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Differential equations for Jacobi theta functions

Table of contents: Fundamentals - Heat equation - Jacobi's differential equation - Relations at zero - Derivatives of ratios

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

Fundamentals

Notation for argument derivatives

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drdzrθj ⁣(z,τ)=θj(r) ⁣(z,τ)\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

37e644
drdτrθj(s) ⁣(z,τ)=1(4πi)rθj(2r+s) ⁣(z,τ)\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

ebc673
θj ⁣(z,τ)4πiddτθj ⁣(z,τ)=0\theta''_{j}\!\left(z , \tau\right) - 4 \pi i \frac{d}{d \tau}\, \theta_{j}\!\left(z , \tau\right) = 0

Jacobi's differential equation

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(30D1315D0D1D2+D02D3)2+32(D0D23D12)3+π2(D0D23D12)2D010=0   where Dr=drdτrθj ⁣(0,τ){\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

d967af
θ1(2r) ⁣(0,τ)=0\theta^{(2 r)}_{1}\!\left(0 , \tau\right) = 0
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θ2(2r+1) ⁣(0,τ)=0\theta^{(2 r + 1)}_{2}\!\left(0 , \tau\right) = 0
a19141
θ3(2r+1) ⁣(0,τ)=0\theta^{(2 r + 1)}_{3}\!\left(0 , \tau\right) = 0
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θ4(2r+1) ⁣(0,τ)=0\theta^{(2 r + 1)}_{4}\!\left(0 , \tau\right) = 0
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θ1 ⁣(0,τ)=πθ2 ⁣(0,τ)θ3 ⁣(0,τ)θ4 ⁣(0,τ)\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)
278274
θ1 ⁣(0,τ)θ1 ⁣(0,τ)=θ2 ⁣(0,τ)θ2 ⁣(0,τ)+θ3 ⁣(0,τ)θ3 ⁣(0,τ)+θ4 ⁣(0,τ)θ4 ⁣(0,τ)\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)}
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θ1 ⁣(0,τ2)θ2 ⁣(0,τ2)=2θ1 ⁣(0,τ)θ4 ⁣(0,τ)\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)
46f244
2θ1 ⁣(0,2τ)θ4 ⁣(0,2τ)=θ1 ⁣(0,τ)θ2 ⁣(0,τ)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

cb493d
ddzθ1 ⁣(z,τ)θ2 ⁣(z,τ)=πθ22 ⁣(0,τ)θ3 ⁣(z,τ)θ4 ⁣(z,τ)θ22 ⁣(z,τ)\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)}
d41a95
ddzθ1 ⁣(z,τ)θ3 ⁣(z,τ)=πθ32 ⁣(0,τ)θ2 ⁣(z,τ)θ4 ⁣(z,τ)θ32 ⁣(z,τ)\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)}
a4eecf
ddzθ1 ⁣(z,τ)θ4 ⁣(z,τ)=πθ42 ⁣(0,τ)θ2 ⁣(z,τ)θ3 ⁣(z,τ)θ42 ⁣(z,τ)\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)}
713b6b
ddzθ2 ⁣(z,τ)θ1 ⁣(z,τ)=πθ22 ⁣(0,τ)θ3 ⁣(z,τ)θ4 ⁣(z,τ)θ12 ⁣(z,τ)\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)}
64b65d
ddzθ2 ⁣(z,τ)θ3 ⁣(z,τ)=πθ42 ⁣(0,τ)θ1 ⁣(z,τ)θ4 ⁣(z,τ)θ32 ⁣(z,τ)\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)}
89985a
ddzθ2 ⁣(z,τ)θ4 ⁣(z,τ)=πθ32 ⁣(0,τ)θ1 ⁣(z,τ)θ3 ⁣(z,τ)θ42 ⁣(z,τ)\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)}
0373dc
ddzθ3 ⁣(z,τ)θ1 ⁣(z,τ)=πθ32 ⁣(0,τ)θ2 ⁣(z,τ)θ4 ⁣(z,τ)θ12 ⁣(z,τ)\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)}
2853d4
ddzθ3 ⁣(z,τ)θ2 ⁣(z,τ)=πθ42 ⁣(0,τ)θ1 ⁣(z,τ)θ4 ⁣(z,τ)θ22 ⁣(z,τ)\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)}
378949
ddzθ3 ⁣(z,τ)θ4 ⁣(z,τ)=πθ22 ⁣(0,τ)θ1 ⁣(z,τ)θ2 ⁣(z,τ)θ42 ⁣(z,τ)\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)}
a0552b
ddzθ4 ⁣(z,τ)θ1 ⁣(z,τ)=πθ42 ⁣(0,τ)θ2 ⁣(z,τ)θ3 ⁣(z,τ)θ12 ⁣(z,τ)\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)}
775637
ddzθ4 ⁣(z,τ)θ2 ⁣(z,τ)=πθ32 ⁣(0,τ)θ1 ⁣(z,τ)θ3 ⁣(z,τ)θ22 ⁣(z,τ)\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)}
23077c
ddzθ4 ⁣(z,τ)θ3 ⁣(z,τ)=πθ22 ⁣(0,τ)θ1 ⁣(z,τ)θ2 ⁣(z,τ)θ32 ⁣(z,τ)\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.

2019-09-20 18:07:53.062439 UTC