Differential equations for Jacobi theta functions

This topic lists identities involving derivatives of Jacobi theta functions $\theta_{j}\!\left(z , \tau\right)$. See the topic Jacobi theta functions for other properties of these functions.

Fundamentals

Notation for argument derivatives

$\frac{d^{r}}{{d z}^{r}} \theta_{j}\!\left(z , \tau\right) = \theta^{(r)}_{j}\!\left(z , \tau\right)$

Conversion of parameter derivatives to argument derivatives

$\frac{d^{r}}{{d \tau}^{r}} \theta^{(s)}_{j}\!\left(z , \tau\right) = \frac{1}{{\left(4 \pi i\right)}^{r}} \theta^{(2 r + s)}_{j}\!\left(z , \tau\right)$

Heat equation

$\theta''_{j}\!\left(z , \tau\right) - 4 \pi i \frac{d}{d \tau}\, \theta_{j}\!\left(z , \tau\right) = 0$

Jacobi's differential equation

${\left(30 D_{1}^{3} - 15 D_{0} D_{1} D_{2} + D_{0}^{2} D_{3}\right)}^{2} + 32 {\left(D_{0} D_{2} - 3 D_{1}^{2}\right)}^{3} + {\pi}^{2} {\left(D_{0} D_{2} - 3 D_{1}^{2}\right)}^{2} D_{0}^{10} = 0\; \text{ where } D_{r} = \frac{d^{r}}{{d \tau}^{r}} \theta_{j}\!\left(0 , \tau\right)$

Relations at zero

$\theta^{(2 r)}_{1}\!\left(0 , \tau\right) = 0$
$\theta^{(2 r + 1)}_{2}\!\left(0 , \tau\right) = 0$
$\theta^{(2 r + 1)}_{3}\!\left(0 , \tau\right) = 0$
$\theta^{(2 r + 1)}_{4}\!\left(0 , \tau\right) = 0$
$\theta'_{1}\!\left(0 , \tau\right) = \pi \theta_{2}\!\left(0 , \tau\right) \theta_{3}\!\left(0 , \tau\right) \theta_{4}\!\left(0 , \tau\right)$
$\frac{\theta'''_{1}\!\left(0 , \tau\right)}{\theta'_{1}\!\left(0 , \tau\right)} = \frac{\theta''_{2}\!\left(0 , \tau\right)}{\theta_{2}\!\left(0 , \tau\right)} + \frac{\theta''_{3}\!\left(0 , \tau\right)}{\theta_{3}\!\left(0 , \tau\right)} + \frac{\theta''_{4}\!\left(0 , \tau\right)}{\theta_{4}\!\left(0 , \tau\right)}$
$\theta'_{1}\!\left(0 , \frac{\tau}{2}\right) \theta_{2}\!\left(0 , \frac{\tau}{2}\right) = 2 \theta'_{1}\!\left(0 , \tau\right) \theta_{4}\!\left(0 , \tau\right)$
$2 \theta'_{1}\!\left(0 , 2 \tau\right) \theta_{4}\!\left(0 , 2 \tau\right) = \theta'_{1}\!\left(0 , \tau\right) \theta_{2}\!\left(0 , \tau\right)$

Derivatives of ratios

$\frac{d}{d z}\, \frac{\theta_{1}\!\left(z , \tau\right)}{\theta_{2}\!\left(z , \tau\right)} = \pi \theta_{2}^{2}\!\left(0, \tau\right) \frac{\theta_{3}\!\left(z , \tau\right) \theta_{4}\!\left(z , \tau\right)}{\theta_{2}^{2}\!\left(z, \tau\right)}$
$\frac{d}{d z}\, \frac{\theta_{1}\!\left(z , \tau\right)}{\theta_{3}\!\left(z , \tau\right)} = \pi \theta_{3}^{2}\!\left(0, \tau\right) \frac{\theta_{2}\!\left(z , \tau\right) \theta_{4}\!\left(z , \tau\right)}{\theta_{3}^{2}\!\left(z, \tau\right)}$
$\frac{d}{d z}\, \frac{\theta_{1}\!\left(z , \tau\right)}{\theta_{4}\!\left(z , \tau\right)} = \pi \theta_{4}^{2}\!\left(0, \tau\right) \frac{\theta_{2}\!\left(z , \tau\right) \theta_{3}\!\left(z , \tau\right)}{\theta_{4}^{2}\!\left(z, \tau\right)}$
$\frac{d}{d z}\, \frac{\theta_{2}\!\left(z , \tau\right)}{\theta_{1}\!\left(z , \tau\right)} = -\pi \theta_{2}^{2}\!\left(0, \tau\right) \frac{\theta_{3}\!\left(z , \tau\right) \theta_{4}\!\left(z , \tau\right)}{\theta_{1}^{2}\!\left(z, \tau\right)}$
$\frac{d}{d z}\, \frac{\theta_{2}\!\left(z , \tau\right)}{\theta_{3}\!\left(z , \tau\right)} = -\pi \theta_{4}^{2}\!\left(0, \tau\right) \frac{\theta_{1}\!\left(z , \tau\right) \theta_{4}\!\left(z , \tau\right)}{\theta_{3}^{2}\!\left(z, \tau\right)}$
$\frac{d}{d z}\, \frac{\theta_{2}\!\left(z , \tau\right)}{\theta_{4}\!\left(z , \tau\right)} = -\pi \theta_{3}^{2}\!\left(0, \tau\right) \frac{\theta_{1}\!\left(z , \tau\right) \theta_{3}\!\left(z , \tau\right)}{\theta_{4}^{2}\!\left(z, \tau\right)}$
$\frac{d}{d z}\, \frac{\theta_{3}\!\left(z , \tau\right)}{\theta_{1}\!\left(z , \tau\right)} = -\pi \theta_{3}^{2}\!\left(0, \tau\right) \frac{\theta_{2}\!\left(z , \tau\right) \theta_{4}\!\left(z , \tau\right)}{\theta_{1}^{2}\!\left(z, \tau\right)}$
$\frac{d}{d z}\, \frac{\theta_{3}\!\left(z , \tau\right)}{\theta_{2}\!\left(z , \tau\right)} = \pi \theta_{4}^{2}\!\left(0, \tau\right) \frac{\theta_{1}\!\left(z , \tau\right) \theta_{4}\!\left(z , \tau\right)}{\theta_{2}^{2}\!\left(z, \tau\right)}$
$\frac{d}{d z}\, \frac{\theta_{3}\!\left(z , \tau\right)}{\theta_{4}\!\left(z , \tau\right)} = -\pi \theta_{2}^{2}\!\left(0, \tau\right) \frac{\theta_{1}\!\left(z , \tau\right) \theta_{2}\!\left(z , \tau\right)}{\theta_{4}^{2}\!\left(z, \tau\right)}$
$\frac{d}{d z}\, \frac{\theta_{4}\!\left(z , \tau\right)}{\theta_{1}\!\left(z , \tau\right)} = -\pi \theta_{4}^{2}\!\left(0, \tau\right) \frac{\theta_{2}\!\left(z , \tau\right) \theta_{3}\!\left(z , \tau\right)}{\theta_{1}^{2}\!\left(z, \tau\right)}$
$\frac{d}{d z}\, \frac{\theta_{4}\!\left(z , \tau\right)}{\theta_{2}\!\left(z , \tau\right)} = \pi \theta_{3}^{2}\!\left(0, \tau\right) \frac{\theta_{1}\!\left(z , \tau\right) \theta_{3}\!\left(z , \tau\right)}{\theta_{2}^{2}\!\left(z, \tau\right)}$
$\frac{d}{d z}\, \frac{\theta_{4}\!\left(z , \tau\right)}{\theta_{3}\!\left(z , \tau\right)} = \pi \theta_{2}^{2}\!\left(0, \tau\right) \frac{\theta_{1}\!\left(z , \tau\right) \theta_{2}\!\left(z , \tau\right)}{\theta_{3}^{2}\!\left(z, \tau\right)}$

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