Fungrim entry: 0207dc

\zeta\!\left(z, \tau\right) = -\frac{z}{3} \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}} + \frac{\frac{d}{d z}\, \theta_1\!\left(z, \tau\right)}{\theta_1\!\left(z, \tau\right)}

Assumptions:

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{\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}} + \frac{\frac{d}{d z}\, \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
`WeierstrassZeta`	$\zeta\!\left(z, \tau\right)$	Weierstrass zeta function
`Derivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Derivative
`JacobiTheta1`	$\theta_1\!\left(z, \tau\right)$	Jacobi theta function
`CC`	$\mathbb{C}$	Complex numbers
`HH`	$\mathbb{H}$	Upper complex half-plane
`Lattice`	$\Lambda_{(a, b)}$	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(Derivative(JacobiTheta1(z, tau), Tuple(z, 0, 3)), Derivative(JacobiTheta1(z, tau), Tuple(z, 0, 1)))), Div(Derivative(JacobiTheta1(z, tau), Tuple(z, z, 1)), JacobiTheta1(z, tau))))),
    Variables(z, tau),
    Assumptions(And(Element(z, CC), Element(tau, HH), NotElement(z, Lattice(1, tau)))))

Topics using this entry

Weierstrass elliptic functions