Fungrim home page

Jacobi theta functions

Table of contents: Definitions - Illustrations - Series and product representations - Zeros - Specific values - Argument transformations - Lattice transformations - Sums and products - Differential equations - Integrals - Representation of other functions - Representation by other functions - Approximations

Definitions

f96eac
Symbol: JacobiTheta θj ⁣(z,τ)\theta_{j}\!\left(z , \tau\right) Jacobi theta function

Illustrations

Main topic: Illustrations of Jacobi theta functions

Variable argument

bac5fb
Image: X-ray of θ3 ⁣(z,i)\theta_{3}\!\left(z , i\right) on z[2,2]+[2,2]iz \in \left[-2, 2\right] + \left[-2, 2\right] i

Variable lattice parameter

e2035a
Image: X-ray of θ1 ⁣(13+34i,τ)\theta_{1}\!\left(\frac{1}{3} + \frac{3}{4} i , \tau\right) on τ[52,52]+[0,2]i\tau \in \left[-\frac{5}{2}, \frac{5}{2}\right] + \left[0, 2\right] i

Series and product representations

Main topic: Series and product representations of Jacobi theta functions

Trigonometric Fourier series

2ba423
θ1 ⁣(z,τ)=2eπiτ/4n=0(1)nqn(n+1)sin ⁣((2n+1)πz)   where q=eπiτ\theta_{1}\!\left(z , \tau\right) = 2 {e}^{\pi i \tau / 4} \sum_{n=0}^{\infty} {\left(-1\right)}^{n} {q}^{n \left(n + 1\right)} \sin\!\left(\left(2 n + 1\right) \pi z\right)\; \text{ where } q = {e}^{\pi i \tau}
06633e
θ2 ⁣(z,τ)=2eπiτ/4n=0qn(n+1)cos ⁣((2n+1)πz)   where q=eπiτ\theta_{2}\!\left(z , \tau\right) = 2 {e}^{\pi i \tau / 4} \sum_{n=0}^{\infty} {q}^{n \left(n + 1\right)} \cos\!\left(\left(2 n + 1\right) \pi z\right)\; \text{ where } q = {e}^{\pi i \tau}
f3e75c
θ3 ⁣(z,τ)=1+2n=1qn2cos ⁣(2nπz)   where q=eπiτ\theta_{3}\!\left(z , \tau\right) = 1 + 2 \sum_{n=1}^{\infty} {q}^{{n}^{2}} \cos\!\left(2 n \pi z\right)\; \text{ where } q = {e}^{\pi i \tau}
8a34d1
θ4 ⁣(z,τ)=1+2n=1(1)nqn2cos ⁣(2nπz)   where q=eπiτ\theta_{4}\!\left(z , \tau\right) = 1 + 2 \sum_{n=1}^{\infty} {\left(-1\right)}^{n} {q}^{{n}^{2}} \cos\!\left(2 n \pi z\right)\; \text{ where } q = {e}^{\pi i \tau}

Jacobi triple product

13d2a1
θ3 ⁣(z,τ)=n=qn2w2n=n=1(1q2n)(1+q2n1w2)(1+q2n1w2)   where q=eπiτ,w=eπiz\theta_{3}\!\left(z , \tau\right) = \sum_{n=-\infty}^{\infty} {q}^{{n}^{2}} {w}^{2 n} = \prod_{n=1}^{\infty} \left(1 - {q}^{2 n}\right) \left(1 + {q}^{2 n - 1} {w}^{2}\right) \left(1 + {q}^{2 n - 1} {w}^{-2}\right)\; \text{ where } q = {e}^{\pi i \tau},\,w = {e}^{\pi i z}

Zeros

154c44
zeroszCθ1 ⁣(z,τ)={m+nτ:mZandnZ}\mathop{\operatorname{zeros}\,}\limits_{z \in \mathbb{C}} \theta_{1}\!\left(z , \tau\right) = \left\{ m + n \tau : m \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, n \in \mathbb{Z} \right\}
ad1eaf
zeroszCθ2 ⁣(z,τ)={(m+12)+nτ:mZandnZ}\mathop{\operatorname{zeros}\,}\limits_{z \in \mathbb{C}} \theta_{2}\!\left(z , \tau\right) = \left\{ \left(m + \frac{1}{2}\right) + n \tau : m \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, n \in \mathbb{Z} \right\}
caf10a
zeroszCθ3 ⁣(z,τ)={(m+12)+(n+12)τ:mZandnZ}\mathop{\operatorname{zeros}\,}\limits_{z \in \mathbb{C}} \theta_{3}\!\left(z , \tau\right) = \left\{ \left(m + \frac{1}{2}\right) + \left(n + \frac{1}{2}\right) \tau : m \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, n \in \mathbb{Z} \right\}
926b2c
zeroszCθ4 ⁣(z,τ)={m+(n+12)τ:mZandnZ}\mathop{\operatorname{zeros}\,}\limits_{z \in \mathbb{C}} \theta_{4}\!\left(z , \tau\right) = \left\{ m + \left(n + \frac{1}{2}\right) \tau : m \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, n \in \mathbb{Z} \right\}

Specific values

Main topic: Specific values of Jacobi theta functions
8f43ab
θ1 ⁣(0,τ)=0\theta_{1}\!\left(0 , \tau\right) = 0
d15f11
θ3 ⁣(0,i)=π1/4Γ ⁣(34)\theta_{3}\!\left(0 , i\right) = \frac{{\pi}^{1 / 4}}{\Gamma\!\left(\frac{3}{4}\right)}
7d7c65
θ2 ⁣(0,i)=θ4 ⁣(0,i)=[21/4]θ3 ⁣(0,i)\theta_{2}\!\left(0 , i\right) = \theta_{4}\!\left(0 , i\right) = \left[{2}^{-1 / 4}\right] \theta_{3}\!\left(0 , i\right)

Argument transformations

Main topic: Argument transformations for Jacobi theta functions

Even-odd symmetry

59f8e1
θ1 ⁣(z,τ)=θ1 ⁣(z,τ)\theta_{1}\!\left(-z , \tau\right) = -\theta_{1}\!\left(z , \tau\right)
fb55cb
θ2 ⁣(z,τ)=θ2 ⁣(z,τ)\theta_{2}\!\left(-z , \tau\right) = \theta_{2}\!\left(z , \tau\right)
380076
θ3 ⁣(z,τ)=θ3 ⁣(z,τ)\theta_{3}\!\left(-z , \tau\right) = \theta_{3}\!\left(z , \tau\right)
4f939e
θ4 ⁣(z,τ)=θ4 ⁣(z,τ)\theta_{4}\!\left(-z , \tau\right) = \theta_{4}\!\left(z , \tau\right)

Conjugate symmetry

a891da
θj ⁣(z,τ)=θj ⁣(z,τ)\theta_{j}\!\left(\overline{z} , \tau\right) = \overline{\theta_{j}\!\left(z , -\overline{\tau}\right)}

Periodicity

2faeb9
θ1 ⁣(z+2n,τ)=θ1 ⁣(z,τ)\theta_{1}\!\left(z + 2 n , \tau\right) = \theta_{1}\!\left(z , \tau\right)
b46534
θ2 ⁣(z+2n,τ)=θ2 ⁣(z,τ)\theta_{2}\!\left(z + 2 n , \tau\right) = \theta_{2}\!\left(z , \tau\right)
e56f77
θ3 ⁣(z+n,τ)=θ3 ⁣(z,τ)\theta_{3}\!\left(z + n , \tau\right) = \theta_{3}\!\left(z , \tau\right)
4448f1
θ4 ⁣(z+n,τ)=θ4 ⁣(z,τ)\theta_{4}\!\left(z + n , \tau\right) = \theta_{4}\!\left(z , \tau\right)

Quasi-periodicity

43fa0e
θ1 ⁣(z+m+nτ,τ)=(1)m+neπi(τn2+2nz)θ1 ⁣(z,τ)\theta_{1}\!\left(z + m + n \tau , \tau\right) = {\left(-1\right)}^{m + n} {e}^{-\pi i \left(\tau {n}^{2} + 2 n z\right)} \theta_{1}\!\left(z , \tau\right)
d29148
θ2 ⁣(z+m+nτ,τ)=(1)meπi(τn2+2nz)θ2 ⁣(z,τ)\theta_{2}\!\left(z + m + n \tau , \tau\right) = {\left(-1\right)}^{m} {e}^{-\pi i \left(\tau {n}^{2} + 2 n z\right)} \theta_{2}\!\left(z , \tau\right)
2e4da0
θ3 ⁣(z+m+nτ,τ)=eπi(τn2+2nz)θ3 ⁣(z,τ)\theta_{3}\!\left(z + m + n \tau , \tau\right) = {e}^{-\pi i \left(\tau {n}^{2} + 2 n z\right)} \theta_{3}\!\left(z , \tau\right)
8d6a1d
θ4 ⁣(z+m+nτ,τ)=(1)neπi(τn2+2nz)θ4 ⁣(z,τ)\theta_{4}\!\left(z + m + n \tau , \tau\right) = {\left(-1\right)}^{n} {e}^{-\pi i \left(\tau {n}^{2} + 2 n z\right)} \theta_{4}\!\left(z , \tau\right)

Lattice transformations

Main topic: Lattice transformations for Jacobi theta functions

Basic modular transformations

6b2078
θ1 ⁣(z,τ+1)=eπi/4θ1 ⁣(z,τ)\theta_{1}\!\left(z , \tau + 1\right) = {e}^{\pi i / 4} \theta_{1}\!\left(z , \tau\right)
cde93e
θ2 ⁣(z,τ+1)=eπi/4θ2 ⁣(z,τ)\theta_{2}\!\left(z , \tau + 1\right) = {e}^{\pi i / 4} \theta_{2}\!\left(z , \tau\right)
9c1e9a
θ3 ⁣(z,τ+1)=θ4 ⁣(z,τ)\theta_{3}\!\left(z , \tau + 1\right) = \theta_{4}\!\left(z , \tau\right)
a5c258
θ4 ⁣(z,τ+1)=θ3 ⁣(z,τ)\theta_{4}\!\left(z , \tau + 1\right) = \theta_{3}\!\left(z , \tau\right)
e8ce0b
θ1 ⁣(z,1τ)=iτieπiτz2θ1 ⁣(τz,τ)\theta_{1}\!\left(z , \frac{-1}{\tau}\right) = -i \sqrt{\frac{\tau}{i}} {e}^{\pi i \tau {z}^{2}} \theta_{1}\!\left(\tau z , \tau\right)
06319a
θ2 ⁣(z,1τ)=τieπiτz2θ4 ⁣(τz,τ)\theta_{2}\!\left(z , \frac{-1}{\tau}\right) = \sqrt{\frac{\tau}{i}} {e}^{\pi i \tau {z}^{2}} \theta_{4}\!\left(\tau z , \tau\right)
c4b16c
θ3 ⁣(z,1τ)=τieπiτz2θ3 ⁣(τz,τ)\theta_{3}\!\left(z , \frac{-1}{\tau}\right) = \sqrt{\frac{\tau}{i}} {e}^{\pi i \tau {z}^{2}} \theta_{3}\!\left(\tau z , \tau\right)
ed8ba7
θ4 ⁣(z,1τ)=τieπiτz2θ2 ⁣(τz,τ)\theta_{4}\!\left(z , \frac{-1}{\tau}\right) = \sqrt{\frac{\tau}{i}} {e}^{\pi i \tau {z}^{2}} \theta_{2}\!\left(\tau z , \tau\right)

General modular transformations

4d8b0f
θj ⁣(z,aτ+bcτ+d)=εj ⁣(a,b,c,d)vieπicvz2θSj ⁣(a,b,c,d) ⁣(vz,τ)   where v=cτ+d\theta_{j}\!\left(z , \frac{a \tau + b}{c \tau + d}\right) = \varepsilon_{j}\!\left(a, b, c, d\right) \sqrt{\frac{v}{i}} {e}^{\pi i c v {z}^{2}} \theta_{S_{j}\!\left(a, b, c, d\right)}\!\left(v z , \tau\right)\; \text{ where } v = c \tau + d

Multiplication of the lattice parameter

69b32e
θ3 ⁣(z,τ2)=θ22 ⁣(z,τ)+θ32 ⁣(z,τ)θ3 ⁣(0,τ2)\theta_{3}\!\left(z , \frac{\tau}{2}\right) = \frac{\theta_{2}^{2}\!\left(z, \tau\right) + \theta_{3}^{2}\!\left(z, \tau\right)}{\theta_{3}\!\left(0 , \frac{\tau}{2}\right)}
3479be
θ3 ⁣(2z,2τ)=θ12 ⁣(z,τ)+θ22 ⁣(z,τ)2θ2 ⁣(0,2τ)\theta_{3}\!\left(2 z , 2 \tau\right) = \frac{\theta_{1}^{2}\!\left(z, \tau\right) + \theta_{2}^{2}\!\left(z, \tau\right)}{2 \theta_{2}\!\left(0 , 2 \tau\right)}
53fef4
θ3 ⁣(2z,4τ)=θ3 ⁣(z,τ)+θ4 ⁣(z,τ)2\theta_{3}\!\left(2 z , 4 \tau\right) = \frac{\theta_{3}\!\left(z , \tau\right) + \theta_{4}\!\left(z , \tau\right)}{2}

Sums and products

See also Argument transformations for Jacobi theta functions and Lattice transformations for Jacobi theta functions for sum and product identities involving transformations.

Fourth powers

1fbc09
θ34 ⁣(0,τ)=θ24 ⁣(0,τ)+θ44 ⁣(0,τ)\theta_{3}^{4}\!\left(0, \tau\right) = \theta_{2}^{4}\!\left(0, \tau\right) + \theta_{4}^{4}\!\left(0, \tau\right)
08822c
θ14 ⁣(z,τ)+θ34 ⁣(z,τ)=θ24 ⁣(z,τ)+θ44 ⁣(z,τ)\theta_{1}^{4}\!\left(z, \tau\right) + \theta_{3}^{4}\!\left(z, \tau\right) = \theta_{2}^{4}\!\left(z, \tau\right) + \theta_{4}^{4}\!\left(z, \tau\right)
5a3ebf
θ14 ⁣(z,τ)θ24 ⁣(z,τ)=θ44 ⁣(z,τ)θ34 ⁣(z,τ)\theta_{1}^{4}\!\left(z, \tau\right) - \theta_{2}^{4}\!\left(z, \tau\right) = \theta_{4}^{4}\!\left(z, \tau\right) - \theta_{3}^{4}\!\left(z, \tau\right)
e08bb4
θ14 ⁣(z,τ)θ44 ⁣(z,τ)=θ24 ⁣(z,τ)θ34 ⁣(z,τ)\theta_{1}^{4}\!\left(z, \tau\right) - \theta_{4}^{4}\!\left(z, \tau\right) = \theta_{2}^{4}\!\left(z, \tau\right) - \theta_{3}^{4}\!\left(z, \tau\right)

Sums of squares

fa7251
θ22 ⁣(0,τ)θ32 ⁣(z,τ)=θ42 ⁣(0,τ)θ12 ⁣(z,τ)+θ32 ⁣(0,τ)θ22 ⁣(z,τ)\theta_{2}^{2}\!\left(0, \tau\right) \theta_{3}^{2}\!\left(z, \tau\right) = \theta_{4}^{2}\!\left(0, \tau\right) \theta_{1}^{2}\!\left(z, \tau\right) + \theta_{3}^{2}\!\left(0, \tau\right) \theta_{2}^{2}\!\left(z, \tau\right)
265d9c
θ22 ⁣(0,τ)θ42 ⁣(z,τ)=θ32 ⁣(0,τ)θ12 ⁣(z,τ)+θ42 ⁣(0,τ)θ22 ⁣(z,τ)\theta_{2}^{2}\!\left(0, \tau\right) \theta_{4}^{2}\!\left(z, \tau\right) = \theta_{3}^{2}\!\left(0, \tau\right) \theta_{1}^{2}\!\left(z, \tau\right) + \theta_{4}^{2}\!\left(0, \tau\right) \theta_{2}^{2}\!\left(z, \tau\right)
0e2635
θ32 ⁣(0,τ)θ22 ⁣(z,τ)=θ22 ⁣(0,τ)θ32 ⁣(z,τ)θ42 ⁣(0,τ)θ12 ⁣(z,τ)\theta_{3}^{2}\!\left(0, \tau\right) \theta_{2}^{2}\!\left(z, \tau\right) = \theta_{2}^{2}\!\left(0, \tau\right) \theta_{3}^{2}\!\left(z, \tau\right) - \theta_{4}^{2}\!\left(0, \tau\right) \theta_{1}^{2}\!\left(z, \tau\right)
6fad93
θ32 ⁣(0,τ)θ32 ⁣(z,τ)=θ42 ⁣(0,τ)θ42 ⁣(z,τ)+θ22 ⁣(0,τ)θ22 ⁣(z,τ)\theta_{3}^{2}\!\left(0, \tau\right) \theta_{3}^{2}\!\left(z, \tau\right) = \theta_{4}^{2}\!\left(0, \tau\right) \theta_{4}^{2}\!\left(z, \tau\right) + \theta_{2}^{2}\!\left(0, \tau\right) \theta_{2}^{2}\!\left(z, \tau\right)
abbe42
θ32 ⁣(0,τ)θ42 ⁣(z,τ)=θ22 ⁣(0,τ)θ12 ⁣(z,τ)+θ42 ⁣(0,τ)θ32 ⁣(z,τ)\theta_{3}^{2}\!\left(0, \tau\right) \theta_{4}^{2}\!\left(z, \tau\right) = \theta_{2}^{2}\!\left(0, \tau\right) \theta_{1}^{2}\!\left(z, \tau\right) + \theta_{4}^{2}\!\left(0, \tau\right) \theta_{3}^{2}\!\left(z, \tau\right)
1c67c8
θ2 ⁣(0,2τ)(θ12 ⁣(z,τ)θ22 ⁣(z,τ))=θ3 ⁣(0,2τ)(θ42 ⁣(z,τ)θ32 ⁣(z,τ))\theta_{2}\!\left(0 , 2 \tau\right) \left(\theta_{1}^{2}\!\left(z, \tau\right) - \theta_{2}^{2}\!\left(z, \tau\right)\right) = \theta_{3}\!\left(0 , 2 \tau\right) \left(\theta_{4}^{2}\!\left(z, \tau\right) - \theta_{3}^{2}\!\left(z, \tau\right)\right)

Differential equations

Main topic: Differential equations for Jacobi theta functions

Notation and conversion to argument derivatives

a222ed
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)
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

936694
(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)

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)}

Integrals

Main topic: Integrals of Jacobi theta functions

Laplace transforms

8a857c
0eatθ1 ⁣(x,ibt)dt=πabsinh ⁣(2xπab)cosh ⁣(πab)\int_{0}^{\infty} {e}^{-a t} \theta_{1}\!\left(x , i b t\right) \, dt = \sqrt{\frac{\pi}{a b}} \frac{\sinh\!\left(2 x \sqrt{\frac{\pi a}{b}}\right)}{\cosh\!\left(\sqrt{\frac{\pi a}{b}}\right)}
74be8f
0eatθ2 ⁣(x,ibt)dt=πabsinh ⁣((2x1)πab)cosh ⁣(πab)\int_{0}^{\infty} {e}^{-a t} \theta_{2}\!\left(x , i b t\right) \, dt = -\sqrt{\frac{\pi}{a b}} \frac{\sinh\!\left(\left(2 x - 1\right) \sqrt{\frac{\pi a}{b}}\right)}{\cosh\!\left(\sqrt{\frac{\pi a}{b}}\right)}
026e44
0eatθ3 ⁣(x,ibt)dt=πabcosh ⁣((2x1)πab)sinh ⁣(πab)\int_{0}^{\infty} {e}^{-a t} \theta_{3}\!\left(x , i b t\right) \, dt = \sqrt{\frac{\pi}{a b}} \frac{\cosh\!\left(\left(2 x - 1\right) \sqrt{\frac{\pi a}{b}}\right)}{\sinh\!\left(\sqrt{\frac{\pi a}{b}}\right)}
a46f94
0eatθ4 ⁣(x,ibt)dt=πabcosh ⁣(2xπab)sinh ⁣(πab)\int_{0}^{\infty} {e}^{-a t} \theta_{4}\!\left(x , i b t\right) \, dt = \sqrt{\frac{\pi}{a b}} \frac{\cosh\!\left(2 x \sqrt{\frac{\pi a}{b}}\right)}{\sinh\!\left(\sqrt{\frac{\pi a}{b}}\right)}