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Series and product representations of Jacobi theta functions

Table of contents: Fourier series - Fourier series for derivatives - Infinite products - Series for logarithmic derivatives - Taylor series - Theta constants

See Jacobi theta functions for an introduction to these functions.

Fourier series

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}

Exponential Fourier series

700d94
θ1 ⁣(z,τ)=ieπiτ/4n=(1)nqn(n+1)w2n+1   where q=eπiτ,w=eπiz\theta_{1}\!\left(z , \tau\right) = -i {e}^{\pi i \tau / 4} \sum_{n=-\infty}^{\infty} {\left(-1\right)}^{n} {q}^{n \left(n + 1\right)} {w}^{2 n + 1}\; \text{ where } q = {e}^{\pi i \tau},\,w = {e}^{\pi i z}
495a98
θ2 ⁣(z,τ)=eπiτ/4n=qn(n+1)w2n+1   where q=eπiτ,w=eπiz\theta_{2}\!\left(z , \tau\right) = {e}^{\pi i \tau / 4} \sum_{n=-\infty}^{\infty} {q}^{n \left(n + 1\right)} {w}^{2 n + 1}\; \text{ where } q = {e}^{\pi i \tau},\,w = {e}^{\pi i z}
2f97f5
θ3 ⁣(z,τ)=n=qn2w2n   where q=eπiτ,w=eπiz\theta_{3}\!\left(z , \tau\right) = \sum_{n=-\infty}^{\infty} {q}^{{n}^{2}} {w}^{2 n}\; \text{ where } q = {e}^{\pi i \tau},\,w = {e}^{\pi i z}
d923de
θ4 ⁣(z,τ)=n=(1)nqn2w2n   where q=eπiτ,w=eπiz\theta_{4}\!\left(z , \tau\right) = \sum_{n=-\infty}^{\infty} {\left(-1\right)}^{n} {q}^{{n}^{2}} {w}^{2 n}\; \text{ where } q = {e}^{\pi i \tau},\,w = {e}^{\pi i z}

Pure exponential series

ed4ce5
θ1 ⁣(z,τ)=n=eπi((n+1/2)2τ+(2n+1)z+n1/2)\theta_{1}\!\left(z , \tau\right) = \sum_{n=-\infty}^{\infty} {e}^{\pi i \left({\left(n + 1 / 2\right)}^{2} \tau + \left(2 n + 1\right) z + n - 1 / 2\right)}
7cb651
θ2 ⁣(z,τ)=n=eπi((n+1/2)2τ+(2n+1)z)\theta_{2}\!\left(z , \tau\right) = \sum_{n=-\infty}^{\infty} {e}^{\pi i \left({\left(n + 1 / 2\right)}^{2} \tau + \left(2 n + 1\right) z\right)}
580ba0
θ3 ⁣(z,τ)=n=eπi(n2τ+2nz)\theta_{3}\!\left(z , \tau\right) = \sum_{n=-\infty}^{\infty} {e}^{\pi i \left({n}^{2} \tau + 2 n z\right)}
27c319
θ4 ⁣(z,τ)=n=eπi(n2τ+2nz+n)\theta_{4}\!\left(z , \tau\right) = \sum_{n=-\infty}^{\infty} {e}^{\pi i \left({n}^{2} \tau + 2 n z + n\right)}

Fourier series for derivatives

Exponential Fourier series

2ae142
θ1(r) ⁣(z,τ)=i(πi)reπiτ/4n=(1)n(2n+1)rqn(n+1)w2n+1   where q=eπiτ,w=eπiz\theta^{(r)}_{1}\!\left(z , \tau\right) = -i {\left(\pi i\right)}^{r} {e}^{\pi i \tau / 4} \sum_{n=-\infty}^{\infty} {\left(-1\right)}^{n} {\left(2 n + 1\right)}^{r} {q}^{n \left(n + 1\right)} {w}^{2 n + 1}\; \text{ where } q = {e}^{\pi i \tau},\,w = {e}^{\pi i z}
42d832
θ2(r) ⁣(z,τ)=(πi)reπiτ/4n=(2n+1)rqn(n+1)w2n+1   where q=eπiτ,w=eπiz\theta^{(r)}_{2}\!\left(z , \tau\right) = {\left(\pi i\right)}^{r} {e}^{\pi i \tau / 4} \sum_{n=-\infty}^{\infty} {\left(2 n + 1\right)}^{r} {q}^{n \left(n + 1\right)} {w}^{2 n + 1}\; \text{ where } q = {e}^{\pi i \tau},\,w = {e}^{\pi i z}
f551ca
θ3(r) ⁣(z,τ)=(2πi)rn=nrqn2w2n   where q=eπiτ,w=eπiz\theta^{(r)}_{3}\!\left(z , \tau\right) = {\left(2 \pi i\right)}^{r} \sum_{n=-\infty}^{\infty} {n}^{r} {q}^{{n}^{2}} {w}^{2 n}\; \text{ where } q = {e}^{\pi i \tau},\,w = {e}^{\pi i z}
1842d9
θ4(r) ⁣(z,τ)=(2πi)rn=(1)nnrqn2w2n   where q=eπiτ,w=eπiz\theta^{(r)}_{4}\!\left(z , \tau\right) = {\left(2 \pi i\right)}^{r} \sum_{n=-\infty}^{\infty} {\left(-1\right)}^{n} {n}^{r} {q}^{{n}^{2}} {w}^{2 n}\; \text{ where } q = {e}^{\pi i \tau},\,w = {e}^{\pi i z}

Infinite products

Infinite q-products with trigonometric factors

024a84
θ1 ⁣(z,τ)=2eπiτ/4sin ⁣(πz)n=1(1q2n)(12q2ncos ⁣(2πz)+q4n)   where q=eπiτ\theta_{1}\!\left(z , \tau\right) = 2 {e}^{\pi i \tau / 4} \sin\!\left(\pi z\right) \prod_{n=1}^{\infty} \left(1 - {q}^{2 n}\right) \left(1 - 2 {q}^{2 n} \cos\!\left(2 \pi z\right) + {q}^{4 n}\right)\; \text{ where } q = {e}^{\pi i \tau}
d6a799
θ2 ⁣(z,τ)=2eπiτ/4cos ⁣(πz)n=1(1q2n)(1+2q2ncos ⁣(2πz)+q4n)   where q=eπiτ\theta_{2}\!\left(z , \tau\right) = 2 {e}^{\pi i \tau / 4} \cos\!\left(\pi z\right) \prod_{n=1}^{\infty} \left(1 - {q}^{2 n}\right) \left(1 + 2 {q}^{2 n} \cos\!\left(2 \pi z\right) + {q}^{4 n}\right)\; \text{ where } q = {e}^{\pi i \tau}
77aed2
θ3 ⁣(z,τ)=n=1(1q2n)(1+2q2n1cos ⁣(2πz)+q4n2)   where q=eπiτ\theta_{3}\!\left(z , \tau\right) = \prod_{n=1}^{\infty} \left(1 - {q}^{2 n}\right) \left(1 + 2 {q}^{2 n - 1} \cos\!\left(2 \pi z\right) + {q}^{4 n - 2}\right)\; \text{ where } q = {e}^{\pi i \tau}
2a2a38
θ4 ⁣(z,τ)=n=1(1q2n)(12q2n1cos ⁣(2πz)+q4n2)   where q=eπiτ\theta_{4}\!\left(z , \tau\right) = \prod_{n=1}^{\infty} \left(1 - {q}^{2 n}\right) \left(1 - 2 {q}^{2 n - 1} \cos\!\left(2 \pi z\right) + {q}^{4 n - 2}\right)\; \text{ where } q = {e}^{\pi i \tau}

Infinite q-products with exponential factors

39b699
θ2 ⁣(z,τ)=ieπiτ/4(ww1)n=1(1q2n)(1q2nw2)(1q2nw2)   where q=eπiτ,w=eπiz\theta_{2}\!\left(z , \tau\right) = -i {e}^{\pi i \tau / 4} \left(w - {w}^{-1}\right) \prod_{n=1}^{\infty} \left(1 - {q}^{2 n}\right) \left(1 - {q}^{2 n} {w}^{2}\right) \left(1 - {q}^{2 n} {w}^{-2}\right)\; \text{ where } q = {e}^{\pi i \tau},\,w = {e}^{\pi i z}
465810
θ2 ⁣(z,τ)=eπiτ/4(w+w1)n=1(1q2n)(1+q2nw2)(1+q2nw2)   where q=eπiτ,w=eπiz\theta_{2}\!\left(z , \tau\right) = {e}^{\pi i \tau / 4} \left(w + {w}^{-1}\right) \prod_{n=1}^{\infty} \left(1 - {q}^{2 n}\right) \left(1 + {q}^{2 n} {w}^{2}\right) \left(1 + {q}^{2 n} {w}^{-2}\right)\; \text{ where } q = {e}^{\pi i \tau},\,w = {e}^{\pi i z}
21851b
θ3 ⁣(z,τ)=n=1(1q2n)(1+q2n1w2)(1+q2n1w2)   where q=eπiτ,w=eπiz\theta_{3}\!\left(z , \tau\right) = \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}
d45548
θ4 ⁣(z,τ)=n=1(1q2n)(1q2n1w2)(1q2n1w2)   where q=eπiτ,w=eπiz\theta_{4}\!\left(z , \tau\right) = \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}

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}

Trigonometric infinite products

d2f183
θ1 ⁣(z,τ)=θ1 ⁣(0,τ)πsin ⁣(πz)n=1sin ⁣(π(nτ+z))sin ⁣(π(nτz))sin2 ⁣(πnτ)\theta_{1}\!\left(z , \tau\right) = \frac{\theta'_{1}\!\left(0 , \tau\right)}{\pi} \sin\!\left(\pi z\right) \prod_{n=1}^{\infty} \frac{\sin\!\left(\pi \left(n \tau + z\right)\right) \sin\!\left(\pi \left(n \tau - z\right)\right)}{\sin^{2}\!\left(\pi n \tau\right)}
64081c
θ2 ⁣(z,τ)=θ2 ⁣(0,τ)cos ⁣(πz)n=1cos ⁣(π(nτ+z))cos ⁣(π(nτz))cos2 ⁣(πnτ)\theta_{2}\!\left(z , \tau\right) = \theta_{2}\!\left(0 , \tau\right) \cos\!\left(\pi z\right) \prod_{n=1}^{\infty} \frac{\cos\!\left(\pi \left(n \tau + z\right)\right) \cos\!\left(\pi \left(n \tau - z\right)\right)}{\cos^{2}\!\left(\pi n \tau\right)}
816057
θ3 ⁣(z,τ)=θ3 ⁣(0,τ)n=1cos ⁣(π((n12)τ+z))cos ⁣(π((n12)τz))cos2 ⁣(π(n12)τ)\theta_{3}\!\left(z , \tau\right) = \theta_{3}\!\left(0 , \tau\right) \prod_{n=1}^{\infty} \frac{\cos\!\left(\pi \left(\left(n - \frac{1}{2}\right) \tau + z\right)\right) \cos\!\left(\pi \left(\left(n - \frac{1}{2}\right) \tau - z\right)\right)}{\cos^{2}\!\left(\pi \left(n - \frac{1}{2}\right) \tau\right)}
3c88a7
θ4 ⁣(z,τ)=θ4 ⁣(0,τ)n=1sin ⁣(π((n12)τ+z))sin ⁣(π((n12)τz))sin2 ⁣(π(n12)τ)\theta_{4}\!\left(z , \tau\right) = \theta_{4}\!\left(0 , \tau\right) \prod_{n=1}^{\infty} \frac{\sin\!\left(\pi \left(\left(n - \frac{1}{2}\right) \tau + z\right)\right) \sin\!\left(\pi \left(\left(n - \frac{1}{2}\right) \tau - z\right)\right)}{\sin^{2}\!\left(\pi \left(n - \frac{1}{2}\right) \tau\right)}

Series for logarithmic derivatives

Lambert series with trigonometric factors

dfbddd
1πθ1 ⁣(z,τ)θ1 ⁣(z,τ)=cot ⁣(πz)+4n=1q2n1q2nsin ⁣(2πnz)   where q=eπiτ\frac{1}{\pi} \frac{\theta'_{1}\!\left(z , \tau\right)}{\theta_{1}\!\left(z , \tau\right)} = \cot\!\left(\pi z\right) + 4 \sum_{n=1}^{\infty} \frac{{q}^{2 n}}{1 - {q}^{2 n}} \sin\!\left(2 \pi n z\right)\; \text{ where } q = {e}^{\pi i \tau}
c7f7a5
1πθ2 ⁣(z,τ)θ2 ⁣(z,τ)=tan ⁣(πz)+4n=1(1)nq2n1q2nsin ⁣(2πnz)   where q=eπiτ\frac{1}{\pi} \frac{\theta'_{2}\!\left(z , \tau\right)}{\theta_{2}\!\left(z , \tau\right)} = -\tan\!\left(\pi z\right) + 4 \sum_{n=1}^{\infty} {\left(-1\right)}^{n} \frac{{q}^{2 n}}{1 - {q}^{2 n}} \sin\!\left(2 \pi n z\right)\; \text{ where } q = {e}^{\pi i \tau}
44e8fb
1πθ3 ⁣(z,τ)θ3 ⁣(z,τ)=4n=1(1)nqn1q2nsin ⁣(2πnz)   where q=eπiτ\frac{1}{\pi} \frac{\theta'_{3}\!\left(z , \tau\right)}{\theta_{3}\!\left(z , \tau\right)} = 4 \sum_{n=1}^{\infty} {\left(-1\right)}^{n} \frac{{q}^{n}}{1 - {q}^{2 n}} \sin\!\left(2 \pi n z\right)\; \text{ where } q = {e}^{\pi i \tau}
1848f1
1πθ4 ⁣(z,τ)θ4 ⁣(z,τ)=4n=1qn1q2nsin ⁣(2πnz)   where q=eπiτ\frac{1}{\pi} \frac{\theta'_{4}\!\left(z , \tau\right)}{\theta_{4}\!\left(z , \tau\right)} = 4 \sum_{n=1}^{\infty} \frac{{q}^{n}}{1 - {q}^{2 n}} \sin\!\left(2 \pi n z\right)\; \text{ where } q = {e}^{\pi i \tau}

Reciprocal trigonometric series

d81f05
d2dz2log ⁣(θ1 ⁣(z,τ))=π2n=1sin2 ⁣(π(z+nτ))\frac{d^{2}}{{d z}^{2}} \log\!\left(\theta_{1}\!\left(z , \tau\right)\right) = {\pi}^{2} \sum_{n=-\infty}^{\infty} \frac{1}{\sin^{2}\!\left(\pi \left(z + n \tau\right)\right)}
561d75
d2dz2log ⁣(θ2 ⁣(z,τ))=π2n=1cos2 ⁣(π(z+nτ))\frac{d^{2}}{{d z}^{2}} \log\!\left(\theta_{2}\!\left(z , \tau\right)\right) = {\pi}^{2} \sum_{n=-\infty}^{\infty} \frac{1}{\cos^{2}\!\left(\pi \left(z + n \tau\right)\right)}

Taylor series

1cdd7b
θj ⁣(z+x,τ)=n=0θj(n) ⁣(z,τ)n!xn\theta_{j}\!\left(z + x , \tau\right) = \sum_{n=0}^{\infty} \frac{\theta^{(n)}_{j}\!\left(z , \tau\right)}{n !} {x}^{n}
d637c5
θj ⁣(z,τ+x)=n=01(4πi)nθj(2n) ⁣(z,τ)n!xn\theta_{j}\!\left(z , \tau + x\right) = \sum_{n=0}^{\infty} \frac{1}{{\left(4 \pi i\right)}^{n}} \frac{\theta^{(2 n)}_{j}\!\left(z , \tau\right)}{n !} {x}^{n}

Theta constants

Fourier series (q-series) with linear exponents

a5e568
θ3k ⁣(0,τ)=n=0rk ⁣(n)qn   where q=eπiτ\theta_{3}^{k}\!\left(0, \tau\right) = \sum_{n=0}^{\infty} r_{k}\!\left(n\right) {q}^{n}\; \text{ where } q = {e}^{\pi i \tau}
7c90eb
θ4k ⁣(0,τ)=n=0(1)nrk ⁣(n)qn   where q=eπiτ\theta_{4}^{k}\!\left(0, \tau\right) = \sum_{n=0}^{\infty} {\left(-1\right)}^{n} r_{k}\!\left(n\right) {q}^{n}\; \text{ where } q = {e}^{\pi i \tau}
df88a0
θ32 ⁣(0,τ)+θ42 ⁣(0,τ)=2n=0r2 ⁣(2n)q2n   where q=eπiτ\theta_{3}^{2}\!\left(0, \tau\right) + \theta_{4}^{2}\!\left(0, \tau\right) = 2 \sum_{n=0}^{\infty} r_{2}\!\left(2 n\right) {q}^{2 n}\; \text{ where } q = {e}^{\pi i \tau}
290f36
θ32 ⁣(0,τ)θ32 ⁣(0,2τ)=n=0r2 ⁣(2n+1)q2n+1   where q=eπiτ\theta_{3}^{2}\!\left(0, \tau\right) - \theta_{3}^{2}\!\left(0, 2 \tau\right) = \sum_{n=0}^{\infty} r_{2}\!\left(2 n + 1\right) {q}^{2 n + 1}\; \text{ where } q = {e}^{\pi i \tau}

Infinite products for quotients

a0ba58
θ3 ⁣(0,τ)θ4 ⁣(0,τ)=n=1(1+q2n11q2n1)2   where q=eπiτ\frac{\theta_{3}\!\left(0 , \tau\right)}{\theta_{4}\!\left(0 , \tau\right)} = \prod_{n=1}^{\infty} {\left(\frac{1 + {q}^{2 n - 1}}{1 - {q}^{2 n - 1}}\right)}^{2}\; \text{ where } q = {e}^{\pi i \tau}
f1f42f
θ2 ⁣(0,τ)θ3 ⁣(0,τ)=2eπiτ/4n=1(1+q2n1+q2n1)2   where q=eπiτ\frac{\theta_{2}\!\left(0 , \tau\right)}{\theta_{3}\!\left(0 , \tau\right)} = 2 {e}^{\pi i \tau / 4} \prod_{n=1}^{\infty} {\left(\frac{1 + {q}^{2 n}}{1 + {q}^{2 n - 1}}\right)}^{2}\; \text{ where } q = {e}^{\pi i \tau}

Lambert series

e4e707
θ3 ⁣(0,τ)=1+2n=1λ(n)qn1qn   where q=eπiτ\theta_{3}\!\left(0 , \tau\right) = 1 + 2 \sum_{n=1}^{\infty} \frac{\lambda(n) {q}^{n}}{1 - {q}^{n}}\; \text{ where } q = {e}^{\pi i \tau}
0650f8
θ32 ⁣(0,τ)=1+4n=1qn1+q2n   where q=eπiτ\theta_{3}^{2}\!\left(0, \tau\right) = 1 + 4 \sum_{n=1}^{\infty} \frac{{q}^{n}}{1 + {q}^{2 n}}\; \text{ where } q = {e}^{\pi i \tau}
c743eb
θ24 ⁣(0,τ)=8n=0(2n+1)q2n+11+q2n+1+8n=0(2n+1)q2n+11q2n+1   where q=eπiτ\theta_{2}^{4}\!\left(0, \tau\right) = 8 \sum_{n=0}^{\infty} \frac{\left(2 n + 1\right) {q}^{2 n + 1}}{1 + {q}^{2 n + 1}} + 8 \sum_{n=0}^{\infty} \frac{\left(2 n + 1\right) {q}^{2 n + 1}}{1 - {q}^{2 n + 1}}\; \text{ where } q = {e}^{\pi i \tau}
8a316c
θ34 ⁣(0,τ)=1+8n=02nq2n1+q2n+8n=0(2n+1)q2n+11q2n+1   where q=eπiτ\theta_{3}^{4}\!\left(0, \tau\right) = 1 + 8 \sum_{n=0}^{\infty} \frac{2 n {q}^{2 n}}{1 + {q}^{2 n}} + 8 \sum_{n=0}^{\infty} \frac{\left(2 n + 1\right) {q}^{2 n + 1}}{1 - {q}^{2 n + 1}}\; \text{ where } q = {e}^{\pi i \tau}
dc7c83
θ44 ⁣(0,τ)=1+8n=02nq2n1+q2n8n=0(2n+1)q2n+11+q2n+1   where q=eπiτ\theta_{4}^{4}\!\left(0, \tau\right) = 1 + 8 \sum_{n=0}^{\infty} \frac{2 n {q}^{2 n}}{1 + {q}^{2 n}} - 8 \sum_{n=0}^{\infty} \frac{\left(2 n + 1\right) {q}^{2 n + 1}}{1 + {q}^{2 n + 1}}\; \text{ where } q = {e}^{\pi i \tau}
1cec67
θ44 ⁣(0,τ)θ24 ⁣(0,τ)=124n=0(2n+1)q2n+11+q2n+1   where q=eπiτ\theta_{4}^{4}\!\left(0, \tau\right) - \theta_{2}^{4}\!\left(0, \tau\right) = 1 - 24 \sum_{n=0}^{\infty} \frac{\left(2 n + 1\right) {q}^{2 n + 1}}{1 + {q}^{2 n + 1}}\; \text{ where } q = {e}^{\pi i \tau}