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Fungrim entry: 0207dc

ζ ⁣(z,τ)=z3θ1 ⁣(0,τ)θ1 ⁣(0,τ)+θ1 ⁣(z,τ)θ1 ⁣(z,τ)\zeta\!\left(z, \tau\right) = -\frac{z}{3} \frac{\theta'''_{1}\!\left(0 , \tau\right)}{\theta'_{1}\!\left(0 , \tau\right)} + \frac{\theta'_{1}\!\left(z , \tau\right)}{\theta_{1}\!\left(z , \tau\right)}
Assumptions:zC  and  τH  and  zΛ(1,τ)z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H} \;\mathbin{\operatorname{and}}\; z \notin \Lambda_{(1, \tau)}
TeX:
\zeta\!\left(z, \tau\right) = -\frac{z}{3} \frac{\theta'''_{1}\!\left(0 , \tau\right)}{\theta'_{1}\!\left(0 , \tau\right)} + \frac{\theta'_{1}\!\left(z , \tau\right)}{\theta_{1}\!\left(z , \tau\right)}

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H} \;\mathbin{\operatorname{and}}\; z \notin \Lambda_{(1, \tau)}
Definitions:
Fungrim symbol Notation Short description
WeierstrassZetaζ ⁣(z,τ)\zeta\!\left(z, \tau\right) Weierstrass zeta function
JacobiThetaθj ⁣(z,τ)\theta_{j}\!\left(z , \tau\right) Jacobi theta function
CCC\mathbb{C} Complex numbers
HHH\mathbb{H} Upper complex half-plane
LatticeΛ(a,b)\Lambda_{(a, b)} Complex lattice with periods a, b
Source code for this entry:
Entry(ID("0207dc"),
    Formula(Equal(WeierstrassZeta(z, tau), Add(Mul(Neg(Div(z, 3)), Div(JacobiTheta(1, 0, tau, 3), JacobiTheta(1, 0, tau, 1))), Div(JacobiTheta(1, z, tau, 1), JacobiTheta(1, z, tau))))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH), NotElement(z, Lattice(1, tau)))))

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2021-03-15 19:12:00.328586 UTC