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Fungrim entry: 3c56c7

εj ⁣(a,b,c,d)=1ε1 ⁣(d,b,c,a){exp ⁣(πi4[(c2)d2+2(1c)δd+1]),j=2exp ⁣(πi4[(a+c2)(b+d)3+2(1ac)δb+d+1]),j=3exp ⁣(πi4[(a2)b4+2(1a)δb+1]),j=4   where δn=nmod2\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{2,3,4}and(abcd)SL2(Z)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
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("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."))

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2019-11-19 15:10:20.037976 UTC