Fungrim entry: 03356b

\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:

\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
`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("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."))

Topics using this entry

Lattice transformations for Jacobi theta functions