)
`Pow`	${a}^{b}$	Power
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("a4eecf"),
    Formula(Equal(MeromorphicDerivative(Div(JacobiTheta(1, z, tau), JacobiTheta(4, z, tau)), For(z, z)), Mul(Mul(Pi, Pow(JacobiTheta(4, 0, tau), 2)), Div(Mul(JacobiTheta(2, z, tau), JacobiTheta(3, z, tau)), Pow(JacobiTheta(4, z, tau), 2))))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH))))

713b6b

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

Assumptions:

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

TeX:

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

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

Definitions:

Fungrim symbol	Notation	Short description
`MeromorphicDerivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Complex derivative, allowing poles
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	Jacobi theta function
`Pi`	$\pi$	The constant pi (3.14...)
`Pow`	${a}^{b}$	Power
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("713b6b"),
    Formula(Equal(MeromorphicDerivative(Div(JacobiTheta(2, z, tau), JacobiTheta(1, z, tau)), For(z, z)), Neg(Mul(Mul(Pi, Pow(JacobiTheta(2, 0, tau), 2)), Div(Mul(JacobiTheta(3, z, tau), JacobiTheta(4, z, tau)), Pow(JacobiTheta(1, z, tau), 2)))))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH))))

64b65d

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

Assumptions:

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

TeX:

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

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

Definitions:

Fungrim symbol	Notation	Short description
`MeromorphicDerivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Complex derivative, allowing poles
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	Jacobi theta function
`Pi`	$\pi$	The constant pi (3.14...)
`Pow`	${a}^{b}$	Power
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("64b65d"),
    Formula(Equal(MeromorphicDerivative(Div(JacobiTheta(2, z, tau), JacobiTheta(3, z, tau)), For(z, z)), Neg(Mul(Mul(Pi, Pow(JacobiTheta(4, 0, tau), 2)), Div(Mul(JacobiTheta(1, z, tau), JacobiTheta(4, z, tau)), Pow(JacobiTheta(3, z, tau), 2)))))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH))))

89985a

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

Assumptions:

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

TeX:

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

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

Definitions:

Fungrim symbol	Notation	Short description
`MeromorphicDerivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Complex derivative, allowing poles
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	Jacobi theta function
`Pi`	$\pi$	The constant pi (3.14...)
`Pow`	${a}^{b}$	Power
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("89985a"),
    Formula(Equal(MeromorphicDerivative(Div(JacobiTheta(2, z, tau), JacobiTheta(4, z, tau)), For(z, z)), Neg(Mul(Mul(Pi, Pow(JacobiTheta(3, 0, tau), 2)), Div(Mul(JacobiTheta(1, z, tau), JacobiTheta(3, z, tau)), Pow(JacobiTheta(4, z, tau), 2)))))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH))))

0373dc

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

Assumptions:

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

TeX:

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

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

Definitions:

Fungrim symbol	Notation	Short description
`MeromorphicDerivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Complex derivative, allowing poles
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	Jacobi theta function
`Pi`	$\pi$	The constant pi (3.14...)
`Pow`	${a}^{b}$	Power
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("0373dc"),
    Formula(Equal(MeromorphicDerivative(Div(JacobiTheta(3, z, tau), JacobiTheta(1, z, tau)), For(z, z)), Neg(Mul(Mul(Pi, Pow(JacobiTheta(3, 0, tau), 2)), Div(Mul(JacobiTheta(2, z, tau), JacobiTheta(4, z, tau)), Pow(JacobiTheta(1, z, tau), 2)))))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH))))

2853d4

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

Assumptions:

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

TeX:

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

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

Definitions:

Fungrim symbol	Notation	Short description
`MeromorphicDerivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Complex derivative, allowing poles
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	Jacobi theta function
`Pi`	$\pi$	The constant pi (3.14...)
`Pow`	${a}^{b}$	Power
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("2853d4"),
    Formula(Equal(MeromorphicDerivative(Div(JacobiTheta(3, z, tau), JacobiTheta(2, z, tau)), For(z, z)), Mul(Mul(Pi, Pow(JacobiTheta(4, 0, tau), 2)), Div(Mul(JacobiTheta(1, z, tau), JacobiTheta(4, z, tau)), Pow(JacobiTheta(2, z, tau), 2))))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH))))

378949

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

Assumptions:

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

TeX:

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

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

Definitions:

Fungrim symbol	Notation	Short description
`MeromorphicDerivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Complex derivative, allowing poles
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	Jacobi theta function
`Pi`	$\pi$	The constant pi (3.14...)
`Pow`	${a}^{b}$	Power
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("378949"),
    Formula(Equal(MeromorphicDerivative(Div(JacobiTheta(3, z, tau), JacobiTheta(4, z, tau)), For(z, z)), Neg(Mul(Mul(Pi, Pow(JacobiTheta(2, 0, tau), 2)), Div(Mul(JacobiTheta(1, z, tau), JacobiTheta(2, z, tau)), Pow(JacobiTheta(4, z, tau), 2)))))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH))))

a0552b

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

Assumptions:

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

TeX:

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

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

Definitions:

Fungrim symbol	Notation	Short description
`MeromorphicDerivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Complex derivative, allowing poles
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	Jacobi theta function
`Pi`	$\pi$	The constant pi (3.14...)
`Pow`	${a}^{b}$	Power
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("a0552b"),
    Formula(Equal(MeromorphicDerivative(Div(JacobiTheta(4, z, tau), JacobiTheta(1, z, tau)), For(z, z)), Neg(Mul(Mul(Pi, Pow(JacobiTheta(4, 0, tau), 2)), Div(Mul(JacobiTheta(2, z, tau), JacobiTheta(3, z, tau)), Pow(JacobiTheta(1, z, tau), 2)))))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH))))

775637

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

Assumptions:

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

TeX:

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

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

Definitions:

Fungrim symbol	Notation	Short description
`MeromorphicDerivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Complex derivative, allowing poles
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	Jacobi theta function
`Pi`	$\pi$	The constant pi (3.14...)
`Pow`	${a}^{b}$	Power
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("775637"),
    Formula(Equal(MeromorphicDerivative(Div(JacobiTheta(4, z, tau), JacobiTheta(2, z, tau)), For(z, z)), Mul(Mul(Pi, Pow(JacobiTheta(3, 0, tau), 2)), Div(Mul(JacobiTheta(1, z, tau), JacobiTheta(3, z, tau)), Pow(JacobiTheta(2, z, tau), 2))))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH))))

23077c

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

Assumptions:

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

TeX:

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

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

Definitions:

Fungrim symbol	Notation	Short description
`MeromorphicDerivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Complex derivative, allowing poles
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	Jacobi theta function
`Pi`	$\pi$	The constant pi (3.14...)
`Pow`	${a}^{b}$	Power
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("23077c"),
    Formula(Equal(MeromorphicDerivative(Div(JacobiTheta(4, z, tau), JacobiTheta(3, z, tau)), For(z, z)), Mul(Mul(Pi, Pow(JacobiTheta(2, 0, tau), 2)), Div(Mul(JacobiTheta(1, z, tau), JacobiTheta(2, z, tau)), Pow(JacobiTheta(3, z, tau), 2))))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH))))

Fundamentals

Notation for argument derivatives

Conversion of parameter derivatives to argument derivatives

Heat equation

Jacobi's differential equation

Relations at zero

Derivatives of ratios