# 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:$z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \tau \in \mathbb{H} \,\mathbin{\operatorname{and}}\, \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \operatorname{PSL}_2(\mathbb{Z})$
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}

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(z, tau, a, b, c, d),
Assumptions(And(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

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2019-12-11 23:01:54.699850 UTC