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Fungrim entry: 4d8b0f

θj ⁣(z,aτ+bcτ+d)=εj ⁣(a,b,c,d)vieπicvz2θSj ⁣(a,b,c,d) ⁣(vz,τ)   where v=cτ+d\theta_{j}\!\left(z , \frac{a \tau + b}{c \tau + d}\right) = \varepsilon_{j}\!\left(a, b, c, d\right) \sqrt{\frac{v}{i}} {e}^{\pi i c v {z}^{2}} \theta_{S_{j}\!\left(a, b, c, d\right)}\!\left(v z , \tau\right)\; \text{ where } v = c \tau + d
Assumptions:zCandτHand(abcd)PSL2(Z)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 , \frac{a \tau + b}{c \tau + d}\right) = \varepsilon_{j}\!\left(a, b, c, d\right) \sqrt{\frac{v}{i}} {e}^{\pi i c v {z}^{2}} \theta_{S_{j}\!\left(a, b, c, d\right)}\!\left(v z , \tau\right)\; \text{ where } v = c \tau + d

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("4d8b0f"),
    Formula(Equal(JacobiTheta(j, z, Div(Add(Mul(a, tau), b), Add(Mul(c, tau), d))), Where(Mul(Mul(Mul(JacobiThetaEpsilon(j, a, b, c, d), Sqrt(Div(v, ConstI))), Exp(Mul(Mul(Mul(Mul(Pi, ConstI), c), v), Pow(z, 2)))), JacobiTheta(JacobiThetaPermutation(j, a, b, c, d), Mul(v, z), tau)), Equal(v, Add(Mul(c, tau), d))))),
    Variables(z, tau, a, b, c, d),
    Assumptions(And(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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Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2019-12-11 23:01:54.699850 UTC