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Fungrim entry: 100d3c

θj ⁣(z,τ)=εj ⁣(d,b,c,a)vieπicvz2θSj ⁣(d,b,c,a) ⁣(vz,aτ+bcτ+d)   where v=1cτ+d\theta_{j}\!\left(z , \tau\right) = \varepsilon_{j}\!\left(-d, b, c, -a\right) \sqrt{\frac{v}{i}} {e}^{\pi i c v {z}^{2}} \theta_{S_{j}\!\left(-d, b, c, -a\right)}\!\left(v z , \frac{a \tau + b}{c \tau + d}\right)\; \text{ where } v = -\frac{1}{c \tau + d}
Assumptions:j{1,2,3,4}  and  zC  and  τH  and  (abcd)PSL2(Z)j \in \left\{1, 2, 3, 4\right\} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H} \;\mathbin{\operatorname{and}}\; \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \operatorname{PSL}_2(\mathbb{Z})
References:
  • Hans Rademacher (1973), Topics in Analytic Number Theory, Springer. Sections 80, 81.
TeX:
\theta_{j}\!\left(z , \tau\right) = \varepsilon_{j}\!\left(-d, b, c, -a\right) \sqrt{\frac{v}{i}} {e}^{\pi i c v {z}^{2}} \theta_{S_{j}\!\left(-d, b, c, -a\right)}\!\left(v z , \frac{a \tau + b}{c \tau + d}\right)\; \text{ where } v = -\frac{1}{c \tau + d}

j \in \left\{1, 2, 3, 4\right\} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H} \;\mathbin{\operatorname{and}}\; \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \operatorname{PSL}_2(\mathbb{Z})
Definitions:
Fungrim symbol Notation Short description
JacobiThetaθj ⁣(z,τ)\theta_{j}\!\left(z , \tau\right) Jacobi theta function
JacobiThetaEpsilonεj ⁣(a,b,c,d)\varepsilon_{j}\!\left(a, b, c, d\right) Root of unity in modular transformation of Jacobi theta functions
Sqrtz\sqrt{z} Principal square root
ConstIii Imaginary unit
Expez{e}^{z} Exponential function
Piπ\pi The constant pi (3.14...)
Powab{a}^{b} Power
JacobiThetaPermutationSj ⁣(a,b,c,d)S_{j}\!\left(a, b, c, d\right) Index permutation in modular transformation of Jacobi theta functions
CCC\mathbb{C} Complex numbers
HHH\mathbb{H} Upper complex half-plane
Matrix2x2(abcd)\begin{pmatrix} a & b \\ c & d \end{pmatrix} Two by two matrix
PSL2ZPSL2(Z)\operatorname{PSL}_2(\mathbb{Z}) Modular group (canonical representatives)
Source code for this entry:
Entry(ID("100d3c"),
    Formula(Equal(JacobiTheta(j, z, tau), Where(Mul(Mul(Mul(JacobiThetaEpsilon(j, Neg(d), b, c, Neg(a)), Sqrt(Div(v, ConstI))), Exp(Mul(Mul(Mul(Mul(Pi, ConstI), c), v), Pow(z, 2)))), JacobiTheta(JacobiThetaPermutation(j, Neg(d), b, c, Neg(a)), Mul(v, z), Div(Add(Mul(a, tau), b), Add(Mul(c, tau), d)))), Equal(v, Neg(Div(1, Add(Mul(c, tau), d))))))),
    Variables(j, z, tau, a, b, c, d),
    Assumptions(And(Element(j, Set(1, 2, 3, 4)), Element(z, CC), Element(tau, HH), Element(Matrix2x2(a, b, c, d), PSL2Z))),
    References("Hans Rademacher (1973), Topics in Analytic Number Theory, Springer. Sections 80, 81."))

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2021-03-15 19:12:00.328586 UTC