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Fungrim entry: 8c4ab4

θ4 ⁣(n4,i)={θ4 ⁣(0,i),n0(mod4)θ3 ⁣(0,i),n2(mod4)[27/16(2+1)1/4]θ3 ⁣(0,i),otherwise\theta_{4}\!\left(\frac{n}{4} , i\right) = \begin{cases} \theta_{4}\!\left(0 , i\right), & n \equiv 0 \pmod {4}\\\theta_{3}\!\left(0 , i\right), & n \equiv 2 \pmod {4}\\\left[{2}^{-7 / 16} {\left(\sqrt{2} + 1\right)}^{1 / 4}\right] \theta_{3}\!\left(0 , i\right), & \text{otherwise}\\ \end{cases}
Assumptions:nZn \in \mathbb{Z}
TeX:
\theta_{4}\!\left(\frac{n}{4} , i\right) = \begin{cases} \theta_{4}\!\left(0 , i\right), & n \equiv 0 \pmod {4}\\\theta_{3}\!\left(0 , i\right), & n \equiv 2 \pmod {4}\\\left[{2}^{-7 / 16} {\left(\sqrt{2} + 1\right)}^{1 / 4}\right] \theta_{3}\!\left(0 , i\right), & \text{otherwise}\\ \end{cases}

n \in \mathbb{Z}
Definitions:
Fungrim symbol Notation Short description
JacobiThetaθj ⁣(z,τ)\theta_{j}\!\left(z , \tau\right) Jacobi theta function
ConstIii Imaginary unit
Powab{a}^{b} Power
Sqrtz\sqrt{z} Principal square root
ZZZ\mathbb{Z} Integers
Source code for this entry:
Entry(ID("8c4ab4"),
    Formula(Equal(JacobiTheta(4, Div(n, 4), ConstI), Cases(Tuple(JacobiTheta(4, 0, ConstI), CongruentMod(n, 0, 4)), Tuple(JacobiTheta(3, 0, ConstI), CongruentMod(n, 2, 4)), Tuple(Mul(Brackets(Mul(Pow(2, Neg(Div(7, 16))), Pow(Add(Sqrt(2), 1), Div(1, 4)))), JacobiTheta(3, 0, ConstI)), Otherwise)))),
    Variables(n),
    Assumptions(Element(n, ZZ)))

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Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2019-09-20 18:07:53.062439 UTC