Fungrim entry: 100d3c

\theta_{j}\!\left(z , \tau\right) = \varepsilon_{j}\!\left(-d, b, c, -a\right) \sqrt{\frac{v}{i}} {e}^{\pi i c v {z}^{2}} \theta_{S_{j}\!\left(-d, b, c, -a\right)}\!\left(v z , \frac{a \tau + b}{c \tau + d}\right)\; \text{ where } v = -\frac{1}{c \tau + d}

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}}\; \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \operatorname{PSL}_2(\mathbb{Z})

References:

Hans Rademacher (1973), Topics in Analytic Number Theory, Springer. Sections 80, 81.

TeX:

\theta_{j}\!\left(z , \tau\right) = \varepsilon_{j}\!\left(-d, b, c, -a\right) \sqrt{\frac{v}{i}} {e}^{\pi i c v {z}^{2}} \theta_{S_{j}\!\left(-d, b, c, -a\right)}\!\left(v z , \frac{a \tau + b}{c \tau + d}\right)\; \text{ where } v = -\frac{1}{c \tau + d}

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}}\; \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \operatorname{PSL}_2(\mathbb{Z})

Definitions:

Fungrim symbol	Notation	Short description
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	Jacobi theta function
`JacobiThetaEpsilon`	$\varepsilon_{j}\!\left(a, b, c, d\right)$	Root of unity in modular transformation of Jacobi theta functions
`Sqrt`	$\sqrt{z}$	Principal square root
`ConstI`	$i$	Imaginary unit
`Exp`	${e}^{z}$	Exponential function
`Pi`	$\pi$	The constant pi (3.14...)
`Pow`	${a}^{b}$	Power
`JacobiThetaPermutation`	$S_{j}\!\left(a, b, c, d\right)$	Index permutation in modular transformation of Jacobi theta functions
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane
`Matrix2x2`	$\begin{pmatrix} a & b \\ c & d \end{pmatrix}$	Two by two matrix
`PSL2Z`	$\operatorname{PSL}_2(\mathbb{Z})$	Modular group (canonical representatives)

Source code for this entry:

Entry(ID("100d3c"),
    Formula(Equal(JacobiTheta(j, z, tau), Where(Mul(Mul(Mul(JacobiThetaEpsilon(j, Neg(d), b, c, Neg(a)), Sqrt(Div(v, ConstI))), Exp(Mul(Mul(Mul(Mul(Pi, ConstI), c), v), Pow(z, 2)))), JacobiTheta(JacobiThetaPermutation(j, Neg(d), b, c, Neg(a)), Mul(v, z), Div(Add(Mul(a, tau), b), Add(Mul(c, tau), d)))), Equal(v, Neg(Div(1, Add(Mul(c, tau), d))))))),
    Variables(j, z, tau, a, b, c, d),
    Assumptions(And(Element(j, Set(1, 2, 3, 4)), Element(z, CC), Element(tau, HH), Element(Matrix2x2(a, b, c, d), PSL2Z))),
    References("Hans Rademacher (1973), Topics in Analytic Number Theory, Springer. Sections 80, 81."))

Topics using this entry

Lattice transformations for Jacobi theta functions