Fungrim entry: 5f9e54

$\theta_{1}\!\left(z , \tau\right) \theta_{2}\!\left(w , \tau\right) = \theta_{1}\!\left(z + w , 2 \tau\right) \theta_{4}\!\left(z - w , 2 \tau\right) + \theta_{4}\!\left(z + w , 2 \tau\right) \theta_{1}\!\left(z - w , 2 \tau\right)$
Assumptions:$z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; w \in \tau \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H}$
TeX:
\theta_{1}\!\left(z , \tau\right) \theta_{2}\!\left(w , \tau\right) = \theta_{1}\!\left(z + w , 2 \tau\right) \theta_{4}\!\left(z - w , 2 \tau\right) + \theta_{4}\!\left(z + w , 2 \tau\right) \theta_{1}\!\left(z - w , 2 \tau\right)

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; w \in \tau \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H}
Definitions:
Fungrim symbol Notation Short description
JacobiTheta$\theta_{j}\!\left(z , \tau\right)$ Jacobi theta function
CC$\mathbb{C}$ Complex numbers
HH$\mathbb{H}$ Upper complex half-plane
Source code for this entry:
Entry(ID("5f9e54"),
Formula(Equal(Mul(JacobiTheta(1, z, tau), JacobiTheta(2, w, tau)), Add(Mul(JacobiTheta(1, Add(z, w), Mul(2, tau)), JacobiTheta(4, Sub(z, w), Mul(2, tau))), Mul(JacobiTheta(4, Add(z, w), Mul(2, tau)), JacobiTheta(1, Sub(z, w), Mul(2, tau)))))),
Variables(z, w, tau),
Assumptions(And(Element(z, CC), Element(w, tau), Element(tau, HH))))

Topics using this entry

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

2021-03-15 19:12:00.328586 UTC