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Fungrim entry: b96c9d

σ ⁣(z,τ)=exp ⁣(z26[d3dz3θ1 ⁣(z,τ)]z=0[ddzθ1 ⁣(z,τ)]z=0)θ1 ⁣(z,τ)[ddzθ1 ⁣(z,τ)]z=0\sigma\!\left(z, \tau\right) = \exp\!\left(-\frac{{z}^{2}}{6} \frac{\left[ \frac{d^{3}}{{d z}^{3}} \theta_1\!\left(z, \tau\right) \right]_{z = 0}}{\left[ \frac{d}{d z}\, \theta_1\!\left(z, \tau\right) \right]_{z = 0}}\right) \frac{\theta_1\!\left(z, \tau\right)}{\left[ \frac{d}{d z}\, \theta_1\!\left(z, \tau\right) \right]_{z = 0}}
Assumptions:zCandτHandzΛ(1,τ)z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \tau \in \mathbb{H} \,\mathbin{\operatorname{and}}\, z \notin \Lambda_{(1, \tau)}
TeX:
\sigma\!\left(z, \tau\right) = \exp\!\left(-\frac{{z}^{2}}{6} \frac{\left[ \frac{d^{3}}{{d z}^{3}} \theta_1\!\left(z, \tau\right) \right]_{z = 0}}{\left[ \frac{d}{d z}\, \theta_1\!\left(z, \tau\right) \right]_{z = 0}}\right) \frac{\theta_1\!\left(z, \tau\right)}{\left[ \frac{d}{d z}\, \theta_1\!\left(z, \tau\right) \right]_{z = 0}}

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
WeierstrassSigmaσ ⁣(z,τ)\sigma\!\left(z, \tau\right) Weierstrass sigma function
Expez{e}^{z} Exponential function
Powab{a}^{b} Power
Derivativeddzf ⁣(z)\frac{d}{d z}\, f\!\left(z\right) Derivative
JacobiTheta1θ1 ⁣(z,τ)\theta_1\!\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("b96c9d"),
    Formula(Equal(WeierstrassSigma(z, tau), Mul(Exp(Mul(Neg(Div(Pow(z, 2), 6)), Div(Derivative(JacobiTheta1(z, tau), Tuple(z, 0, 3)), Derivative(JacobiTheta1(z, tau), Tuple(z, 0, 1))))), Div(JacobiTheta1(z, tau), Derivative(JacobiTheta1(z, tau), Tuple(z, 0, 1)))))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH), NotElement(z, Lattice(1, tau)))))

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

2019-06-18 07:49:59.356594 UTC