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Fungrim entry: 816057

θ3 ⁣(z,τ)=θ3 ⁣(0,τ)n=1cos ⁣(π((n12)τ+z))cos ⁣(π((n12)τz))cos2 ⁣(π(n12)τ)\theta_{3}\!\left(z , \tau\right) = \theta_{3}\!\left(0 , \tau\right) \prod_{n=1}^{\infty} \frac{\cos\!\left(\pi \left(\left(n - \frac{1}{2}\right) \tau + z\right)\right) \cos\!\left(\pi \left(\left(n - \frac{1}{2}\right) \tau - z\right)\right)}{\cos^{2}\!\left(\pi \left(n - \frac{1}{2}\right) \tau\right)}
Assumptions:zC  and  τHz \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H}
\theta_{3}\!\left(z , \tau\right) = \theta_{3}\!\left(0 , \tau\right) \prod_{n=1}^{\infty} \frac{\cos\!\left(\pi \left(\left(n - \frac{1}{2}\right) \tau + z\right)\right) \cos\!\left(\pi \left(\left(n - \frac{1}{2}\right) \tau - z\right)\right)}{\cos^{2}\!\left(\pi \left(n - \frac{1}{2}\right) \tau\right)}

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H}
Fungrim symbol Notation Short description
JacobiThetaθj ⁣(z,τ)\theta_{j}\!\left(z , \tau\right) Jacobi theta function
Productnf(n)\prod_{n} f(n) Product
Coscos(z)\cos(z) Cosine
Piπ\pi The constant pi (3.14...)
Powab{a}^{b} Power
Infinity\infty Positive infinity
CCC\mathbb{C} Complex numbers
HHH\mathbb{H} Upper complex half-plane
Source code for this entry:
    Formula(Equal(JacobiTheta(3, z, tau), Mul(JacobiTheta(3, 0, tau), Product(Div(Mul(Cos(Mul(Pi, Add(Mul(Sub(n, Div(1, 2)), tau), z))), Cos(Mul(Pi, Sub(Mul(Sub(n, Div(1, 2)), tau), z)))), Pow(Cos(Mul(Pi, Mul(Sub(n, Div(1, 2)), tau))), 2)), For(n, 1, Infinity))))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH))))

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2021-03-15 19:12:00.328586 UTC