# 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)), 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)), 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)), delta_(Add(b, 1))))))), Equal(j, 4)))), Def(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."))

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

2021-03-15 19:12:00.328586 UTC