Fungrim entry: 3c56c7

$\varepsilon_{j}\!\left(a, b, c, d\right) = \frac{1}{\varepsilon_{1}\!\left(-d, b, c, -a\right)} \begin{cases} \exp\!\left(\frac{\pi i}{4} \left[\left(c - 2\right) d - 2 + 2 \left(1 - c\right) {\delta}_{d + 1}\right]\right), & j = 2\\\exp\!\left(\frac{\pi i}{4} \left[\left(a + c - 2\right) \left(b + d\right) - 3 + 2 \left(1 - a - c\right) {\delta}_{b + d + 1}\right]\right), & j = 3\\\exp\!\left(\frac{\pi i}{4} \left[\left(a - 2\right) b - 4 + 2 \left(1 - a\right) {\delta}_{b + 1}\right]\right), & j = 4\\ \end{cases}\; \text{ where } {\delta}_{n} = n \bmod 2$
Assumptions:$j \in \left\{2, 3, 4\right\} \,\mathbin{\operatorname{and}}\, \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \operatorname{SL}_2(\mathbb{Z})$
References:
• Hans Rademacher (1973), Topics in Analytic Number Theory, Springer. Section 81.
TeX:
\varepsilon_{j}\!\left(a, b, c, d\right) = \frac{1}{\varepsilon_{1}\!\left(-d, b, c, -a\right)} \begin{cases} \exp\!\left(\frac{\pi i}{4} \left[\left(c - 2\right) d - 2 + 2 \left(1 - c\right) {\delta}_{d + 1}\right]\right), & j = 2\\\exp\!\left(\frac{\pi i}{4} \left[\left(a + c - 2\right) \left(b + d\right) - 3 + 2 \left(1 - a - c\right) {\delta}_{b + d + 1}\right]\right), & j = 3\\\exp\!\left(\frac{\pi i}{4} \left[\left(a - 2\right) b - 4 + 2 \left(1 - a\right) {\delta}_{b + 1}\right]\right), & j = 4\\ \end{cases}\; \text{ where } {\delta}_{n} = n \bmod 2

j \in \left\{2, 3, 4\right\} \,\mathbin{\operatorname{and}}\, \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \operatorname{SL}_2(\mathbb{Z})
Definitions:
Fungrim symbol Notation Short description
JacobiThetaEpsilon$\varepsilon_{j}\!\left(a, b, c, d\right)$ Root of unity in modular transformation of Jacobi theta functions
Exp${e}^{z}$ Exponential function
Pi$\pi$ The constant pi (3.14...)
ConstI$i$ Imaginary unit
Matrix2x2$\begin{pmatrix} a & b \\ c & d \end{pmatrix}$ Two by two matrix
SL2Z$\operatorname{SL}_2(\mathbb{Z})$ Modular group
Source code for this entry:
Entry(ID("3c56c7"),
Formula(Equal(JacobiThetaEpsilon(j, a, b, c, d), Where(Mul(Div(1, JacobiThetaEpsilon(1, Neg(d), b, c, Neg(a))), Cases(Tuple(Call(Exp, Mul(Div(Mul(Pi, ConstI), 4), Brackets(Add(Sub(Mul(Sub(c, 2), d), 2), Mul(Mul(2, Sub(1, c)), Subscript(delta, Add(d, 1))))))), Equal(j, 2)), Tuple(Call(Exp, Mul(Div(Mul(Pi, ConstI), 4), Brackets(Add(Sub(Mul(Sub(Add(a, c), 2), Add(b, d)), 3), Mul(Mul(2, Sub(Sub(1, a), c)), Subscript(delta, Add(Add(b, d), 1))))))), Equal(j, 3)), Tuple(Call(Exp, Mul(Div(Mul(Pi, ConstI), 4), Brackets(Add(Sub(Mul(Sub(a, 2), b), 4), Mul(Mul(2, Sub(1, a)), Subscript(delta, Add(b, 1))))))), Equal(j, 4)))), Equal(Subscript(delta, n), Mod(n, 2))))),
Variables(j, a, b, c, d),
Assumptions(And(Element(j, Set(2, 3, 4)), Element(Matrix2x2(a, b, c, d), SL2Z))),
References("Hans Rademacher (1973), Topics in Analytic Number Theory, Springer. Section 81."))

Topics using this entry

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

2020-01-31 18:09:28.494564 UTC