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Fungrim entry: 03356b

ε1 ⁣(a,b,c,d)={(cd)exp ⁣(πi4[d(bc1)+2]),c even(dc)exp ⁣(πi4[c(a+d+1)3]),c odd\varepsilon_{1}\!\left(a, b, c, d\right) = \begin{cases} \left( \frac{c}{d} \right) \exp\!\left(\frac{\pi i}{4} \left[d \left(b - c - 1\right) + 2\right]\right), & c \text{ even}\\\left( \frac{d}{c} \right) \exp\!\left(\frac{\pi i}{4} \left[c \left(a + d + 1\right) - 3\right]\right), & c \text{ odd}\\ \end{cases}
Assumptions:(abcd)SL2(Z)\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 80.
TeX:
\varepsilon_{1}\!\left(a, b, c, d\right) = \begin{cases} \left( \frac{c}{d} \right) \exp\!\left(\frac{\pi i}{4} \left[d \left(b - c - 1\right) + 2\right]\right), & c \text{ even}\\\left( \frac{d}{c} \right) \exp\!\left(\frac{\pi i}{4} \left[c \left(a + d + 1\right) - 3\right]\right), & c \text{ odd}\\ \end{cases}

\begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \operatorname{SL}_2(\mathbb{Z})
Definitions:
Fungrim symbol Notation Short description
JacobiThetaEpsilonεj ⁣(a,b,c,d)\varepsilon_{j}\!\left(a, b, c, d\right) Root of unity in modular transformation of Jacobi theta functions
Expez{e}^{z} Exponential function
Piπ\pi The constant pi (3.14...)
ConstIii Imaginary unit
Matrix2x2(abcd)\begin{pmatrix} a & b \\ c & d \end{pmatrix} Two by two matrix
SL2ZSL2(Z)\operatorname{SL}_2(\mathbb{Z}) Modular group
Source code for this entry:
Entry(ID("03356b"),
    Formula(Equal(JacobiThetaEpsilon(1, a, b, c, d), Cases(Tuple(Mul(KroneckerSymbol(c, d), Call(Exp, Mul(Div(Mul(Pi, ConstI), 4), Brackets(Add(Mul(d, Sub(Sub(b, c), 1)), 2))))), Even(c)), Tuple(Mul(KroneckerSymbol(d, c), Call(Exp, Mul(Div(Mul(Pi, ConstI), 4), Brackets(Sub(Mul(c, Add(Add(a, d), 1)), 3))))), Odd(c))))),
    Variables(a, b, c, d),
    Assumptions(Element(Matrix2x2(a, b, c, d), SL2Z)),
    References("Hans Rademacher (1973), Topics in Analytic Number Theory, Springer. Section 80."))

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2021-03-15 19:12:00.328586 UTC