Modular forms and theta functions

Part of the Fungrim Identities reference — 314 identities for modular forms and theta functions.

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Dedekind eta function (23)
Illustrations of Eisenstein series (41)
Jacobi theta functions (206)
Modular j-invariant (20)
Modular lambda function (24)

Dedekind eta function

$36\tau\mapsto\tau\mapsto\mathrm{DedekindEta}(\tau)^{\prime}(\tau)/\mathrm{DedekindEta}(\tau)^{\prime}(\tau)^2-\frac{1}{\mathrm{DedekindEta}(\tau)}(24\tau\mapsto\mathrm{DedekindEta}(\tau)^{\prime}(\tau)\tau\mapsto\tau\mapsto\mathrm{DedekindEta}(\tau)^{\prime}(\tau)/\mathrm{DedekindEta}(\tau)^{\doubleprime}(\tau))+\tau\mapsto\frac{1}{\mathrm{DedekindEta}(\tau)}(\tau\mapsto\mathrm{DedekindEta}(\tau)^{\prime}(\tau))^{\tripleprime}(\tau)=0$

Holds when $\Im(\tau)\gt0$ . Symbols: DedekindEta — Dedekind eta function. Used by the Compute Engine for simplification. Reference: functions.wolfram.com 02d14f · Fungrim entry ↗

$\mathrm{DedekindEta}(16\imaginaryI)=\frac{\mathrm{DedekindEta}(\imaginaryI)\sqrt[4]{2^{1/4}-1}\sqrt{(1+2^{1/2})^{1/2}-2^{5/8}}}{2^{\frac{113}{64}}\sqrt[16]{1+\sqrt{2}}}$

Symbols: DedekindEta — Dedekind eta function. Used by the Compute Engine for simplification. Reference: math.stackexchange.com 0701dc · Fungrim entry ↗

$\mathrm{DedekindEta}(\tau+1)=\exp(\frac{\pi\imaginaryI}{12})\mathrm{DedekindEta}(\tau)$

Holds when $\Im(\tau)\gt0$ . Symbols: DedekindEta — Dedekind eta function. Used by the Compute Engine for simplification. 1bae52 · Fungrim entry ↗

$\tau\mapsto\mathrm{DedekindEta}(\tau)^{\prime}(\tau)=\frac{1}{\pi}((\frac{1}{2}\imaginaryI)\mathrm{DedekindEta}(\tau)\mathrm{WeierstrassZeta}(\frac{1}{2}, \tau))$

Holds when $\Im(\tau)\gt0$ . Symbols: DedekindEta — Dedekind eta function; WeierstrassZeta — Weierstrass zeta function. Used by the Compute Engine for simplification. 1c25d3 · Fungrim entry ↗

$\mathrm{DedekindEta}(\exp(\frac{2\pi\imaginaryI}{3}))=\frac{\sqrt[8]{3}\exp(-((1/24\imaginaryI)\pi))\sqrt{\Gamma(1/3)}^{3}}{2\pi}$

Symbols: DedekindEta — Dedekind eta function. Used by the Compute Engine for simplification. 204acd · Fungrim entry ↗

$\mathrm{DedekindEta}(4\imaginaryI)=(\mathrm{DedekindEta}(\imaginaryI))(2^{\frac{13}{16}}\sqrt[4]{1+\sqrt{2}})^{-1}$

Symbols: DedekindEta — Dedekind eta function. Used by the Compute Engine for simplification. 3a56d8 · Fungrim entry ↗

$\mathrm{DedekindEta}(-(\frac{1}{\tau}))=\sqrt{-(\imaginaryI\tau)}\mathrm{DedekindEta}(\tau)$

Holds when $\Im(\tau)\gt0$ . Symbols: DedekindEta — Dedekind eta function. Used by the Compute Engine for simplification. 3b806f · Fungrim entry ↗

$\mathrm{DedekindEta}(6\imaginaryI)=\frac{\mathrm{DedekindEta}(\imaginaryI)\sqrt[6]{\frac{5-3^{1/2}}{2}-\frac{1}{2}(2^{1/2}\times3^{3/4})}}{6^{\frac{3}{8}}}$

Symbols: DedekindEta — Dedekind eta function. Used by the Compute Engine for simplification. Reference: math.stackexchange.com 62ffb3 · Fungrim entry ↗

$\mathrm{DedekindEta}(\tau)=\exp(\frac{\pi\imaginaryI\tau}{12})\mathrm{JacobiTheta}(3, \frac{\tau+1}{2}, 3\tau)$

Holds when $\Im(\tau)\gt0$ . Symbols: DedekindEta — Dedekind eta function; JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. 737805 · Fungrim entry ↗

$\mathrm{DedekindEta}(7\imaginaryI)=\sqrt[4]{7^{1/2}-7/2+\frac{1}{2}((4\times7^{1/2}-7)^{1/2})}/\sqrt{7}\mathrm{DedekindEta}(\imaginaryI)$

Symbols: DedekindEta — Dedekind eta function. Used by the Compute Engine for simplification. Reference: math.stackexchange.com 7cc3d3 · Fungrim entry ↗

$\tau\mapsto\mathrm{DedekindEta}(\tau)^{\prime}(\tau)=\frac{1}{12}(\imaginaryI\pi\mathrm{DedekindEta}(\tau))\mathrm{EisensteinE}(2, \tau)$

$\mathrm{DedekindEta}(2\imaginaryI)=\frac{\mathrm{DedekindEta}(\imaginaryI)}{2^{\frac{3}{8}}}$

Symbols: DedekindEta — Dedekind eta function. Used by the Compute Engine for simplification. 87e9ed · Fungrim entry ↗

$\mathrm{DedekindEtaEpsilon}(a, b, c, d)=\exp(\pi\imaginaryI((a+d)/(12c)-\mathrm{DedekindSum}(d, c)-\frac{1}{4}))$

Holds when $a\in\Z\land b\in\Z\land c\in\Z\land d\in\Z\land ad-bc=1\land c\gt0$ . Symbols: DedekindEtaEpsilon — Root of unity in the functional equation of the Dedekind eta function; DedekindSum — Dedekind sum. Used by the Compute Engine for expansion. 921ef0 · Fungrim entry ↗

$\mathrm{DedekindEta}(\imaginaryI)=\frac{\Gamma(\frac{1}{4})}{2\pi^{\frac{3}{4}}}$

Symbols: DedekindEta — Dedekind eta function. Used by the Compute Engine for simplification. 9b8c9f · Fungrim entry ↗

$\mathrm{DedekindEta}(3\imaginaryI)=(\mathrm{DedekindEta}(\imaginaryI))(3^{\frac{3}{8}}\sqrt[12]{2+\sqrt{3}})^{-1}$

Symbols: DedekindEta — Dedekind eta function. Used by the Compute Engine for simplification. 9ce413 · Fungrim entry ↗

$\mathrm{DedekindEta}(\tau+\frac{1}{2})=\frac{\exp((1/24\imaginaryI)\pi)\mathrm{DedekindEta}(2\tau)^3}{\mathrm{DedekindEta}(\tau)\mathrm{DedekindEta}(4\tau)}$

Holds when $\Im(\tau)\gt0$ . Symbols: DedekindEta — Dedekind eta function. Used by the Compute Engine for simplification. a1a3d4 · Fungrim entry ↗

$\mathrm{DedekindEta}(\tau+24)=\mathrm{DedekindEta}(\tau)$

Holds when $\Im(\tau)\gt0$ . Symbols: DedekindEta — Dedekind eta function. Used by the Compute Engine for simplification. acee1a · Fungrim entry ↗

$\mathrm{DedekindEta}(8\imaginaryI)=\frac{\mathrm{DedekindEta}(\imaginaryI)\sqrt{2^{1/4}-1}}{2^{\frac{41}{32}}\sqrt[8]{1+\sqrt{2}}}$

Symbols: DedekindEta — Dedekind eta function. Used by the Compute Engine for simplification. Reference: math.stackexchange.com be2f32 · Fungrim entry ↗

$\mathrm{DedekindEta}(5\imaginaryI)=\frac{\mathrm{DedekindEta}(\imaginaryI)}{\sqrt{5\varphi}}$

Symbols: DedekindEta — Dedekind eta function. Used by the Compute Engine for simplification. Reference: math.stackexchange.com d2900f · Fungrim entry ↗

$(\mathrm{DedekindEta}(\tau)^2(33\tau\mapsto\mathrm{DedekindEta}(\tau)^{\doubleprime}(\tau)^2+\mathrm{DedekindEta}(\tau)\tau\mapsto\mathrm{DedekindEta}(\tau)^{(4)}(\tau))-18{\tau\mapsto\mathrm{DedekindEta}(\tau)^{\prime}(\tau)}^4+12\mathrm{DedekindEta}(\tau)\tau\mapsto\mathrm{DedekindEta}(\tau)^{\doubleprime}(\tau)\tau\mapsto\mathrm{DedekindEta}(\tau)^{\prime}(\tau)^2)-28\mathrm{DedekindEta}(\tau)^2\tau\mapsto\mathrm{DedekindEta}(\tau)^{\tripleprime}(\tau)\tau\mapsto\mathrm{DedekindEta}(\tau)^{\prime}(\tau)=0$

$\mathrm{DedekindEta}(\sqrt{3}\imaginaryI)=\frac{\sqrt[8]{3}\sqrt{\Gamma(1/3)}^{3}}{\pi\times2^{\frac{4}{3}}}$

Symbols: DedekindEta — Dedekind eta function. Used by the Compute Engine for simplification. Reference: math.stackexchange.com e3e4c5 · Fungrim entry ↗

$\mathrm{DedekindEtaEpsilon}(1, b, 0, 1)=\exp(\frac{\pi\imaginaryI b}{12})$

Symbols: DedekindEtaEpsilon — Root of unity in the functional equation of the Dedekind eta function. Used by the Compute Engine for simplification. f04e01 · Fungrim entry ↗

$\mathrm{DedekindEta}(\tau)=\exp(\frac{\pi\imaginaryI\tau}{12})\mathrm{EulerQSeries}(\exp(2\pi\imaginaryI\tau))$

Holds when $\Im(\tau)\gt0$ . Symbols: DedekindEta — Dedekind eta function; EulerQSeries — Euler's q-series. Used by the Compute Engine for simplification. ff587a · Fungrim entry ↗

Illustrations of Eisenstein series

$\mathrm{EisensteinE}(2, \tau)=-(\frac{(12\imaginaryI)\tau\mapsto\mathrm{DedekindEta}(\tau)^{\prime}(\tau)}{\pi\mathrm{DedekindEta}(\tau)})$

$\mathrm{EisensteinE}(8, \tau)=\mathrm{EisensteinE}(4, \tau)^2$

Holds when $\Im(\tau)\gt0$ . Symbols: EisensteinE — Normalized Eisenstein series. Used by the Compute Engine for simplification. 044128 · Fungrim entry ↗

$\mathrm{EisensteinE}(2k, \tau)=\frac{\mathrm{EisensteinG}(2k, \tau)}{2\Zeta(2k)}$

Holds when $k\in\N^*\land\Im(\tau)\gt0$ . Symbols: EisensteinE — Normalized Eisenstein series; EisensteinG — Eisenstein series. Used by the Compute Engine for simplification. 0a2120 · Fungrim entry ↗

$\mathrm{EisensteinE}(6, \tau)=\frac{\mathrm{DedekindEta}(\tau)^{24}}{\mathrm{DedekindEta}(2\tau)^{12}}-480\mathrm{DedekindEta}(2\tau)^{12}-(16\,896\mathrm{DedekindEta}(2\tau)^{12}\mathrm{DedekindEta}(4\tau)^8)/\mathrm{DedekindEta}(\tau)^8+\frac{8\,192\mathrm{DedekindEta}(4\tau)^{24}}{\mathrm{DedekindEta}(2\tau)^{12}}$

Holds when $\Im(\tau)\gt0$ . Symbols: DedekindEta — Dedekind eta function; EisensteinE — Normalized Eisenstein series. Used by the Compute Engine for simplification. Reference: K. Ono (2004), Web of Modularity: Arithmetic of the Coefficients of Modular Forms and q-series, American Mathematical Society. Theorem 1.67. 0a5ef4 · Fungrim entry ↗

$\mathrm{EisensteinG}(6, \exp(\frac{2\pi\imaginaryI}{3}))=\frac{\Gamma(1/3)^{18}}{8\,960\pi^6}$

Symbols: EisensteinG — Eisenstein series. Used by the Compute Engine for simplification. 0fda1b · Fungrim entry ↗

$\mathrm{EisensteinE}(6, \tau)=\frac{1}{2}((\mathrm{JacobiTheta}(3, 0, \tau)^{12}+\mathrm{JacobiTheta}(4, 0, \tau)^{12})-3\mathrm{JacobiTheta}(2, 0, \tau)^8(\mathrm{JacobiTheta}(3, 0, \tau)^4+\mathrm{JacobiTheta}(4, 0, \tau)^4))$

Holds when $\Im(\tau)\gt0$ . Symbols: EisensteinE — Normalized Eisenstein series; JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. 10f3b2 · Fungrim entry ↗

$\mathrm{EisensteinE}(2k, \tau)=1-\frac{4k(\sum_{n=1}^{\infty}\sum_{m=1}^{\infty}\exp(2\imaginaryI\pi mn\tau)n^{2k-1})}{\mathrm{BernoulliB}(2k)}$

Holds when $k\in\N^*\land\Im(\tau)\gt0$ . Symbols: BernoulliB — Bernoulli number; EisensteinE — Normalized Eisenstein series. Used by the Compute Engine for simplification. 15b347 · Fungrim entry ↗

$\mathrm{EisensteinE}(6, \tau)=1+63(\sum_{m=1}^{\infty}\frac{2\cos(\pi m\tau)^4+11\cos(\pi m\tau)^2+2}{\sin(\pi m\tau)^6})$

Holds when $\Im(\tau)\gt0$ . Symbols: EisensteinE — Normalized Eisenstein series. Used by the Compute Engine for simplification. 171724 · Fungrim entry ↗

$\mathrm{EisensteinE}(2, \tau)=1+6(\sum_{m=1}^{\infty}(\sin(\pi m\tau)^2)^{-1})$

Holds when $\Im(\tau)\gt0$ . Symbols: EisensteinE — Normalized Eisenstein series. Used by the Compute Engine for simplification. 18a4d1 · Fungrim entry ↗

$\mathrm{EisensteinG}(2k, \tau+n)=\mathrm{EisensteinG}(2k, \tau)$

Holds when $k\in\N^*\land\Im(\tau)\gt0\land n\in\Z$ . Symbols: EisensteinG — Eisenstein series. Used by the Compute Engine for simplification. 23a5e0 · Fungrim entry ↗

$\mathrm{EisensteinE}(2, \exp(\frac{2\pi\imaginaryI}{3}))=\frac{2\sqrt{3}}{\pi}$

Symbols: EisensteinE — Normalized Eisenstein series. Used by the Compute Engine for simplification. 30a054 · Fungrim entry ↗

$\mathrm{EisensteinG}(4, \exp(\frac{2\pi\imaginaryI}{3}))=\mathrm{EisensteinE}(4, \exp(\frac{2\pi\imaginaryI}{3}))=0$

Symbols: EisensteinE — Normalized Eisenstein series; EisensteinG — Eisenstein series. Used by the Compute Engine for simplification. 3102a7 · Fungrim entry ↗

$\mathrm{EisensteinE}(12, \tau)=\frac{1}{691}(441\mathrm{EisensteinE}(4, \tau)^3+250\mathrm{EisensteinE}(6, \tau)^2)$

Holds when $\Im(\tau)\gt0$ . Symbols: EisensteinE — Normalized Eisenstein series. Used by the Compute Engine for simplification. 36fff2 · Fungrim entry ↗

$\mathrm{EisensteinE}(2, \tau)=(6\mathrm{WeierstrassZeta}(\frac{1}{2}, \tau))/\pi^2$

Holds when $\Im(\tau)\gt0$ . Symbols: EisensteinE — Normalized Eisenstein series; WeierstrassZeta — Weierstrass zeta function. Used by the Compute Engine for simplification. 3bf702 · Fungrim entry ↗

$\tau\mapsto\mathrm{EisensteinE}(6, \tau)^{\prime}(\tau)=\frac{1}{2}(2\pi\imaginaryI(\mathrm{EisensteinE}(2, \tau)\mathrm{EisensteinE}(6, \tau)-\mathrm{EisensteinE}(4, \tau)^2))$

Holds when $\Im(\tau)\gt0$ . Symbols: EisensteinE — Normalized Eisenstein series. Used by the Compute Engine for simplification. Reference: B. C. Berndt and A. J. Yee (2002) Ramanujan's Contributions to Eisenstein Series, Especially in His Lost Notebook. In: Kanemitsu S., Jia C. (eds) Number Theoretic Methods. Developments in Mathematics, vol 8. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-3675-5_3 3bfced · Fungrim entry ↗

$\mathrm{EisensteinE}(4, \tau)=\frac{\mathrm{DedekindEta}(\tau)^{16}}{\mathrm{DedekindEta}(2\tau)^8}+(256\mathrm{DedekindEta}(2\tau)^{16})/\mathrm{DedekindEta}(\tau)^8$

$\mathrm{EisensteinE}(4, \imaginaryI)=\frac{3\Gamma(1/4)^8}{64\pi^6}$

Symbols: EisensteinE — Normalized Eisenstein series. Used by the Compute Engine for simplification. 53fcdd · Fungrim entry ↗

$\mathrm{EisensteinG}(2, \imaginaryI)=\pi$

Symbols: EisensteinG — Eisenstein series. Used by the Compute Engine for simplification. 570399 · Fungrim entry ↗

$\mathrm{EisensteinE}(6, \exp(\frac{2\pi\imaginaryI}{3}))=\frac{27\Gamma(1/3)^{18}}{512\pi^{12}}$

Symbols: EisensteinE — Normalized Eisenstein series. Used by the Compute Engine for simplification. 6c71c0 · Fungrim entry ↗

$\mathrm{EisensteinE}(8, \tau)=\frac{1}{2}(\mathrm{JacobiTheta}(2, 0, \tau)^{16}+\mathrm{JacobiTheta}(3, 0, \tau)^{16}+\mathrm{JacobiTheta}(4, 0, \tau)^{16})$

Holds when $\Im(\tau)\gt0$ . Symbols: EisensteinE — Normalized Eisenstein series; JacobiTheta — Jacobi theta function. Used by the Compute Engine for expansion. 6d2880 · Fungrim entry ↗

$\mathrm{EisensteinE}(2, \tau)=1-12(\sum_{m=1}^{\infty}(\cos(2\pi m\tau)-1)^{-1})$

Holds when $\Im(\tau)\gt0$ . Symbols: EisensteinE — Normalized Eisenstein series. Used by the Compute Engine for simplification. 7b62e4 · Fungrim entry ↗

$\mathrm{EisensteinE}(2k, \tau)=1-\frac{4k(\sum_{n=1}^{\infty}\mathrm{DivisorSigma}(2k-1, n)\exp(2\imaginaryI\pi n\tau))}{\mathrm{BernoulliB}(2k)}$

$\tau\mapsto\mathrm{EisensteinE}(2, \tau)^{\prime}(\tau)=\frac{1}{12}(2\pi\imaginaryI(\mathrm{EisensteinE}(2, \tau)^2-\mathrm{EisensteinE}(4, \tau)))$

$\mathrm{EisensteinE}(2k, \tau)=1-\frac{4k(\sum_{n=1}^{\infty}(\exp(2\imaginaryI\pi n\tau)n^{2k-1})/(1-\exp(2\imaginaryI\pi n\tau)))}{\mathrm{BernoulliB}(2k)}$

$\mathrm{EisensteinE}(14, \tau)=\mathrm{EisensteinE}(4, \tau)\mathrm{EisensteinE}(10, \tau)$

Holds when $\Im(\tau)\gt0$ . Symbols: EisensteinE — Normalized Eisenstein series. Used by the Compute Engine for simplification. 9e1f83 · Fungrim entry ↗

$\mathrm{EisensteinG}(2, \exp(\frac{2\pi\imaginaryI}{3}))=\frac{2\pi}{\sqrt{3}}$

Symbols: EisensteinG — Eisenstein series. Used by the Compute Engine for simplification. 9ea739 · Fungrim entry ↗

$\mathrm{EisensteinE}(6, \tau)^2=\frac{1}{8}({(\mathrm{JacobiTheta}(2, 0, \tau)^8+\mathrm{JacobiTheta}(3, 0, \tau)^8+\mathrm{JacobiTheta}(4, 0, \tau)^8)}^3-54(\mathrm{JacobiTheta}(2, 0, \tau)\mathrm{JacobiTheta}(3, 0, \tau)\mathrm{JacobiTheta}(4, 0, \tau))^8)$

$\mathrm{EisensteinG}(6, \imaginaryI)=\mathrm{EisensteinE}(6, \imaginaryI)=0$

Symbols: EisensteinE — Normalized Eisenstein series; EisensteinG — Eisenstein series. Used by the Compute Engine for simplification. a4109c · Fungrim entry ↗

$\mathrm{EisensteinE}(2, \imaginaryI)=\frac{3}{\pi}$

Symbols: EisensteinE — Normalized Eisenstein series. Used by the Compute Engine for simplification. a691b3 · Fungrim entry ↗

$\mathrm{EisensteinE}(4, \tau)=1+30(\sum_{m=1}^{\infty}\frac{\cos(\pi m\tau)^2+1}{\sin(\pi m\tau)^4})$

Holds when $\Im(\tau)\gt0$ . Symbols: EisensteinE — Normalized Eisenstein series. Used by the Compute Engine for simplification. a92c1a · Fungrim entry ↗

$\mathrm{EisensteinE}(10, \tau)=\mathrm{EisensteinE}(4, \tau)\mathrm{EisensteinE}(6, \tau)$

Holds when $\Im(\tau)\gt0$ . Symbols: EisensteinE — Normalized Eisenstein series. Used by the Compute Engine for simplification. adaf5a · Fungrim entry ↗

$\tau\mapsto\mathrm{EisensteinE}(4, \tau)^{\prime}(\tau)=\frac{1}{3}(2\pi\imaginaryI(\mathrm{EisensteinE}(2, \tau)\mathrm{EisensteinE}(4, \tau)-\mathrm{EisensteinE}(6, \tau)))$

Holds when $\Im(\tau)\gt0$ . Symbols: EisensteinE — Normalized Eisenstein series. Used by the Compute Engine for expansion. Reference: B. C. Berndt and A. J. Yee (2002) Ramanujan's Contributions to Eisenstein Series, Especially in His Lost Notebook. In: Kanemitsu S., Jia C. (eds) Number Theoretic Methods. Developments in Mathematics, vol 8. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-3675-5_3 af2ea9 · Fungrim entry ↗

$\mathrm{EisensteinG}(2k, \tau)=2\Zeta(2k)+2(\sum_{m=1}^{\infty}\sum_{n\in \Z}((m\tau+n)^{2k})^{-1})$

Holds when $k\in\N^*\land\Im(\tau)\gt0$ . Symbols: EisensteinG — Eisenstein series. Used by the Compute Engine for simplification. b07750 · Fungrim entry ↗

$\mathrm{EisensteinG}(2, \tau)=2\mathrm{WeierstrassZeta}(\frac{1}{2}, \tau)$

Holds when $\Im(\tau)\gt0$ . Symbols: EisensteinG — Eisenstein series; WeierstrassZeta — Weierstrass zeta function. Used by the Compute Engine for simplification. b52b6f · Fungrim entry ↗

$\mathrm{EisensteinE}(4, \tau)^3-\mathrm{EisensteinE}(6, \tau)^2=\frac{1}{4}(27(\mathrm{JacobiTheta}(2, 0, \tau)\mathrm{JacobiTheta}(3, 0, \tau)\mathrm{JacobiTheta}(4, 0, \tau))^8)$

$\mathrm{EisensteinE}(4, \tau)=\frac{1}{2}(\mathrm{JacobiTheta}(2, 0, \tau)^8+\mathrm{JacobiTheta}(3, 0, \tau)^8+\mathrm{JacobiTheta}(4, 0, \tau)^8)$

$\mathrm{EisensteinE}(2k, \tau+n)=\mathrm{EisensteinE}(2k, \tau)$

Holds when $k\in\N^*\land\Im(\tau)\gt0\land n\in\Z$ . Symbols: EisensteinE — Normalized Eisenstein series. Used by the Compute Engine for simplification. d56eb6 · Fungrim entry ↗

$\mathrm{EisensteinG}(2, \tau)=-(\frac{1}{\mathrm{DedekindEta}(\tau)}((4\imaginaryI)\pi\tau\mapsto\mathrm{DedekindEta}(\tau)^{\prime}(\tau)))$

Holds when $\Im(\tau)\gt0$ . Symbols: DedekindEta — Dedekind eta function; EisensteinG — Eisenstein series. Used by the Compute Engine for simplification. dbf388 · Fungrim entry ↗

$\mathrm{EisensteinG}(4, \imaginaryI)=\frac{\Gamma(1/4)^8}{960\pi^2}$

Symbols: EisensteinG — Eisenstein series. Used by the Compute Engine for simplification. e03b7c · Fungrim entry ↗

$\mathrm{EisensteinE}(14, \tau)=\mathrm{EisensteinE}(4, \tau)^2\mathrm{EisensteinE}(6, \tau)$

Holds when $\Im(\tau)\gt0$ . Symbols: EisensteinE — Normalized Eisenstein series. Used by the Compute Engine for simplification. e60fd4 · Fungrim entry ↗

$\mathrm{EisensteinE}(14, \tau)=\mathrm{EisensteinE}(6, \tau)\mathrm{EisensteinE}(8, \tau)$

Holds when $\Im(\tau)\gt0$ . Symbols: EisensteinE — Normalized Eisenstein series. Used by the Compute Engine for simplification. feb95e · Fungrim entry ↗

Jacobi theta functions

$\mathrm{JacobiTheta}(3, 4z, 4\tau)=\frac{\mathrm{JacobiTheta}(3, 1/8-z, \tau)\mathrm{JacobiTheta}(3, \frac{1}{8}+z, \tau)\mathrm{JacobiTheta}(3, 3/8-z, \tau)\mathrm{JacobiTheta}(3, \frac{3}{8}+z, \tau)}{\mathrm{JacobiTheta}(3, 0, \tau)\mathrm{JacobiTheta}(4, 0, \tau)\mathrm{JacobiTheta}(3, \frac{1}{4}, \tau)}$

Holds when $z\in\C\land\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. 0096a8 · Fungrim entry ↗

$z\mapsto\frac{\mathrm{JacobiTheta}(3, z, \tau)}{\mathrm{JacobiTheta}(1, z, \tau)}^{\prime}(z)=-(\frac{\pi\mathrm{JacobiTheta}(2, z, \tau)\mathrm{JacobiTheta}(4, z, \tau)\mathrm{JacobiTheta}(3, 0, \tau)^2}{\mathrm{JacobiTheta}(1, z, \tau)^2})$

Holds when $z\in\C\land\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. 0373dc · Fungrim entry ↗

$\mathrm{JacobiTheta}(4, 0, \tau, 2r+1)=0$

Holds when $\Im(\tau)\gt0\land r\in\N$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. 055b0a · Fungrim entry ↗

$\mathrm{JacobiTheta}(2, z, \frac{-1}{\tau})=\sqrt{\frac{\tau}{\imaginaryI}}\exp(\pi\imaginaryI\tau z^2)\mathrm{JacobiTheta}(4, \tau z, \tau)$

Holds when $z\in\C\land\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. 06319a · Fungrim entry ↗

$\mathrm{JacobiTheta}(3, 0, \tau)^2=1+4(\sum_{n=1}^{\infty}\frac{\exp(\pi\imaginaryI\tau)^{n}}{1+\exp(\pi\imaginaryI\tau)^{2n}})$

Holds when $\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. 0650f8 · Fungrim entry ↗

$\mathrm{JacobiTheta}(3, 0, \tau)\mathrm{JacobiTheta}(4, 0, \tau)\mathrm{JacobiTheta}(3, z+w, \tau)\mathrm{JacobiTheta}(4, z-w, \tau)=\mathrm{JacobiTheta}(3, z, \tau)\mathrm{JacobiTheta}(4, z, \tau)\mathrm{JacobiTheta}(3, w, \tau)\mathrm{JacobiTheta}(4, w, \tau)-\mathrm{JacobiTheta}(1, z, \tau)\mathrm{JacobiTheta}(2, z, \tau)\mathrm{JacobiTheta}(1, w, \tau)\mathrm{JacobiTheta}(2, w, \tau)$

Holds when $z\in\C\land w\in\C\land\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. 077394 · Fungrim entry ↗

$\mathrm{JacobiTheta}(2, z, \tau)=\exp(\pi\imaginaryI(z+\frac{\tau}{4}))\mathrm{JacobiTheta}(3, z+\frac{\tau}{2}, \tau)$

Holds when $z\in\C\land\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. 0878a4 · Fungrim entry ↗

$\mathrm{JacobiTheta}(1, z, \tau)^4+\mathrm{JacobiTheta}(3, z, \tau)^4=\mathrm{JacobiTheta}(2, z, \tau)^4+\mathrm{JacobiTheta}(4, z, \tau)^4$

Holds when $z\in\C\land\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for expansion. 08822c · Fungrim entry ↗

$\mathrm{JacobiTheta}(3, 2z, 2\tau)=\frac{\mathrm{JacobiTheta}(3, 1/4-z, \tau)\mathrm{JacobiTheta}(3, \frac{1}{4}+z, \tau)}{\mathrm{JacobiTheta}(4, 0, 2\tau)}$

Holds when $z\in\C\land\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. 0a9ec2 · Fungrim entry ↗

$\mathrm{JacobiTheta}(3, 0, \tau)^2\mathrm{JacobiTheta}(2, z, \tau)^2=\mathrm{JacobiTheta}(2, 0, \tau)^2\mathrm{JacobiTheta}(3, z, \tau)^2-\mathrm{JacobiTheta}(4, 0, \tau)^2\mathrm{JacobiTheta}(1, z, \tau)^2$

Holds when $z\in\C\land\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. 0e2635 · Fungrim entry ↗

$\mathrm{JacobiTheta}(3, z+\tau, \tau)=\exp(-(\pi\imaginaryI(2z+\tau)))\mathrm{JacobiTheta}(3, z, \tau)$

Holds when $z\in\C\land\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. 103bfb · Fungrim entry ↗

$\mathrm{JacobiTheta}(4, z, \tau)=\imaginaryI\exp(\pi\imaginaryI(z+\frac{\tau}{4}))\mathrm{JacobiTheta}(2, z+\frac{1}{2}+\frac{\tau}{2}, \tau)$

Holds when $z\in\C\land\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. 10ca40 · Fungrim entry ↗

$\mathrm{JacobiTheta}(3, 0, \imaginaryI)=\frac{\Gamma(\frac{1}{4})}{\sqrt{2}\pi^{\frac{3}{4}}}$

Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. 1403b5 · Fungrim entry ↗

$\mathrm{JacobiTheta}(1, z, \tau)\mathrm{JacobiTheta}(1, w, \tau)=\mathrm{JacobiTheta}(3, z+w, 2\tau)\mathrm{JacobiTheta}(2, z-w, 2\tau)-\mathrm{JacobiTheta}(2, z+w, 2\tau)\mathrm{JacobiTheta}(3, z-w, 2\tau)$

Holds when $z\in\C\land w\in\C\land\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. 1792a9 · Fungrim entry ↗

$\mathrm{JacobiTheta}(4, z, \tau+2n)=\mathrm{JacobiTheta}(4, z, \tau)$

Holds when $z\in\C\land\Im(\tau)\gt0\land n\in\Z$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. 19acd8 · Fungrim entry ↗

$\mathrm{JacobiTheta}(2, 0, 2\tau)(\mathrm{JacobiTheta}(1, z, \tau)^2-\mathrm{JacobiTheta}(2, z, \tau)^2)=\mathrm{JacobiTheta}(3, 0, 2\tau)(\mathrm{JacobiTheta}(4, z, \tau)^2-\mathrm{JacobiTheta}(3, z, \tau)^2)$

Holds when $z\in\C\land\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for expansion. 1c67c8 · Fungrim entry ↗

$\mathrm{JacobiTheta}(4, 0, \tau)^4-\mathrm{JacobiTheta}(2, 0, \tau)^4=1-24(\sum_{n=0}^{\infty}\frac{(2n+1)\exp(\pi\imaginaryI\tau)^{2n+1}}{1+\exp(\pi\imaginaryI\tau)^{2n+1}})$

Holds when $\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. 1cec67 · Fungrim entry ↗

$\mathrm{JacobiTheta}(1, z, \tau+n)=\exp(\frac{\pi\imaginaryI n}{4})\mathrm{JacobiTheta}(1, z, \tau)$

Holds when $z\in\C\land\Im(\tau)\gt0\land n\in\Z$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. 1fa8e7 · Fungrim entry ↗

$\mathrm{JacobiTheta}(3, 0, \tau)^4=\mathrm{JacobiTheta}(2, 0, \tau)^4+\mathrm{JacobiTheta}(4, 0, \tau)^4$

Holds when $\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. 1fbc09 · Fungrim entry ↗

$\mathrm{JacobiTheta}(2, z+w, \tau)\mathrm{JacobiTheta}(2, z-w, \tau)\mathrm{JacobiTheta}(2, 0, \tau)^2=\mathrm{JacobiTheta}(2, z, \tau)^2\mathrm{JacobiTheta}(2, w, \tau)^2-\mathrm{JacobiTheta}(1, z, \tau)^2\mathrm{JacobiTheta}(1, w, \tau)^2=\mathrm{JacobiTheta}(3, z, \tau)^2\mathrm{JacobiTheta}(3, w, \tau)^2-\mathrm{JacobiTheta}(4, z, \tau)^2\mathrm{JacobiTheta}(4, w, \tau)^2$

Holds when $z\in\C\land w\in\C\land\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. 1feda6 · Fungrim entry ↗

$\mathrm{JacobiTheta}(3, 2z, \tau)=\frac{\mathrm{JacobiTheta}(1, z, \tau)^4+\mathrm{JacobiTheta}(3, z, \tau)^4}{\mathrm{JacobiTheta}(3, 0, \tau)^3}$

Holds when $z\in\C\land\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. 20d581 · Fungrim entry ↗

$2\mathrm{JacobiTheta}(2, 0, 2\tau)^2=\mathrm{JacobiTheta}(3, 0, \tau)^2-\mathrm{JacobiTheta}(4, 0, \tau)^2$

Holds when $\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. 21c2f7 · Fungrim entry ↗

$\mathrm{JacobiTheta}(4, 2z, \tau)=\frac{\mathrm{JacobiTheta}(1, z, \tau)^2\mathrm{JacobiTheta}(2, z, \tau)^2+\mathrm{JacobiTheta}(3, z, \tau)^2\mathrm{JacobiTheta}(4, z, \tau)^2}{\mathrm{JacobiTheta}(4, 0, \tau)\mathrm{JacobiTheta}(3, 0, \tau)^2}$

Holds when $z\in\C\land\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. 21dc98 · Fungrim entry ↗

$z\mapsto\frac{\mathrm{JacobiTheta}(4, z, \tau)}{\mathrm{JacobiTheta}(3, z, \tau)}^{\prime}(z)=\frac{\pi\mathrm{JacobiTheta}(1, z, \tau)\mathrm{JacobiTheta}(2, z, \tau)\mathrm{JacobiTheta}(2, 0, \tau)^2}{\mathrm{JacobiTheta}(3, z, \tau)^2}$

Holds when $z\in\C\land\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. 23077c · Fungrim entry ↗

$\mathrm{JacobiTheta}(3, z, \tau)=\exp(\pi\imaginaryI(z+\frac{\tau}{4}))\mathrm{JacobiTheta}(1, z+\frac{1}{2}+\frac{\tau}{2}, \tau)$

Holds when $z\in\C\land\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. 235d0d · Fungrim entry ↗

$\mathrm{JacobiTheta}(2, 0, \tau)^2\mathrm{JacobiTheta}(4, z, \tau)^2=\mathrm{JacobiTheta}(3, 0, \tau)^2\mathrm{JacobiTheta}(1, z, \tau)^2+\mathrm{JacobiTheta}(4, 0, \tau)^2\mathrm{JacobiTheta}(2, z, \tau)^2$

Holds when $z\in\C\land\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. 265d9c · Fungrim entry ↗

$\frac{\mathrm{JacobiTheta}(1, 0, \tau, 3)}{\mathrm{JacobiTheta}(1, 0, \tau, 1)}=\frac{\mathrm{JacobiTheta}(2, 0, \tau, 2)}{\mathrm{JacobiTheta}(2, 0, \tau)}+\frac{\mathrm{JacobiTheta}(3, 0, \tau, 2)}{\mathrm{JacobiTheta}(3, 0, \tau)}+\frac{\mathrm{JacobiTheta}(4, 0, \tau, 2)}{\mathrm{JacobiTheta}(4, 0, \tau)}$

Holds when $\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. 278274 · Fungrim entry ↗

$\mathrm{JacobiTheta}(1, 4z, 4\tau)=\frac{\mathrm{JacobiTheta}(1, z, \tau)\mathrm{JacobiTheta}(1, 1/4-z, \tau)\mathrm{JacobiTheta}(1, \frac{1}{4}+z, \tau)\mathrm{JacobiTheta}(2, z, \tau)}{\mathrm{JacobiTheta}(3, 0, \tau)\mathrm{JacobiTheta}(4, 0, \tau)\mathrm{JacobiTheta}(3, \frac{1}{4}, \tau)}$

Holds when $z\in\C\land\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. 27b169 · Fungrim entry ↗

$z\mapsto\frac{\mathrm{JacobiTheta}(3, z, \tau)}{\mathrm{JacobiTheta}(2, z, \tau)}^{\prime}(z)=\frac{\pi\mathrm{JacobiTheta}(1, z, \tau)\mathrm{JacobiTheta}(4, z, \tau)\mathrm{JacobiTheta}(4, 0, \tau)^2}{\mathrm{JacobiTheta}(2, z, \tau)^2}$

Holds when $z\in\C\land\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. 2853d4 · Fungrim entry ↗

$\mathrm{JacobiTheta}(3, z, \tau+n)=\begin{cases}\mathrm{JacobiTheta}(3, z, \tau)&\mathrm{IsEven}(n)\\\mathrm{JacobiTheta}(4, z, \tau)&\mathrm{IsOdd}(n)\end{cases}$

Holds when $z\in\C\land\Im(\tau)\gt0\land n\in\Z$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. 28b4c3 · Fungrim entry ↗

$\mathrm{JacobiTheta}(3, z+\frac{\tau}{2}, \tau)=\exp(-(\pi\imaginaryI(z+\frac{\tau}{4})))\mathrm{JacobiTheta}(2, z, \tau)$

Holds when $z\in\C\land\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. 2d2dde · Fungrim entry ↗

$\mathrm{JacobiTheta}(3, z+m+n\tau, \tau)=\exp(-(\pi\imaginaryI(\tau n^2+2nz)))\mathrm{JacobiTheta}(3, z, \tau)$

Holds when $z\in\C\land\Im(\tau)\gt0\land m\in\Z\land n\in\Z$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for expansion. 2e4da0 · Fungrim entry ↗

$\mathrm{JacobiTheta}(1, \frac{n}{4}, \imaginaryI)=\begin{cases}0&\mathrm{CongruentMod}(n, 0, 4)\\(-1)^{\lfloor n/4\rfloor}\mathrm{JacobiTheta}(4, 0, \imaginaryI)&\mathrm{CongruentMod}(n, 2, 4)\\(-1)^{\lfloor n/4\rfloor}\frac{\sqrt{2^{1/2}-1}}{2^{7/16}}\sqrt[4]{\sqrt{2}+1}\mathrm{JacobiTheta}(3, 0, \imaginaryI)&\top\end{cases}$

Holds when $n\in\Z$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. 2f3ed3 · Fungrim entry ↗

$\mathrm{JacobiTheta}(1, z+2n, \tau)=\mathrm{JacobiTheta}(1, z, \tau)$

Holds when $z\in\C\land\Im(\tau)\gt0\land n\in\Z$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. 2faeb9 · Fungrim entry ↗

$\mathrm{JacobiTheta}(3, 0, 1+\frac{\imaginaryI}{2})=\frac{\mathrm{JacobiTheta}(3, 0, \imaginaryI){(2^{1/2}-1)}^{\frac{2}{3}}\sqrt[12]{4+3\sqrt{2}}}{2^{\frac{7}{24}}}$

Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. Reference: doi.org 324483 · Fungrim entry ↗

$\mathrm{JacobiTheta}(3, 2z, 2\tau)=\frac{\mathrm{JacobiTheta}(1, z, \tau)^2+\mathrm{JacobiTheta}(2, z, \tau)^2}{2\mathrm{JacobiTheta}(2, 0, 2\tau)}$

Holds when $z\in\C\land\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. 3479be · Fungrim entry ↗

$\mathrm{JacobiTheta}(3, 0, \tau)\mathrm{JacobiTheta}(4, 0, \tau)\mathrm{JacobiTheta}(1, z+w, \tau)\mathrm{JacobiTheta}(2, z-w, \tau)=\mathrm{JacobiTheta}(1, z, \tau)\mathrm{JacobiTheta}(2, z, \tau)\mathrm{JacobiTheta}(3, w, \tau)\mathrm{JacobiTheta}(4, w, \tau)+\mathrm{JacobiTheta}(3, z, \tau)\mathrm{JacobiTheta}(4, z, \tau)\mathrm{JacobiTheta}(1, w, \tau)\mathrm{JacobiTheta}(2, w, \tau)$

Holds when $z\in\C\land w\in\C\land\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. 34d1c6 · Fungrim entry ↗

$z\mapsto\frac{\mathrm{JacobiTheta}(3, z, \tau)}{\mathrm{JacobiTheta}(4, z, \tau)}^{\prime}(z)=-(\frac{\pi\mathrm{JacobiTheta}(1, z, \tau)\mathrm{JacobiTheta}(2, z, \tau)\mathrm{JacobiTheta}(2, 0, \tau)^2}{\mathrm{JacobiTheta}(4, z, \tau)^2})$

Holds when $z\in\C\land\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. 378949 · Fungrim entry ↗

$\tau\mapsto\mathrm{JacobiTheta}(j, z, \tau, s)^{\prime}(\tau)=\frac{\mathrm{JacobiTheta}(j, z, \tau, 2r+s)}{(4\imaginaryI\pi)^{r}}$

Holds when $j\in\lbrace1, 2, 3, 4\rbrace\land z\in\C\land\Im(\tau)\gt0\land r\in\N\land s\in\N$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. 37e644 · Fungrim entry ↗

$\mathrm{JacobiTheta}(3, -z, \tau)=\mathrm{JacobiTheta}(3, z, \tau)$

Holds when $z\in\C\land\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. 380076 · Fungrim entry ↗

$\mathrm{JacobiTheta}(3, 0, 1+10\imaginaryI)=\frac{\sqrt{5}\mathrm{JacobiTheta}(3, 0, \imaginaryI)\times2^{\frac{7}{8}}}{5(\sqrt[4]{5}-1)\sqrt{1+\sqrt{5}}}$

Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. Reference: doi.org 390158 · Fungrim entry ↗

$\mathrm{JacobiTheta}(2, 2z, \tau)=\frac{\mathrm{JacobiTheta}(2, z, \tau)^4-\mathrm{JacobiTheta}(1, z, \tau)^4}{\mathrm{JacobiTheta}(2, 0, \tau)^3}$

Holds when $z\in\C\land\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for expansion. 3a77e0 · Fungrim entry ↗

$\mathrm{JacobiTheta}(4, z+w, \tau)\mathrm{JacobiTheta}(4, z-w, \tau)\mathrm{JacobiTheta}(3, 0, \tau)^2=\mathrm{JacobiTheta}(1, z, \tau)^2\mathrm{JacobiTheta}(2, w, \tau)^2+\mathrm{JacobiTheta}(3, z, \tau)^2\mathrm{JacobiTheta}(4, w, \tau)^2=\mathrm{JacobiTheta}(2, z, \tau)^2\mathrm{JacobiTheta}(1, w, \tau)^2+\mathrm{JacobiTheta}(4, z, \tau)^2\mathrm{JacobiTheta}(3, w, \tau)^2$

Holds when $z\in\C\land w\in\C\land\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. 3cac28 · Fungrim entry ↗

$\mathrm{JacobiTheta}(3, \frac{n}{4}, \imaginaryI)=\begin{cases}\mathrm{JacobiTheta}(3, 0, \imaginaryI)&\mathrm{CongruentMod}(n, 0, 4)\\\mathrm{JacobiTheta}(4, 0, \imaginaryI)&\mathrm{CongruentMod}(n, 2, 4)\\\frac{(2^{1/2}+1)^{1/4}}{2^{7/16}}\mathrm{JacobiTheta}(3, 0, \imaginaryI)&\top\end{cases}$

Holds when $n\in\Z$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. 3fb309 · Fungrim entry ↗

$\mathrm{JacobiTheta}(3, 0, \frac{\imaginaryI}{2})=(\mathrm{JacobiTheta}(3, 0, \imaginaryI)\sqrt[4]{2}\sqrt{1+\sqrt{2}})/\sqrt{2}$

Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. Reference: doi.org 4256f0 · Fungrim entry ↗

$\mathrm{JacobiTheta}(4, z+\frac{\tau}{2}, \tau)=\exp(-(\pi\imaginaryI(z+\frac{\tau}{4})))\imaginaryI\mathrm{JacobiTheta}(1, z, \tau)$

Holds when $z\in\C\land\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. 429093 · Fungrim entry ↗

$\mathrm{JacobiTheta}(1, z+m+n\tau, \tau)=(-1)^{m+n}\exp(-(\pi\imaginaryI(\tau n^2+2nz)))\mathrm{JacobiTheta}(1, z, \tau)$

Holds when $z\in\C\land\Im(\tau)\gt0\land m\in\Z\land n\in\Z$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. 43fa0e · Fungrim entry ↗

$\mathrm{JacobiTheta}(4, z+n, \tau)=\mathrm{JacobiTheta}(4, z, \tau)$

Holds when $z\in\C\land\Im(\tau)\gt0\land n\in\Z$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. 4448f1 · Fungrim entry ↗

$\mathrm{JacobiTheta}(1, z+w, \tau)\mathrm{JacobiTheta}(1, z-w, \tau)\mathrm{JacobiTheta}(2, 0, \tau)^2=\mathrm{JacobiTheta}(1, z, \tau)^2\mathrm{JacobiTheta}(2, w, \tau)^2-\mathrm{JacobiTheta}(2, z, \tau)^2\mathrm{JacobiTheta}(1, w, \tau)^2=\mathrm{JacobiTheta}(4, z, \tau)^2\mathrm{JacobiTheta}(3, w, \tau)^2-\mathrm{JacobiTheta}(3, z, \tau)^2\mathrm{JacobiTheta}(4, w, \tau)^2$

Holds when $z\in\C\land w\in\C\land\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. 45165c · Fungrim entry ↗

$\mathrm{JacobiTheta}(4, z+w, \tau)\mathrm{JacobiTheta}(4, z-w, \tau)\mathrm{JacobiTheta}(4, 0, \tau)^2=\mathrm{JacobiTheta}(3, z, \tau)^2\mathrm{JacobiTheta}(3, w, \tau)^2-\mathrm{JacobiTheta}(2, z, \tau)^2\mathrm{JacobiTheta}(2, w, \tau)^2=\mathrm{JacobiTheta}(4, z, \tau)^2\mathrm{JacobiTheta}(4, w, \tau)^2-\mathrm{JacobiTheta}(1, z, \tau)^2\mathrm{JacobiTheta}(1, w, \tau)^2$

Holds when $z\in\C\land w\in\C\land\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. 45a130 · Fungrim entry ↗

$2\mathrm{JacobiTheta}(1, 0, 2\tau, 1)\mathrm{JacobiTheta}(4, 0, 2\tau)=\mathrm{JacobiTheta}(1, 0, \tau, 1)\mathrm{JacobiTheta}(2, 0, \tau)$

Holds when $\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. 46f244 · Fungrim entry ↗

$\mathrm{JacobiTheta}(2, 0, \tau, 2r+1)=0$

Holds when $\Im(\tau)\gt0\land r\in\N$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. 474c51 · Fungrim entry ↗

$\mathrm{JacobiTheta}(3, 0, \frac{\tau}{2})\mathrm{JacobiTheta}(4, 0, \frac{\tau}{2})=\mathrm{JacobiTheta}(4, 0, \tau)^2$

Holds when $\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. 476642 · Fungrim entry ↗

$\mathrm{JacobiTheta}(2, 0, \tau)\mathrm{JacobiTheta}(4, 0, \tau)\mathrm{JacobiTheta}(1, z+w, \tau)\mathrm{JacobiTheta}(3, z-w, \tau)=\mathrm{JacobiTheta}(1, z, \tau)\mathrm{JacobiTheta}(3, z, \tau)\mathrm{JacobiTheta}(2, w, \tau)\mathrm{JacobiTheta}(4, w, \tau)+\mathrm{JacobiTheta}(2, z, \tau)\mathrm{JacobiTheta}(4, z, \tau)\mathrm{JacobiTheta}(1, w, \tau)\mathrm{JacobiTheta}(3, w, \tau)$

Holds when $z\in\C\land w\in\C\land\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. 47e587 · Fungrim entry ↗

$\mathrm{JacobiTheta}(2, 0, y\imaginaryI)=\mathrm{JacobiTheta}(3, 0, \frac{\imaginaryI}{y}+1)/\sqrt{y}$

Holds when $y\in\lparen0, \infty\rparen$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. 47f4ba · Fungrim entry ↗

$\mathrm{JacobiTheta}(2, z+\frac{1}{2}, \tau)=-\mathrm{JacobiTheta}(1, z, \tau)$

Holds when $z\in\C\land\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for expansion. 47f6dd · Fungrim entry ↗

$\mathrm{JacobiTheta}(3, 0, 5\imaginaryI)=\frac{\sqrt{5}\mathrm{JacobiTheta}(3, 0, \imaginaryI)}{5\sqrt{5^{1/2}-2}}$

Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. Reference: doi.org 483e7e · Fungrim entry ↗

$\mathrm{JacobiTheta}(2, z+w, \tau)\mathrm{JacobiTheta}(2, z-w, \tau)\mathrm{JacobiTheta}(4, 0, \tau)^2=\mathrm{JacobiTheta}(4, z, \tau)^2\mathrm{JacobiTheta}(2, w, \tau)^2-\mathrm{JacobiTheta}(1, z, \tau)^2\mathrm{JacobiTheta}(3, w, \tau)^2=\mathrm{JacobiTheta}(2, z, \tau)^2\mathrm{JacobiTheta}(4, w, \tau)^2-\mathrm{JacobiTheta}(3, z, \tau)^2\mathrm{JacobiTheta}(1, w, \tau)^2$

Holds when $z\in\C\land w\in\C\land\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. 48a1c6 · Fungrim entry ↗

$\mathrm{JacobiTheta}(1, z, \tau)=-\imaginaryI\exp(\pi\imaginaryI(z+\frac{\tau}{4}))\mathrm{JacobiTheta}(4, z+\frac{\tau}{2}, \tau)$

Holds when $z\in\C\land\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. 4c462b · Fungrim entry ↗

$\mathrm{JacobiTheta}(3, 0, 1+\imaginaryI)=\frac{\mathrm{JacobiTheta}(3, 0, \imaginaryI)}{\sqrt[4]{2}}$

Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. Reference: doi.org 4c8873 · Fungrim entry ↗

$\mathrm{JacobiTheta}(2, z, \tau+4n)=(-1)^{n}\mathrm{JacobiTheta}(2, z, \tau)$

Holds when $z\in\C\land\Im(\tau)\gt0\land n\in\Z$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for expansion. 4cf228 · Fungrim entry ↗

$\mathrm{JacobiTheta}(4, 0, \tau)^8=1+16(\sum_{n=1}^{\infty}\frac{(-1)^{n}n^3\exp(\pi\imaginaryI\tau)^{n}}{1-\exp(\pi\imaginaryI\tau)^{n}})$

Holds when $\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. 4d26ec · Fungrim entry ↗

$\mathrm{JacobiTheta}(4, -z, \tau)=\mathrm{JacobiTheta}(4, z, \tau)$

Holds when $z\in\C\land\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. 4f939e · Fungrim entry ↗

$\mathrm{JacobiTheta}(3, 0, \frac{\imaginaryI}{3})=\sqrt[4]{2\sqrt{3}+3}\mathrm{JacobiTheta}(3, 0, \imaginaryI)$

Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. Reference: doi.org 52302f · Fungrim entry ↗

$\mathrm{JacobiTheta}(3, 0, 1+6\imaginaryI)=\frac{\mathrm{JacobiTheta}(3, 0, \imaginaryI)\sqrt[3]{1+\sqrt{3}+\sqrt{2}\sqrt[4]{27}}}{2^{\frac{11}{24}}\times3^{\frac{3}{8}}\sqrt[6]{3^{1/2}-1}}$

Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. Reference: doi.org 5384f3 · Fungrim entry ↗

$\mathrm{JacobiTheta}(3, 2z, 4\tau)=\frac{1}{2}(\mathrm{JacobiTheta}(3, z, \tau)+\mathrm{JacobiTheta}(4, z, \tau))$

Holds when $z\in\C\land\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. 53fef4 · Fungrim entry ↗

$\mathrm{JacobiTheta}(2, 0, \tau)\mathrm{JacobiTheta}(3, 0, \tau)\mathrm{JacobiTheta}(4, 0, \tau)=2\mathrm{DedekindEta}(\tau)^3$

$\mathrm{JacobiTheta}(1, z+\frac{1}{2}, \tau)=\mathrm{JacobiTheta}(2, z, \tau)$

Holds when $z\in\C\land\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. 563d18 · Fungrim entry ↗

$\mathrm{JacobiTheta}(3, z+w, \tau)\mathrm{JacobiTheta}(3, z-w, \tau)\mathrm{JacobiTheta}(4, 0, \tau)^2=\mathrm{JacobiTheta}(4, z, \tau)^2\mathrm{JacobiTheta}(3, w, \tau)^2-\mathrm{JacobiTheta}(1, z, \tau)^2\mathrm{JacobiTheta}(2, w, \tau)^2=\mathrm{JacobiTheta}(3, z, \tau)^2\mathrm{JacobiTheta}(4, w, \tau)^2-\mathrm{JacobiTheta}(2, z, \tau)^2\mathrm{JacobiTheta}(1, w, \tau)^2$

Holds when $z\in\C\land w\in\C\land\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. 5752b8 · Fungrim entry ↗

$\mathrm{JacobiTheta}(1, 0, \frac{\tau}{2}, 1)\mathrm{JacobiTheta}(2, 0, \frac{\tau}{2})=2\mathrm{JacobiTheta}(1, 0, \tau, 1)\mathrm{JacobiTheta}(4, 0, \tau)$

Holds when $\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. 59184e · Fungrim entry ↗

$\mathrm{JacobiTheta}(1, -z, \tau)=-\mathrm{JacobiTheta}(1, z, \tau)$

Holds when $z\in\C\land\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for expansion. 59f8e1 · Fungrim entry ↗

$\mathrm{JacobiTheta}(2, 0, \frac{\tau}{2})^2=2\mathrm{JacobiTheta}(2, 0, \tau)\mathrm{JacobiTheta}(3, 0, \tau)$

Holds when $\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. 59fd23 · Fungrim entry ↗

$\mathrm{JacobiTheta}(1, z, \tau)^4-\mathrm{JacobiTheta}(2, z, \tau)^4=\mathrm{JacobiTheta}(4, z, \tau)^4-\mathrm{JacobiTheta}(3, z, \tau)^4$

Holds when $z\in\C\land\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for expansion. 5a3ebf · Fungrim entry ↗

$\mathrm{JacobiTheta}(1, z+n, \tau)=(-1)^{n}\mathrm{JacobiTheta}(1, z, \tau)$

Holds when $z\in\C\land\Im(\tau)\gt0\land n\in\Z$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. 5cdae6 · Fungrim entry ↗

$\mathrm{JacobiTheta}(4, z, \tau)=\mathrm{JacobiTheta}(3, z+\frac{1}{2}, \tau)$

Holds when $z\in\C\land\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. 5d41b1 · Fungrim entry ↗

$\mathrm{JacobiTheta}(1, z, \tau)\mathrm{JacobiTheta}(2, w, \tau)=\mathrm{JacobiTheta}(1, z+w, 2\tau)\mathrm{JacobiTheta}(4, z-w, 2\tau)+\mathrm{JacobiTheta}(4, z+w, 2\tau)\mathrm{JacobiTheta}(1, z-w, 2\tau)$

Holds when $z\in\C\land w\in\C\land\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. 5f9e54 · Fungrim entry ↗

$\mathrm{JacobiTheta}(1, 2z, \tau)=\frac{2\mathrm{JacobiTheta}(1, z, \tau)\mathrm{JacobiTheta}(2, z, \tau)\mathrm{JacobiTheta}(3, z, \tau)\mathrm{JacobiTheta}(4, z, \tau)}{\mathrm{JacobiTheta}(2, 0, \tau)\mathrm{JacobiTheta}(3, 0, \tau)\mathrm{JacobiTheta}(4, 0, \tau)}$

Holds when $z\in\C\land\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. 5fe58d · Fungrim entry ↗

$z\mapsto\frac{\mathrm{JacobiTheta}(2, z, \tau)}{\mathrm{JacobiTheta}(3, z, \tau)}^{\prime}(z)=-(\frac{\pi\mathrm{JacobiTheta}(1, z, \tau)\mathrm{JacobiTheta}(4, z, \tau)\mathrm{JacobiTheta}(4, 0, \tau)^2}{\mathrm{JacobiTheta}(3, z, \tau)^2})$

Holds when $z\in\C\land\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. 64b65d · Fungrim entry ↗

$\mathrm{JacobiTheta}(4, z, \tau+n)=\begin{cases}\mathrm{JacobiTheta}(4, z, \tau)&\mathrm{IsEven}(n)\\\mathrm{JacobiTheta}(3, z, \tau)&\mathrm{IsOdd}(n)\end{cases}$

Holds when $z\in\C\land\Im(\tau)\gt0\land n\in\Z$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. 64f0a5 · Fungrim entry ↗

$\mathrm{JacobiTheta}(1, z+w, \tau)\mathrm{JacobiTheta}(1, z-w, \tau)\mathrm{JacobiTheta}(4, 0, \tau)^2=\mathrm{JacobiTheta}(3, z, \tau)^2\mathrm{JacobiTheta}(2, w, \tau)^2-\mathrm{JacobiTheta}(2, z, \tau)^2\mathrm{JacobiTheta}(3, w, \tau)^2=\mathrm{JacobiTheta}(1, z, \tau)^2\mathrm{JacobiTheta}(4, w, \tau)^2-\mathrm{JacobiTheta}(4, z, \tau)^2\mathrm{JacobiTheta}(1, w, \tau)^2$

Holds when $z\in\C\land w\in\C\land\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. 663a02 · Fungrim entry ↗

$\mathrm{JacobiTheta}(3, 0, 6\imaginaryI)=\frac{\mathrm{JacobiTheta}(3, 0, \imaginaryI)\sqrt[3]{-4+2\sqrt{2}\times3^{3/4}+2\sqrt{3}+3\sqrt{2}-3^{3/4}+3^{5/4}}}{2\times3^{\frac{3}{8}}\sqrt[6]{(2^{1/2}-1)(3^{1/2}-1)}}$

Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. Reference: doi.org 669765 · Fungrim entry ↗

$\mathrm{JacobiTheta}(1, z, \frac{\tau}{2})=\frac{2\mathrm{JacobiTheta}(1, z, \tau)\mathrm{JacobiTheta}(4, z, \tau)}{\mathrm{JacobiTheta}(2, 0, \frac{\tau}{2})}$

Holds when $z\in\C\land\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. 66eb8b · Fungrim entry ↗

$\mathrm{JacobiTheta}(3, z+w, \tau)\mathrm{JacobiTheta}(3, z-w, \tau)\mathrm{JacobiTheta}(2, 0, \tau)^2=\mathrm{JacobiTheta}(3, z, \tau)^2\mathrm{JacobiTheta}(2, w, \tau)^2+\mathrm{JacobiTheta}(4, z, \tau)^2\mathrm{JacobiTheta}(1, w, \tau)^2=\mathrm{JacobiTheta}(2, z, \tau)^2\mathrm{JacobiTheta}(3, w, \tau)^2+\mathrm{JacobiTheta}(1, z, \tau)^2\mathrm{JacobiTheta}(4, w, \tau)^2$

Holds when $z\in\C\land w\in\C\land\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. 66efb8 · Fungrim entry ↗

$\mathrm{JacobiTheta}(3, 0, 1+12\imaginaryI)=\frac{\mathrm{JacobiTheta}(3, 0, \imaginaryI)\times2^{\frac{-19}{48}}\times3^{\frac{-3}{8}}\sqrt[3]{2-3\sqrt{2}+3^{5/4}+3^{3/4}}}{\sqrt[3]{-1+\sqrt{2}\times3^{3/4}-\sqrt{3}}\sqrt[12]{2^{1/2}-1}\sqrt[6]{1+\sqrt{3}}}$

Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. Reference: doi.org 675f23 · Fungrim entry ↗

$\mathrm{JacobiTheta}(4, 2z, 2\tau)=\frac{\mathrm{JacobiTheta}(3, z, \tau)\mathrm{JacobiTheta}(4, z, \tau)}{\mathrm{JacobiTheta}(4, 0, 2\tau)}$

Holds when $z\in\C\land\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. 686ce0 · Fungrim entry ↗

$\mathrm{JacobiTheta}(3, z, \frac{\tau}{2})=\frac{\mathrm{JacobiTheta}(2, z, \tau)^2+\mathrm{JacobiTheta}(3, z, \tau)^2}{\mathrm{JacobiTheta}(3, 0, \frac{\tau}{2})}$

Holds when $z\in\C\land\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. 69b32e · Fungrim entry ↗

$\mathrm{JacobiTheta}(2, z, \tau)=\exp(\pi\imaginaryI(z+\frac{\tau}{4}))\mathrm{JacobiTheta}(4, z+\frac{1}{2}+\frac{\tau}{2}, \tau)$

Holds when $z\in\C\land\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. 6a7704 · Fungrim entry ↗

$\mathrm{JacobiTheta}(3, 0, 45\imaginaryI)=\frac{\sqrt{10}(3+\sqrt{5}+(\sqrt{3}+\sqrt{5}+\sqrt[4]{60})\sqrt[3]{2+\sqrt{3}})\mathrm{JacobiTheta}(3, 0, \imaginaryI)}{30\sqrt{1+\sqrt{5}}}$

Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. Reference: doi.org 6ade92 · Fungrim entry ↗

$\mathrm{JacobiTheta}(1, z, \tau+1)=\exp(\frac{\pi\imaginaryI}{4})\mathrm{JacobiTheta}(1, z, \tau)$

Holds when $z\in\C\land\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. 6b2078 · Fungrim entry ↗

$\mathrm{JacobiTheta}(3, 0, 1+4\imaginaryI)=\frac{\sqrt[4]{\sqrt{2}+1}}{2^{\frac{7}{16}}}\mathrm{JacobiTheta}(3, 0, \imaginaryI)$

Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. Reference: doi.org 6cbce8 · Fungrim entry ↗

$\mathrm{JacobiTheta}(4, z, \tau)=-\imaginaryI\exp(\pi\imaginaryI(z+\frac{\tau}{4}))\mathrm{JacobiTheta}(1, z+\frac{\tau}{2}, \tau)$

Holds when $z\in\C\land\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. 6d918c · Fungrim entry ↗

$\mathrm{JacobiTheta}(3, 0, \tau)^2\mathrm{JacobiTheta}(3, z, \tau)^2=\mathrm{JacobiTheta}(4, 0, \tau)^2\mathrm{JacobiTheta}(4, z, \tau)^2+\mathrm{JacobiTheta}(2, 0, \tau)^2\mathrm{JacobiTheta}(2, z, \tau)^2$

Holds when $z\in\C\land\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. 6fad93 · Fungrim entry ↗

$\mathrm{JacobiTheta}(4, 2z, \tau)=\frac{\mathrm{JacobiTheta}(3, z, \tau)^4-\mathrm{JacobiTheta}(2, z, \tau)^4}{\mathrm{JacobiTheta}(4, 0, \tau)^3}$

Holds when $z\in\C\land\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for expansion. 7131cd · Fungrim entry ↗

$\mathrm{JacobiTheta}(2, 2z, 2\tau)=\frac{\mathrm{JacobiTheta}(2, z, \tau)^2-\mathrm{JacobiTheta}(1, z, \tau)^2}{2\mathrm{JacobiTheta}(3, 0, 2\tau)}$

Holds when $z\in\C\land\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. 7137a2 · Fungrim entry ↗

$z\mapsto\frac{\mathrm{JacobiTheta}(2, z, \tau)}{\mathrm{JacobiTheta}(1, z, \tau)}^{\prime}(z)=-(\frac{\pi\mathrm{JacobiTheta}(3, z, \tau)\mathrm{JacobiTheta}(4, z, \tau)\mathrm{JacobiTheta}(2, 0, \tau)^2}{\mathrm{JacobiTheta}(1, z, \tau)^2})$

Holds when $z\in\C\land\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. 713b6b · Fungrim entry ↗

$\mathrm{JacobiTheta}(3, z, \tau)=\exp(\pi\imaginaryI(z+\frac{\tau}{4}))\mathrm{JacobiTheta}(2, z+\frac{\tau}{2}, \tau)$

Holds when $z\in\C\land\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. 71d5ee · Fungrim entry ↗

$\mathrm{JacobiTheta}(3, 0, 7\imaginaryI)=(\mathrm{JacobiTheta}(3, 0, \imaginaryI)\sqrt{(\sqrt{7+3\times7^{1/2}}+\sqrt{13+7^{1/2}})\sqrt[8]{28}})/\sqrt{14}$

Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. Reference: doi.org 72f583 · Fungrim entry ↗

$\mathrm{JacobiTheta}(4, z, \tau)\mathrm{JacobiTheta}(4, w, \tau)=\mathrm{JacobiTheta}(3, z+w, 2\tau)\mathrm{JacobiTheta}(3, z-w, 2\tau)-\mathrm{JacobiTheta}(2, z+w, 2\tau)\mathrm{JacobiTheta}(2, z-w, 2\tau)$

Holds when $z\in\C\land w\in\C\land\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. 73eb5d · Fungrim entry ↗

$\mathrm{JacobiTheta}(4, 0, \frac{\tau}{2})^2=\mathrm{JacobiTheta}(3, 0, \tau)^2-\mathrm{JacobiTheta}(2, 0, \tau)^2$

Holds when $\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. 7527f1 · Fungrim entry ↗

$\mathrm{JacobiTheta}(1, z+w, \tau)\mathrm{JacobiTheta}(1, z-w, \tau)\mathrm{JacobiTheta}(3, 0, \tau)^2=\mathrm{JacobiTheta}(1, z, \tau)^2\mathrm{JacobiTheta}(3, w, \tau)^2-\mathrm{JacobiTheta}(3, z, \tau)^2\mathrm{JacobiTheta}(1, w, \tau)^2=\mathrm{JacobiTheta}(4, z, \tau)^2\mathrm{JacobiTheta}(2, w, \tau)^2-\mathrm{JacobiTheta}(2, z, \tau)^2\mathrm{JacobiTheta}(4, w, \tau)^2$

Holds when $z\in\C\land w\in\C\land\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. 75cb8c · Fungrim entry ↗

$\mathrm{JacobiTheta}(3, z, \tau+2n)=\mathrm{JacobiTheta}(3, z, \tau)$

Holds when $z\in\C\land\Im(\tau)\gt0\land n\in\Z$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. 772c88 · Fungrim entry ↗

$z\mapsto\frac{\mathrm{JacobiTheta}(4, z, \tau)}{\mathrm{JacobiTheta}(2, z, \tau)}^{\prime}(z)=\frac{\pi\mathrm{JacobiTheta}(1, z, \tau)\mathrm{JacobiTheta}(3, z, \tau)\mathrm{JacobiTheta}(3, 0, \tau)^2}{\mathrm{JacobiTheta}(2, z, \tau)^2}$

Holds when $z\in\C\land\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. 775637 · Fungrim entry ↗

$\mathrm{JacobiTheta}(3, 2z, \tau)=\frac{\mathrm{JacobiTheta}(2, z, \tau)^2\mathrm{JacobiTheta}(3, z, \tau)^2+\mathrm{JacobiTheta}(1, z, \tau)^2\mathrm{JacobiTheta}(4, z, \tau)^2}{\mathrm{JacobiTheta}(3, 0, \tau)\mathrm{JacobiTheta}(2, 0, \tau)^2}$

Holds when $z\in\C\land\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. 794106 · Fungrim entry ↗

$\mathrm{JacobiTheta}(3, 0, \sqrt{6}\imaginaryI)=\sqrt{\frac{1}{\pi}(2\mathrm{EllipticK}((2-3^{1/2})^2(2^{1/2}-3^{1/2})^2))}$

Symbols: EllipticK — Legendre complete elliptic integral of the first kind; JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. Reference: mathworld.wolfram.com 799b5e · Fungrim entry ↗

$\mathrm{JacobiTheta}(2, 0, \imaginaryI)=\mathrm{JacobiTheta}(4, 0, \imaginaryI)=\frac{\mathrm{JacobiTheta}(3, 0, \imaginaryI)}{\sqrt[4]{2}}$

Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. 7d7c65 · Fungrim entry ↗

$\mathrm{JacobiTheta}(3, 2z, 2\tau)=\frac{\mathrm{JacobiTheta}(3, z, \tau)^2+\mathrm{JacobiTheta}(4, z, \tau)^2}{2\mathrm{JacobiTheta}(3, 0, 2\tau)}$

Holds when $z\in\C\land\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. 7e0002 · Fungrim entry ↗

$\mathrm{JacobiTheta}(3, 0, \frac{\imaginaryI}{4})=\frac{(1+\frac{1}{\sqrt[4]{2}})\mathrm{JacobiTheta}(3, 0, \imaginaryI)\sqrt{1+\sqrt{2}}}{\sqrt{1+\sqrt{2}}}$

Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. Reference: doi.org 7f9273 · Fungrim entry ↗

$\mathrm{JacobiTheta}(4, 0, 1+y\imaginaryI)=\mathrm{JacobiTheta}(3, 0, y\imaginaryI)$

Holds when $y\in\lparen0, \infty\rparen$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. 81550a · Fungrim entry ↗

$\mathrm{JacobiTheta}(3, 0, 9\imaginaryI)=\frac{1}{3}((1+\sqrt[3]{2(\sqrt{3}+1)})\mathrm{JacobiTheta}(3, 0, \imaginaryI))$

Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. Reference: doi.org 8356db · Fungrim entry ↗

$\mathrm{JacobiTheta}(3, 0, \tau)=\frac{\mathrm{DedekindEta}((\tau+1)/2)^2}{\mathrm{DedekindEta}(\tau+1)}$

$z\mapsto\frac{\mathrm{JacobiTheta}(2, z, \tau)}{\mathrm{JacobiTheta}(4, z, \tau)}^{\prime}(z)=-(\frac{\pi\mathrm{JacobiTheta}(1, z, \tau)\mathrm{JacobiTheta}(3, z, \tau)\mathrm{JacobiTheta}(3, 0, \tau)^2}{\mathrm{JacobiTheta}(4, z, \tau)^2})$

Holds when $z\in\C\land\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. 89985a · Fungrim entry ↗

$\mathrm{JacobiTheta}(2, z+w, \tau)\mathrm{JacobiTheta}(2, z-w, \tau)\mathrm{JacobiTheta}(3, 0, \tau)^2=\mathrm{JacobiTheta}(2, z, \tau)^2\mathrm{JacobiTheta}(3, w, \tau)^2-\mathrm{JacobiTheta}(4, z, \tau)^2\mathrm{JacobiTheta}(1, w, \tau)^2=\mathrm{JacobiTheta}(3, z, \tau)^2\mathrm{JacobiTheta}(2, w, \tau)^2-\mathrm{JacobiTheta}(1, z, \tau)^2\mathrm{JacobiTheta}(4, w, \tau)^2$

Holds when $z\in\C\land w\in\C\land\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. 89c9e4 · Fungrim entry ↗

$\mathrm{JacobiTheta}(3, 0, \tau)^4=1+8(\sum_{n=0}^{\infty}\frac{2n\exp(\pi\imaginaryI\tau)^{2n}}{1+\exp(\pi\imaginaryI\tau)^{2n}})+8(\sum_{n=0}^{\infty}\frac{(2n+1)\exp(\pi\imaginaryI\tau)^{2n+1}}{1-\exp(\pi\imaginaryI\tau)^{2n+1}})$

Holds when $\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. 8a316c · Fungrim entry ↗

$\mathrm{JacobiTheta}(4, z, \tau)=1+2(\sum_{n=1}^{\infty}(-1)^{n}\exp(\pi\imaginaryI\tau)^{n^2}\cos(2n\pi z))$

Holds when $z\in\C\land\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. 8a34d1 · Fungrim entry ↗

$\mathrm{JacobiTheta}(4, 2z, \tau)=\frac{\mathrm{JacobiTheta}(4, z, \tau)^4-\mathrm{JacobiTheta}(1, z, \tau)^4}{\mathrm{JacobiTheta}(4, 0, \tau)^3}$

Holds when $z\in\C\land\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for expansion. 8b825c · Fungrim entry ↗

$\mathrm{JacobiTheta}(4, \frac{n}{4}, \imaginaryI)=\begin{cases}\mathrm{JacobiTheta}(4, 0, \imaginaryI)&\mathrm{CongruentMod}(n, 0, 4)\\\mathrm{JacobiTheta}(3, 0, \imaginaryI)&\mathrm{CongruentMod}(n, 2, 4)\\\frac{(2^{1/2}+1)^{1/4}}{2^{7/16}}\mathrm{JacobiTheta}(3, 0, \imaginaryI)&\top\end{cases}$

Holds when $n\in\Z$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. 8c4ab4 · Fungrim entry ↗

$\mathrm{JacobiTheta}(4, z+m+n\tau, \tau)=(-1)^{n}\exp(-(\pi\imaginaryI(\tau n^2+2nz)))\mathrm{JacobiTheta}(4, z, \tau)$

Holds when $z\in\C\land\Im(\tau)\gt0\land m\in\Z\land n\in\Z$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for expansion. 8d6a1d · Fungrim entry ↗

$\mathrm{JacobiTheta}(1, 0, \tau)=0$

Holds when $\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. 8f43ab · Fungrim entry ↗

$\mathrm{JacobiTheta}(4, 2z, \tau)=\frac{\mathrm{JacobiTheta}(1, z, \tau)^2\mathrm{JacobiTheta}(3, z, \tau)^2+\mathrm{JacobiTheta}(2, z, \tau)^2\mathrm{JacobiTheta}(4, z, \tau)^2}{\mathrm{JacobiTheta}(4, 0, \tau)\mathrm{JacobiTheta}(2, 0, \tau)^2}$

Holds when $z\in\C\land\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. 931201 · Fungrim entry ↗

$(30{\tau\mapsto\mathrm{JacobiTheta}(j, 0, \tau)^{\prime}(\tau)}^3-15\tau\mapsto\mathrm{JacobiTheta}(j, 0, \tau)^{(0)}(\tau)\tau\mapsto\mathrm{JacobiTheta}(j, 0, \tau)^{\prime}(\tau)\tau\mapsto\mathrm{JacobiTheta}(j, 0, \tau)^{\doubleprime}(\tau)+\tau\mapsto\mathrm{JacobiTheta}(j, 0, \tau)^{(0)}(\tau)^2\tau\mapsto\mathrm{JacobiTheta}(j, 0, \tau)^{\tripleprime}(\tau))^2+32{(\tau\mapsto\mathrm{JacobiTheta}(j, 0, \tau)^{(0)}(\tau)\tau\mapsto\mathrm{JacobiTheta}(j, 0, \tau)^{\doubleprime}(\tau)-3\tau\mapsto\mathrm{JacobiTheta}(j, 0, \tau)^{\prime}(\tau)^2)}^3+\pi^2(\tau\mapsto\mathrm{JacobiTheta}(j, 0, \tau)^{(0)}(\tau)\tau\mapsto\mathrm{JacobiTheta}(j, 0, \tau)^{\doubleprime}(\tau)-3\tau\mapsto\mathrm{JacobiTheta}(j, 0, \tau)^{\prime}(\tau)^2)^2{\tau\mapsto\mathrm{JacobiTheta}(j, 0, \tau)^{(0)}(\tau)}^{10}=0$

Holds when $j\in\lbrace1, 2, 3, 4\rbrace\land\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. 936694 · Fungrim entry ↗

$\mathrm{JacobiTheta}(4, 0, \tau)=\frac{1}{\mathrm{DedekindEta}(\tau)}(\mathrm{DedekindEta}(\tau/2)^2)$

$\mathrm{JacobiTheta}(1, z, \tau)=-\mathrm{JacobiTheta}(2, z+\frac{1}{2}, \tau)$

Holds when $z\in\C\land\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. 95988c · Fungrim entry ↗

$\mathrm{JacobiTheta}(4, z, \frac{\tau}{2})=\frac{\mathrm{JacobiTheta}(3, z, \tau)^2-\mathrm{JacobiTheta}(2, z, \tau)^2}{\mathrm{JacobiTheta}(4, 0, \frac{\tau}{2})}$

Holds when $z\in\C\land\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. 95e508 · Fungrim entry ↗

$\mathrm{JacobiTheta}(3, 0, 4\imaginaryI)=\frac{1}{2}((1+2^{-(1/4)})\mathrm{JacobiTheta}(3, 0, \imaginaryI))$

Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. Reference: doi.org 95e9e4 · Fungrim entry ↗

$\mathrm{JacobiTheta}(2, 0, \tau)\mathrm{JacobiTheta}(4, 0, \tau)\mathrm{JacobiTheta}(2, z+w, \tau)\mathrm{JacobiTheta}(4, z-w, \tau)=\mathrm{JacobiTheta}(2, z, \tau)\mathrm{JacobiTheta}(4, z, \tau)\mathrm{JacobiTheta}(2, w, \tau)\mathrm{JacobiTheta}(4, w, \tau)-\mathrm{JacobiTheta}(1, z, \tau)\mathrm{JacobiTheta}(3, z, \tau)\mathrm{JacobiTheta}(1, w, \tau)\mathrm{JacobiTheta}(3, w, \tau)$

Holds when $z\in\C\land w\in\C\land\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. 9973ef · Fungrim entry ↗

$2\mathrm{JacobiTheta}(2, 0, 2\tau)\mathrm{JacobiTheta}(3, 0, 2\tau)=\mathrm{JacobiTheta}(2, 0, \tau)^2$

Holds when $\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. 9a2054 · Fungrim entry ↗

$\mathrm{JacobiTheta}(2, z, \tau)\mathrm{JacobiTheta}(2, w, \tau)=\mathrm{JacobiTheta}(2, z+w, 2\tau)\mathrm{JacobiTheta}(3, z-w, 2\tau)+\mathrm{JacobiTheta}(3, z+w, 2\tau)\mathrm{JacobiTheta}(2, z-w, 2\tau)$

Holds when $z\in\C\land w\in\C\land\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. 9a9487 · Fungrim entry ↗

$\mathrm{JacobiTheta}(3, z+w, \tau)\mathrm{JacobiTheta}(3, z-w, \tau)\mathrm{JacobiTheta}(3, 0, \tau)^2=\mathrm{JacobiTheta}(1, z, \tau)^2\mathrm{JacobiTheta}(1, w, \tau)^2+\mathrm{JacobiTheta}(3, z, \tau)^2\mathrm{JacobiTheta}(3, w, \tau)^2=\mathrm{JacobiTheta}(2, z, \tau)^2\mathrm{JacobiTheta}(2, w, \tau)^2+\mathrm{JacobiTheta}(4, z, \tau)^2\mathrm{JacobiTheta}(4, w, \tau)^2$

Holds when $z\in\C\land w\in\C\land\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. 9aa437 · Fungrim entry ↗

$\mathrm{JacobiTheta}(3, z, \tau+1)=\mathrm{JacobiTheta}(4, z, \tau)$

Holds when $z\in\C\land\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. 9c1e9a · Fungrim entry ↗

$z\mapsto\frac{\mathrm{JacobiTheta}(4, z, \tau)}{\mathrm{JacobiTheta}(1, z, \tau)}^{\prime}(z)=-(\frac{\pi\mathrm{JacobiTheta}(2, z, \tau)\mathrm{JacobiTheta}(3, z, \tau)\mathrm{JacobiTheta}(4, 0, \tau)^2}{\mathrm{JacobiTheta}(1, z, \tau)^2})$

Holds when $z\in\C\land\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. a0552b · Fungrim entry ↗

$\mathrm{JacobiTheta}(2, 2z, 4\tau)=\frac{1}{2}(\mathrm{JacobiTheta}(3, z, \tau)-\mathrm{JacobiTheta}(4, z, \tau))$

Holds when $z\in\C\land\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. a0a1ee · Fungrim entry ↗

$\mathrm{JacobiTheta}(3, 0, \tau, 2r+1)=0$

Holds when $\Im(\tau)\gt0\land r\in\N$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. a19141 · Fungrim entry ↗

$z\mapsto\mathrm{JacobiTheta}(j, z, \tau)^{\prime}(z)=\mathrm{JacobiTheta}(j, z, \tau, r)$

Holds when $j\in\lbrace1, 2, 3, 4\rbrace\land z\in\C\land\Im(\tau)\gt0\land r\in\N$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. a222ed · Fungrim entry ↗

$\mathrm{JacobiTheta}(2, 4z, 4\tau)=\frac{\mathrm{JacobiTheta}(2, 1/8-z, \tau)\mathrm{JacobiTheta}(2, \frac{1}{8}+z, \tau)\mathrm{JacobiTheta}(2, 3/8-z, \tau)\mathrm{JacobiTheta}(2, \frac{3}{8}+z, \tau)}{\mathrm{JacobiTheta}(3, 0, \tau)\mathrm{JacobiTheta}(4, 0, \tau)\mathrm{JacobiTheta}(3, \frac{1}{4}, \tau)}$

Holds when $z\in\C\land\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. a255e1 · Fungrim entry ↗

$z\mapsto\frac{\mathrm{JacobiTheta}(1, z, \tau)}{\mathrm{JacobiTheta}(4, z, \tau)}^{\prime}(z)=\frac{\pi\mathrm{JacobiTheta}(2, z, \tau)\mathrm{JacobiTheta}(3, z, \tau)\mathrm{JacobiTheta}(4, 0, \tau)^2}{\mathrm{JacobiTheta}(4, z, \tau)^2}$

Holds when $z\in\C\land\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. a4eecf · Fungrim entry ↗

$\mathrm{JacobiTheta}(4, z, \tau+1)=\mathrm{JacobiTheta}(3, z, \tau)$

Holds when $z\in\C\land\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. a5c258 · Fungrim entry ↗

$\mathrm{JacobiTheta}(j, z^\star, \tau)=\mathrm{JacobiTheta}(j, z, -\tau^\star)^\star$

Holds when $j\in\lbrace1, 2, 3, 4\rbrace\land z\in\C\land\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. a891da · Fungrim entry ↗

$\mathrm{JacobiTheta}(3, 2z, \tau)=\frac{\mathrm{JacobiTheta}(3, z, \tau)^2\mathrm{JacobiTheta}(4, z, \tau)^2-\mathrm{JacobiTheta}(1, z, \tau)^2\mathrm{JacobiTheta}(2, z, \tau)^2}{\mathrm{JacobiTheta}(3, 0, \tau)\mathrm{JacobiTheta}(4, 0, \tau)^2}$

Holds when $z\in\C\land\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. a94b43 · Fungrim entry ↗

$\mathrm{JacobiTheta}(2, 0, \tau)=\frac{1}{\mathrm{DedekindEta}(\tau)}(2\mathrm{DedekindEta}(2\tau)^2)$

$\mathrm{JacobiTheta}(2, z, \frac{\tau}{2})=\frac{2\mathrm{JacobiTheta}(2, z, \tau)\mathrm{JacobiTheta}(3, z, \tau)}{\mathrm{JacobiTheta}(2, 0, \frac{\tau}{2})}$

Holds when $z\in\C\land\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. a9cdda · Fungrim entry ↗

$\mathrm{JacobiTheta}(2, 2z, \tau)=\frac{\mathrm{JacobiTheta}(2, z, \tau)^2\mathrm{JacobiTheta}(3, z, \tau)^2-\mathrm{JacobiTheta}(1, z, \tau)^2\mathrm{JacobiTheta}(4, z, \tau)^2}{\mathrm{JacobiTheta}(2, 0, \tau)\mathrm{JacobiTheta}(3, 0, \tau)^2}$

Holds when $z\in\C\land\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. aaa582 · Fungrim entry ↗

$\mathrm{JacobiTheta}(3, 0, \tau)^2\mathrm{JacobiTheta}(4, z, \tau)^2=\mathrm{JacobiTheta}(2, 0, \tau)^2\mathrm{JacobiTheta}(1, z, \tau)^2+\mathrm{JacobiTheta}(4, 0, \tau)^2\mathrm{JacobiTheta}(3, z, \tau)^2$

Holds when $z\in\C\land\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. abbe42 · Fungrim entry ↗

$\mathrm{JacobiTheta}(1, z, \tau+8n)=\mathrm{JacobiTheta}(1, z, \tau)$

Holds when $z\in\C\land\Im(\tau)\gt0\land n\in\Z$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. abc1e7 · Fungrim entry ↗

$\mathrm{JacobiTheta}(2, 2z, \tau)=\frac{\mathrm{JacobiTheta}(2, z, \tau)^2\mathrm{JacobiTheta}(4, z, \tau)^2-\mathrm{JacobiTheta}(1, z, \tau)^2\mathrm{JacobiTheta}(3, z, \tau)^2}{\mathrm{JacobiTheta}(2, 0, \tau)\mathrm{JacobiTheta}(4, 0, \tau)^2}$

Holds when $z\in\C\land\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. b1d07b · Fungrim entry ↗

$\mathrm{JacobiTheta}(1, 0, \tau, 2r)=0$

Holds when $\Im(\tau)\gt0\land r\in\N$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. b3c440 · Fungrim entry ↗

$\mathrm{JacobiTheta}(3, z, \tau)=\mathrm{JacobiTheta}(4, z+\frac{1}{2}, \tau)$

Holds when $z\in\C\land\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. b3fc6d · Fungrim entry ↗

$\mathrm{JacobiTheta}(2, z+2n, \tau)=\mathrm{JacobiTheta}(2, z, \tau)$

Holds when $z\in\C\land\Im(\tau)\gt0\land n\in\Z$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. b46534 · Fungrim entry ↗

$\mathrm{JacobiTheta}(3, 0, 1+2\imaginaryI)=\frac{\mathrm{JacobiTheta}(3, 0, \imaginaryI)}{\sqrt[8]{2}}$

Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. Reference: doi.org b58070 · Fungrim entry ↗

$\mathrm{JacobiTheta}(4, z+\tau, \tau)=-\exp(-(\pi\imaginaryI(2z+\tau)))\mathrm{JacobiTheta}(4, z, \tau)$

Holds when $z\in\C\land\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. b83f63 · Fungrim entry ↗

$\mathrm{JacobiTheta}(1, z, \tau+2n)=\imaginaryI^{n}\mathrm{JacobiTheta}(1, z, \tau)$

Holds when $z\in\C\land\Im(\tau)\gt0\land n\in\Z$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for expansion. b978f0 · Fungrim entry ↗

$\mathrm{JacobiTheta}(1, z, \tau+4n)=(-1)^{n}\mathrm{JacobiTheta}(1, z, \tau)$

Holds when $z\in\C\land\Im(\tau)\gt0\land n\in\Z$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for expansion. b9c650 · Fungrim entry ↗

$2\mathrm{JacobiTheta}(3, 0, 2\tau)^2=\mathrm{JacobiTheta}(3, 0, \tau)^2+\mathrm{JacobiTheta}(4, 0, \tau)^2$

Holds when $\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. c3d8c2 · Fungrim entry ↗

$\mathrm{JacobiTheta}(3, z, \frac{-1}{\tau})=\sqrt{\frac{\tau}{\imaginaryI}}\exp(\pi\imaginaryI\tau z^2)\mathrm{JacobiTheta}(3, \tau z, \tau)$

Holds when $z\in\C\land\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. c4b16c · Fungrim entry ↗

$\mathrm{JacobiTheta}(3, 0, \sqrt{6}\imaginaryI)=\sqrt[4]{\frac{\sqrt{6}\Gamma(1/24)\Gamma(5/24)\Gamma(7/24)\Gamma(11/24)}{96(18-10\sqrt{3}-7\sqrt{6}+12\sqrt{2})\pi^3}}$

Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. Reference: mathworld.wolfram.com c60033 · Fungrim entry ↗

$\mathrm{JacobiTheta}(2, 0, \tau)^4=8(\sum_{n=0}^{\infty}\frac{(2n+1)\exp(\pi\imaginaryI\tau)^{2n+1}}{1+\exp(\pi\imaginaryI\tau)^{2n+1}})+8(\sum_{n=0}^{\infty}\frac{(2n+1)\exp(\pi\imaginaryI\tau)^{2n+1}}{1-\exp(\pi\imaginaryI\tau)^{2n+1}})$

Holds when $\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. c743eb · Fungrim entry ↗

$\frac{\mathrm{JacobiTheta}(2, z, \tau, 1)}{\pi\mathrm{JacobiTheta}(2, z, \tau)}=4(\sum_{n=1}^{\infty}\frac{\sin(2\pi nz)\times(-1)^{n}\exp(2\imaginaryI\pi n\tau)}{1-\exp(2\imaginaryI\pi n\tau)})-\tan(\pi z)$

Holds when $z\in\C\land\Im(\tau)\gt0\land\vert\Im(z)\vert\lt\vert\Im(\tau)\vert\land\cos(\pi z)\ne0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. c7f7a5 · Fungrim entry ↗

$\mathrm{JacobiTheta}(4, z+w, \tau)\mathrm{JacobiTheta}(4, z-w, \tau)\mathrm{JacobiTheta}(2, 0, \tau)^2=\mathrm{JacobiTheta}(4, z, \tau)^2\mathrm{JacobiTheta}(2, w, \tau)^2+\mathrm{JacobiTheta}(3, z, \tau)^2\mathrm{JacobiTheta}(1, w, \tau)^2=\mathrm{JacobiTheta}(1, z, \tau)^2\mathrm{JacobiTheta}(3, w, \tau)^2+\mathrm{JacobiTheta}(2, z, \tau)^2\mathrm{JacobiTheta}(4, w, \tau)^2$

Holds when $z\in\C\land w\in\C\land\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. c891a1 · Fungrim entry ↗

$\mathrm{JacobiTheta}(4, z, \frac{\tau}{2})=\frac{\mathrm{JacobiTheta}(4, z, \tau)^2+\mathrm{JacobiTheta}(1, z, \tau)^2}{\mathrm{JacobiTheta}(3, 0, \frac{\tau}{2})}$

Holds when $z\in\C\land\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. c92a6f · Fungrim entry ↗

$z\mapsto\frac{\mathrm{JacobiTheta}(1, z, \tau)}{\mathrm{JacobiTheta}(2, z, \tau)}^{\prime}(z)=\frac{\pi\mathrm{JacobiTheta}(3, z, \tau)\mathrm{JacobiTheta}(4, z, \tau)\mathrm{JacobiTheta}(2, 0, \tau)^2}{\mathrm{JacobiTheta}(2, z, \tau)^2}$

Holds when $z\in\C\land\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. cb493d · Fungrim entry ↗

$\mathrm{JacobiTheta}(3, 0, 5\imaginaryI)=\frac{\mathrm{JacobiTheta}(3, 0, \imaginaryI)\sqrt{5+2\sqrt{5}}}{5^{\frac{3}{4}}}$

Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. Reference: doi.org cb6c9c · Fungrim entry ↗

$\mathrm{JacobiTheta}(2, z+\frac{\tau}{2}, \tau)=\exp(-(\pi\imaginaryI(z+\frac{\tau}{4})))\mathrm{JacobiTheta}(3, z, \tau)$

Holds when $z\in\C\land\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. cc6d21 · Fungrim entry ↗

$\mathrm{JacobiTheta}(2, z+\tau, \tau)=\exp(-(\pi\imaginaryI(2z+\tau)))\mathrm{JacobiTheta}(2, z, \tau)$

Holds when $z\in\C\land\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. cd5f45 · Fungrim entry ↗

$\mathrm{JacobiTheta}(2, z, \tau+1)=\exp(\frac{\pi\imaginaryI}{4})\mathrm{JacobiTheta}(2, z, \tau)$

Holds when $z\in\C\land\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. cde93e · Fungrim entry ↗

$\mathrm{JacobiTheta}(3, 0, 2\imaginaryI)=\frac{1}{2}(\sqrt{\sqrt{2}+2}\mathrm{JacobiTheta}(3, 0, \imaginaryI))$

Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. Reference: doi.org cf3c8e · Fungrim entry ↗

$\mathrm{JacobiTheta}(4, 0, y\imaginaryI)=\mathrm{JacobiTheta}(3, 0, 1+y\imaginaryI)$

Holds when $y\in\lparen0, \infty\rparen$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. cf7ee3 · Fungrim entry ↗

$\mathrm{JacobiTheta}(2, z, \tau+n)=\exp(\frac{\pi\imaginaryI n}{4})\mathrm{JacobiTheta}(2, z, \tau)$

Holds when $z\in\C\land\Im(\tau)\gt0\land n\in\Z$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. d0dfba · Fungrim entry ↗

$\mathrm{JacobiTheta}(2, z, \tau+2n)=\imaginaryI^{n}\mathrm{JacobiTheta}(2, z, \tau)$

Holds when $z\in\C\land\Im(\tau)\gt0\land n\in\Z$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for expansion. d11b7f · Fungrim entry ↗

$\mathrm{JacobiTheta}(3, 0, \imaginaryI)=\frac{\sqrt[4]{\pi}}{\Gamma(\frac{3}{4})}$

Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. d15f11 · Fungrim entry ↗

$\mathrm{JacobiTheta}(2, z+m+n\tau, \tau)=(-1)^{m}\exp(-(\pi\imaginaryI(\tau n^2+2nz)))\mathrm{JacobiTheta}(2, z, \tau)$

$\mathrm{JacobiTheta}(3, z, \tau)\mathrm{JacobiTheta}(4, w, \tau)=\mathrm{JacobiTheta}(4, z+w, 2\tau)\mathrm{JacobiTheta}(4, z-w, 2\tau)-\mathrm{JacobiTheta}(1, z+w, 2\tau)\mathrm{JacobiTheta}(1, z-w, 2\tau)$

Holds when $z\in\C\land w\in\C\land\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. d36e97 · Fungrim entry ↗

$z\mapsto\frac{\mathrm{JacobiTheta}(1, z, \tau)}{\mathrm{JacobiTheta}(3, z, \tau)}^{\prime}(z)=\frac{\pi\mathrm{JacobiTheta}(2, z, \tau)\mathrm{JacobiTheta}(4, z, \tau)\mathrm{JacobiTheta}(3, 0, \tau)^2}{\mathrm{JacobiTheta}(3, z, \tau)^2}$

Holds when $z\in\C\land\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. d41a95 · Fungrim entry ↗

$\mathrm{JacobiTheta}(1, z+\frac{\tau}{2}, \tau)=\exp(-(\pi\imaginaryI(z+\frac{\tau}{4})))\imaginaryI\mathrm{JacobiTheta}(4, z, \tau)$

Holds when $z\in\C\land\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. d5a29e · Fungrim entry ↗

$\mathrm{JacobiTheta}(1, z+\tau, \tau)=-\exp(-(\pi\imaginaryI(2z+\tau)))\mathrm{JacobiTheta}(1, z, \tau)$

Holds when $z\in\C\land\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. d989cd · Fungrim entry ↗

$\mathrm{JacobiTheta}(2, 2z, 2\tau)=\frac{\mathrm{JacobiTheta}(3, z, \tau)^2-\mathrm{JacobiTheta}(4, z, \tau)^2}{2\mathrm{JacobiTheta}(2, 0, 2\tau)}$

Holds when $z\in\C\land\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. db4e29 · Fungrim entry ↗

$\mathrm{JacobiTheta}(4, 0, \tau)^4=(1+8(\sum_{n=0}^{\infty}\frac{2n\exp(\pi\imaginaryI\tau)^{2n}}{1+\exp(\pi\imaginaryI\tau)^{2n}}))-8(\sum_{n=0}^{\infty}\frac{(2n+1)\exp(\pi\imaginaryI\tau)^{2n+1}}{1+\exp(\pi\imaginaryI\tau)^{2n+1}})$

Holds when $\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. dc7c83 · Fungrim entry ↗

$\mathrm{JacobiTheta}(2, \frac{n}{4}, \imaginaryI)=\begin{cases}(-1)^{\lfloor(n+1)/4\rfloor}\mathrm{JacobiTheta}(4, 0, \imaginaryI)&\mathrm{CongruentMod}(n, 0, 4)\\0&\mathrm{CongruentMod}(n, 2, 4)\\(-1)^{\lfloor(n+1)/4\rfloor}\frac{\sqrt{2^{1/2}-1}}{2^{7/16}}\sqrt[4]{\sqrt{2}+1}\mathrm{JacobiTheta}(3, 0, \imaginaryI)&\top\end{cases}$

Holds when $n\in\Z$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. dd5f43 · Fungrim entry ↗

$\mathrm{JacobiTheta}(3, 0, \frac{\tau}{2})^2=\mathrm{JacobiTheta}(2, 0, \tau)^2+\mathrm{JacobiTheta}(3, 0, \tau)^2$

Holds when $\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. de7918 · Fungrim entry ↗

$\frac{\mathrm{JacobiTheta}(1, z, \tau, 1)}{\pi\mathrm{JacobiTheta}(1, z, \tau)}=\cot(\pi z)+4(\sum_{n=1}^{\infty}\frac{\sin(2\pi nz)\exp(2\imaginaryI\pi n\tau)}{1-\exp(2\imaginaryI\pi n\tau)})$

Holds when $z\in\C\land\Im(\tau)\gt0\land\vert\Im(z)\vert\lt\vert\Im(\tau)\vert\land\sin(\pi z)\ne0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. dfbddd · Fungrim entry ↗

$\mathrm{JacobiTheta}(2, 0, \tau)\mathrm{JacobiTheta}(3, 0, \tau)\mathrm{JacobiTheta}(2, z+w, \tau)\mathrm{JacobiTheta}(3, z-w, \tau)=\mathrm{JacobiTheta}(2, z, \tau)\mathrm{JacobiTheta}(3, z, \tau)\mathrm{JacobiTheta}(2, w, \tau)\mathrm{JacobiTheta}(3, w, \tau)-\mathrm{JacobiTheta}(1, z, \tau)\mathrm{JacobiTheta}(4, z, \tau)\mathrm{JacobiTheta}(1, w, \tau)\mathrm{JacobiTheta}(4, w, \tau)$

Holds when $z\in\C\land w\in\C\land\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. dfea7d · Fungrim entry ↗

$\mathrm{JacobiTheta}(1, z, \tau)^4-\mathrm{JacobiTheta}(4, z, \tau)^4=\mathrm{JacobiTheta}(2, z, \tau)^4-\mathrm{JacobiTheta}(3, z, \tau)^4$

Holds when $z\in\C\land\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for expansion. e08bb4 · Fungrim entry ↗

$\mathrm{JacobiTheta}(1, 2z, 2\tau)=\frac{\mathrm{JacobiTheta}(1, z, \tau)\mathrm{JacobiTheta}(2, z, \tau)}{\mathrm{JacobiTheta}(4, 0, 2\tau)}$

Holds when $z\in\C\land\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. e13fe9 · Fungrim entry ↗

$\mathrm{JacobiTheta}(2, 0, 1+y\imaginaryI)=\frac{\sqrt{2}(1+\imaginaryI)\mathrm{JacobiTheta}(3, 0, \frac{\imaginaryI}{y}+1)}{2\sqrt{y}}$

Holds when $y\in\lparen0, \infty\rparen$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. e2288d · Fungrim entry ↗

$\mathrm{JacobiTheta}(3, 0, 1+8\imaginaryI)=\frac{\sqrt[8]{16+15\sqrt[4]{2}+12\sqrt{2}+9\sqrt[4]{8}}}{2^{\frac{7}{8}}}\mathrm{JacobiTheta}(3, 0, \imaginaryI)$

Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. Reference: doi.org e2bc80 · Fungrim entry ↗

$\mathrm{JacobiTheta}(3, 0, \tau)=1+2(\sum_{n=1}^{\infty}\frac{\mathrm{LiouvilleLambda}(n)\exp(\pi\imaginaryI\tau)^{n}}{1-\exp(\pi\imaginaryI\tau)^{n}})$

Holds when $\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. e4e707 · Fungrim entry ↗

$\mathrm{JacobiTheta}(3, z+n, \tau)=\mathrm{JacobiTheta}(3, z, \tau)$

Holds when $z\in\C\land\Im(\tau)\gt0\land n\in\Z$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. e56f77 · Fungrim entry ↗

$\mathrm{JacobiTheta}(3, z, \frac{\tau}{2})=\frac{\mathrm{JacobiTheta}(4, z, \tau)^2-\mathrm{JacobiTheta}(1, z, \tau)^2}{\mathrm{JacobiTheta}(4, 0, \frac{\tau}{2})}$

Holds when $z\in\C\land\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. e6d333 · Fungrim entry ↗

$\mathrm{JacobiTheta}(2, 2z, \tau)=\frac{\mathrm{JacobiTheta}(3, z, \tau)^4-\mathrm{JacobiTheta}(4, z, \tau)^4}{\mathrm{JacobiTheta}(2, 0, \tau)^3}$

Holds when $z\in\C\land\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for expansion. e6dc09 · Fungrim entry ↗

$\mathrm{JacobiTheta}(1, z, \frac{-1}{\tau})=-\imaginaryI\sqrt{\frac{\tau}{\imaginaryI}}\exp(\pi\imaginaryI\tau z^2)\mathrm{JacobiTheta}(1, \tau z, \tau)$

Holds when $z\in\C\land\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. e8ce0b · Fungrim entry ↗

$\mathrm{JacobiTheta}(j, z, \tau, 2)-4\pi\imaginaryI\tau\mapsto\mathrm{JacobiTheta}(j, z, \tau)^{\prime}(\tau)=0$

$\mathrm{JacobiTheta}(1, z, \tau)=-\imaginaryI\exp(\pi\imaginaryI(z+\frac{\tau}{4}))\mathrm{JacobiTheta}(3, z+\frac{1}{2}+\frac{\tau}{2}, \tau)$

Holds when $z\in\C\land\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. ed0756 · Fungrim entry ↗

$\mathrm{JacobiTheta}(3, 2z, \tau)=\frac{\mathrm{JacobiTheta}(2, z, \tau)^4+\mathrm{JacobiTheta}(4, z, \tau)^4}{\mathrm{JacobiTheta}(3, 0, \tau)^3}$

Holds when $z\in\C\land\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. ed3ff9 · Fungrim entry ↗

$\mathrm{JacobiTheta}(4, z, \frac{-1}{\tau})=\sqrt{\frac{\tau}{\imaginaryI}}\exp(\pi\imaginaryI\tau z^2)\mathrm{JacobiTheta}(2, \tau z, \tau)$

Holds when $z\in\C\land\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. ed8ba7 · Fungrim entry ↗

$\mathrm{JacobiTheta}(2, 0, \tau)\mathrm{JacobiTheta}(3, 0, \tau)\mathrm{JacobiTheta}(1, z+w, \tau)\mathrm{JacobiTheta}(4, z-w, \tau)=\mathrm{JacobiTheta}(1, z, \tau)\mathrm{JacobiTheta}(4, z, \tau)\mathrm{JacobiTheta}(2, w, \tau)\mathrm{JacobiTheta}(3, w, \tau)+\mathrm{JacobiTheta}(2, z, \tau)\mathrm{JacobiTheta}(3, z, \tau)\mathrm{JacobiTheta}(1, w, \tau)\mathrm{JacobiTheta}(4, w, \tau)$

Holds when $z\in\C\land w\in\C\land\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. ee8617 · Fungrim entry ↗

$\mathrm{JacobiTheta}(2, 2z, 2\tau)=\frac{\mathrm{JacobiTheta}(1, 1/4-z, \tau)\mathrm{JacobiTheta}(1, \frac{1}{4}+z, \tau)}{\mathrm{JacobiTheta}(4, 0, 2\tau)}$

Holds when $z\in\C\land\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. f12569 · Fungrim entry ↗

$\mathrm{JacobiTheta}(3, 0, 3\imaginaryI)=\frac{\mathrm{JacobiTheta}(3, 0, \imaginaryI)\sqrt{1+\sqrt{3}}}{\sqrt[4]{2}\times3^{\frac{3}{8}}}$

Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. Reference: doi.org f12e20 · Fungrim entry ↗

$\mathrm{JacobiTheta}(4, 0, 2\tau)^2=\mathrm{JacobiTheta}(3, 0, \tau)\mathrm{JacobiTheta}(4, 0, \tau)$

Holds when $\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. f14471 · Fungrim entry ↗

$\mathrm{JacobiTheta}(1, 0, \tau, 1)=\pi\mathrm{JacobiTheta}(2, 0, \tau)\mathrm{JacobiTheta}(3, 0, \tau)\mathrm{JacobiTheta}(4, 0, \tau)$

Holds when $\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. f2e28a · Fungrim entry ↗

$\mathrm{JacobiTheta}(3, z, \tau)=1+2(\sum_{n=1}^{\infty}\exp(\pi\imaginaryI\tau)^{n^2}\cos(2n\pi z))$

Holds when $z\in\C\land\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. f3e75c · Fungrim entry ↗

$\mathrm{JacobiTheta}(3, z, \tau)\mathrm{JacobiTheta}(3, w, \tau)=\mathrm{JacobiTheta}(3, z+w, 2\tau)\mathrm{JacobiTheta}(3, z-w, 2\tau)+\mathrm{JacobiTheta}(2, z+w, 2\tau)\mathrm{JacobiTheta}(2, z-w, 2\tau)$

Holds when $z\in\C\land w\in\C\land\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. f4554f · Fungrim entry ↗

$\mathrm{JacobiTheta}(2, z+n, \tau)=(-1)^{n}\mathrm{JacobiTheta}(2, z, \tau)$

Holds when $z\in\C\land\Im(\tau)\gt0\land n\in\Z$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. f697d5 · Fungrim entry ↗

$\mathrm{JacobiTheta}(3, 0, \tau)^2=1+2(\sum_{n=1}^{\infty}(\cos(\pi\tau n))^{-1})$

Holds when $\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. f8cd8f · Fungrim entry ↗

$\mathrm{JacobiTheta}(2, 0, \tau)^2\mathrm{JacobiTheta}(3, z, \tau)^2=\mathrm{JacobiTheta}(4, 0, \tau)^2\mathrm{JacobiTheta}(1, z, \tau)^2+\mathrm{JacobiTheta}(3, 0, \tau)^2\mathrm{JacobiTheta}(2, z, \tau)^2$

Holds when $z\in\C\land\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. fa7251 · Fungrim entry ↗

$\mathrm{JacobiTheta}(2, z, \tau+8n)=\mathrm{JacobiTheta}(2, z, \tau)$

Holds when $z\in\C\land\Im(\tau)\gt0\land n\in\Z$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. fb4b1b · Fungrim entry ↗

$\mathrm{JacobiTheta}(2, -z, \tau)=\mathrm{JacobiTheta}(2, z, \tau)$

Holds when $z\in\C\land\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. fb55cb · Fungrim entry ↗

$\mathrm{JacobiTheta}(4, 4z, 4\tau)=\frac{\mathrm{JacobiTheta}(4, z, \tau)\mathrm{JacobiTheta}(4, 1/4-z, \tau)\mathrm{JacobiTheta}(4, \frac{1}{4}+z, \tau)\mathrm{JacobiTheta}(3, z, \tau)}{\mathrm{JacobiTheta}(3, 0, \tau)\mathrm{JacobiTheta}(4, 0, \tau)\mathrm{JacobiTheta}(3, \frac{1}{4}, \tau)}$

Holds when $z\in\C\land\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. fc3c44 · Fungrim entry ↗

$\mathrm{JacobiTheta}(j, z, -\tau^\star)=\mathrm{JacobiTheta}(j, z^\star, \tau)^\star$

Modular j-invariant

$\mathrm{ModularJ}(\sqrt{2}\imaginaryI)=20^3=8\,000$

Symbols: ModularJ — Modular j-invariant. Used by the Compute Engine for simplification. 1356e4 · Fungrim entry ↗

$\mathrm{ModularJ}(\frac{1}{2}(1+\sqrt{163}\imaginaryI))=-640\,320^3$

Symbols: ModularJ — Modular j-invariant. Used by the Compute Engine for simplification. 1cb24e · Fungrim entry ↗

$\mathrm{ModularJ}(2\imaginaryI)=66^3=287\,496$

Symbols: ModularJ — Modular j-invariant. Used by the Compute Engine for simplification. 229c97 · Fungrim entry ↗

$\mathrm{ModularJ}(\frac{1}{2}(1+\sqrt{7}\imaginaryI))=-15^3$

Symbols: ModularJ — Modular j-invariant. Used by the Compute Engine for simplification. 29c095 · Fungrim entry ↗

$\mathrm{ModularJ}(4\imaginaryI)=27(724+513\sqrt{2})^3$

Symbols: ModularJ — Modular j-invariant. Used by the Compute Engine for simplification. 3189b9 · Fungrim entry ↗

$\tau\mapsto\mathrm{ModularJ}(\tau)^{\prime}(\tau)=\frac{(-2\imaginaryI)\pi\mathrm{EisensteinE}(6, \tau)\mathrm{ModularJ}(\tau)}{\mathrm{EisensteinE}(4, \tau)}$

Holds when $\Im(\tau)\gt0\land\mathrm{EisensteinE}(4, \tau)\ne0$ . Symbols: EisensteinE — Normalized Eisenstein series; ModularJ — Modular j-invariant. Used by the Compute Engine for simplification. 348b26 · Fungrim entry ↗

$\mathrm{ModularJ}(\frac{1}{2}(1+\sqrt{19}\imaginaryI))=-96^3$

Symbols: ModularJ — Modular j-invariant. Used by the Compute Engine for simplification. 3ee358 · Fungrim entry ↗

$\mathrm{ModularJ}(-(\frac{1}{\tau}))=\mathrm{ModularJ}(\tau)$

Holds when $\Im(\tau)\gt0$ . Symbols: ModularJ — Modular j-invariant. Used by the Compute Engine for simplification. 42a909 · Fungrim entry ↗

$\mathrm{Count}(\mathrm{Solutions}(\tau\mapsto\mathrm{ModularJ}(\tau)=z, \mathrm{ModularGroupFundamentalDomain}))=1$

Holds when $z\in\C$ . Symbols: ModularGroupFundamentalDomain — Fundamental domain for action of the modular group; ModularJ — Modular j-invariant; Solutions — Solution set. Used by the Compute Engine for simplification. 441301 · Fungrim entry ↗

$\mathrm{ModularJ}(\frac{1}{2}(1+\sqrt{43}\imaginaryI))=-960^3$

Symbols: ModularJ — Modular j-invariant. Used by the Compute Engine for simplification. 5b108e · Fungrim entry ↗

$\mathrm{ModularJ}(\tau)={(\frac{\mathrm{DedekindEta}(\tau)}{\mathrm{DedekindEta}(2\tau)}^8+(256\mathrm{DedekindEta}(2\tau)^{16})/\mathrm{DedekindEta}(\tau)^{16})}^3$

Holds when $\Im(\tau)\gt0$ . Symbols: DedekindEta — Dedekind eta function; ModularJ — Modular j-invariant. Used by the Compute Engine for expansion. 664b4c · Fungrim entry ↗

$\mathrm{ModularJ}(3\imaginaryI)=64(2+\sqrt{3})^2(21+20\sqrt{3})^3$

Symbols: ModularJ — Modular j-invariant. Used by the Compute Engine for simplification. 8be46c · Fungrim entry ↗

$\mathrm{ModularJ}(\frac{1}{2}(1+\sqrt{67}\imaginaryI))=-5\,280^3$

Symbols: ModularJ — Modular j-invariant. Used by the Compute Engine for simplification. 951017 · Fungrim entry ↗

$\mathrm{ModularJ}(\exp(\frac{\pi\imaginaryI}{3}))=0$

Symbols: ModularJ — Modular j-invariant. Used by the Compute Engine for simplification. 9aa62c · Fungrim entry ↗

$\mathrm{ModularJ}(\frac{1}{2}(1+\sqrt{11}\imaginaryI))=-32^3$

Symbols: ModularJ — Modular j-invariant. Used by the Compute Engine for simplification. a498dd · Fungrim entry ↗

$\mathrm{ModularJ}(\tau+1)=\mathrm{ModularJ}(\tau)$

Holds when $\Im(\tau)\gt0$ . Symbols: ModularJ — Modular j-invariant. Used by the Compute Engine for simplification. a997f2 · Fungrim entry ↗

$\mathrm{ModularJ}(\imaginaryI)=1\,728$

Symbols: ModularJ — Modular j-invariant. Used by the Compute Engine for simplification. ad228f · Fungrim entry ↗

$\mathrm{ModularJ}(\tau)=\frac{32{(\mathrm{JacobiTheta}(2, 0, \tau)^8+\mathrm{JacobiTheta}(3, 0, \tau)^8+\mathrm{JacobiTheta}(4, 0, \tau)^8)}^3}{(\mathrm{JacobiTheta}(2, 0, \tau)\mathrm{JacobiTheta}(3, 0, \tau)\mathrm{JacobiTheta}(4, 0, \tau))^8}$

Holds when $\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function; ModularJ — Modular j-invariant. Used by the Compute Engine for simplification. cedcfc · Fungrim entry ↗

$\mathrm{ModularJ}(\tau)=\frac{\mathrm{EisensteinE}(4, \tau)^3}{\mathrm{DedekindEta}(\tau)^{24}}$

Holds when $\Im(\tau)\gt0$ . Symbols: DedekindEta — Dedekind eta function; EisensteinE — Normalized Eisenstein series; ModularJ — Modular j-invariant. Used by the Compute Engine for expansion. dc8251 · Fungrim entry ↗

$\tau\mapsto\mathrm{ModularJ}(\tau)^{\prime}(\tau)=((-2\imaginaryI)\pi\mathrm{EisensteinE}(14, \tau))/\mathrm{DedekindEta}(\tau)^{24}$

Modular lambda function

$\frac{1}{\mathrm{ModularLambda}(\tau)}=\frac{\mathrm{DedekindEta}(\tau/2)^8}{16\mathrm{DedekindEta}(2\tau)^8}+1$

Holds when $\Im(\tau)\gt0$ . Symbols: DedekindEta — Dedekind eta function; ModularLambda — Modular lambda function. Used by the Compute Engine for simplification. 033d39 · Fungrim entry ↗

$1-\mathrm{ModularLambda}(\tau)=\frac{\mathrm{JacobiTheta}(4, 0, \tau)^4}{\mathrm{JacobiTheta}(3, 0, \tau)^4}$

Holds when $\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function; ModularLambda — Modular lambda function. Used by the Compute Engine for simplification. 04d3a6 · Fungrim entry ↗

$\mathrm{ModularLambda}(\frac{1+\imaginaryI}{2})=2$

Symbols: ModularLambda — Modular lambda function. Used by the Compute Engine for simplification. 078869 · Fungrim entry ↗

$\mathrm{ModularLambda}(-(\frac{1}{\tau}))=1-\mathrm{ModularLambda}(\tau)$

Holds when $\Im(\tau)\gt0$ . Symbols: ModularLambda — Modular lambda function. Used by the Compute Engine for simplification. 07bf27 · Fungrim entry ↗

$\mathrm{ModularLambda}(\tau)=\frac{\mathrm{WeierstrassP}((1+\tau)/2, \tau)-\mathrm{WeierstrassP}(\tau/2, \tau)}{\mathrm{WeierstrassP}(1/2, \tau)-\mathrm{WeierstrassP}(\tau/2, \tau)}$

Holds when $\Im(\tau)\gt0$ . Symbols: ModularLambda — Modular lambda function; WeierstrassP — Weierstrass elliptic function. Used by the Compute Engine for simplification. 166402 · Fungrim entry ↗

$\tau\mapsto\mathrm{ModularLambda}(\tau)^{\prime}(\tau)=\frac{1}{3}(\pi\imaginaryI((\mathrm{EisensteinE}(2, \tau/2)+8\mathrm{EisensteinE}(2, 2\tau))-6\mathrm{EisensteinE}(2, \tau)))\mathrm{ModularLambda}(\tau)$

Holds when $\Im(\tau)\gt0$ . Symbols: EisensteinE — Normalized Eisenstein series; ModularLambda — Modular lambda function. Used by the Compute Engine for simplification. 27b2c7 · Fungrim entry ↗

$\mathrm{ModularLambda}(\frac{1}{1-\tau})=\frac{1}{1-\mathrm{ModularLambda}(\tau)}$

Holds when $\Im(\tau)\gt0$ . Symbols: ModularLambda — Modular lambda function. Used by the Compute Engine for expansion. 2ba627 · Fungrim entry ↗

$\mathrm{ModularLambda}(2\imaginaryI)=17-12\sqrt{2}$

Symbols: ModularLambda — Modular lambda function. Used by the Compute Engine for simplification. 35c85f · Fungrim entry ↗

$\mathrm{ModularLambda}(\frac{\tau-1}{\tau})=\frac{\mathrm{ModularLambda}(\tau)-1}{\mathrm{ModularLambda}(\tau)}$

Holds when $\Im(\tau)\gt0$ . Symbols: ModularLambda — Modular lambda function. Used by the Compute Engine for simplification. 3a7a0b · Fungrim entry ↗

$\mathrm{ModularJ}(\tau)=\frac{256{(-\mathrm{ModularLambda}(\tau)+\mathrm{ModularLambda}(\tau)^2+1)}^3}{(1-\mathrm{ModularLambda}(\tau))^2\mathrm{ModularLambda}(\tau)^2}$

Holds when $\Im(\tau)\gt0$ . Symbols: ModularJ — Modular j-invariant; ModularLambda — Modular lambda function. Used by the Compute Engine for simplification. 44a529 · Fungrim entry ↗

$\mathrm{ModularLambda}(\frac{\imaginaryI}{2})=12\sqrt{2}-16$

Symbols: ModularLambda — Modular lambda function. Used by the Compute Engine for simplification. 4877f2 · Fungrim entry ↗

$\mathrm{ModularLambda}(\tau)=\frac{\mathrm{JacobiTheta}(2, 0, \tau)^4}{\mathrm{JacobiTheta}(3, 0, \tau)^4}$

Holds when $\Im(\tau)\gt0$ . Symbols: JacobiTheta — Jacobi theta function; ModularLambda — Modular lambda function. Used by the Compute Engine for expansion. 5b9c02 · Fungrim entry ↗

$\mathrm{ModularLambda}(\tau)=(16\mathrm{DedekindEta}(2\tau)^{16}\mathrm{DedekindEta}(\tau/2)^8)/\mathrm{DedekindEta}(\tau)^{24}$

$\mathrm{ModularLambda}(\tau+2)=\mathrm{ModularLambda}(\tau)$

Holds when $\Im(\tau)\gt0$ . Symbols: ModularLambda — Modular lambda function. Used by the Compute Engine for simplification. 6678af · Fungrim entry ↗

$\mathrm{ModularLambdaFundamentalDomain}=\lbrace\tau, \tau\in\mathrm{HH}\in(\Re(\tau)\in\lparen-1, 1\rparen\land\min(\vert\tau-1/2\vert, \vert z+1/2\vert)\gt1/2\lor\Re(\tau)=-1\lor\vert\tau+1/2\vert=1/2)\rbrace$

Symbols: HH — Upper complex half-plane; ModularLambdaFundamentalDomain — Fundamental domain of the modular lambda function. Used by the Compute Engine for simplification. Reference: J. M. Borwein and P. B. Borwein. Pi and the AGM. Wiley, New York, 1987. p. 113. 737f2b · Fungrim entry ↗

$\frac{\mathrm{ModularLambda}(\tau)}{\mathrm{ModularLambda}(\tau)-1}=-(\frac{\mathrm{JacobiTheta}(2, 0, \tau)^4}{\mathrm{JacobiTheta}(4, 0, \tau)^4})$

$\mathrm{ModularLambda}(\imaginaryI)=\frac{1}{2}$

Symbols: ModularLambda — Modular lambda function. Used by the Compute Engine for simplification. a35b3c · Fungrim entry ↗

$\mathrm{ModularLambda}(\exp(\frac{2\pi\imaginaryI}{3}))=-\exp(\frac{2\pi\imaginaryI}{3})$

Symbols: ModularLambda — Modular lambda function. Used by the Compute Engine for simplification. b0e1cb · Fungrim entry ↗

$\tau=\frac{(\imaginaryI)\mathrm{EllipticK}(1-\mathrm{ModularLambda}(\tau))}{\mathrm{EllipticK}(\mathrm{ModularLambda}(\tau))}$

Holds when $\tau\in\mathrm{Interior}(\mathrm{ModularLambdaFundamentalDomain})\cup\lbrace\tau, \tau\in\mathrm{HH}\in\Re(\tau)=1\rbrace$ . Symbols: EllipticK — Legendre complete elliptic integral of the first kind; ModularLambda — Modular lambda function. Used by the Compute Engine for simplification. b7174d · Fungrim entry ↗

$\mathrm{ModularLambda}(\tau+1)=\frac{\mathrm{ModularLambda}(\tau)}{\mathrm{ModularLambda}(\tau)-1}$

Holds when $\Im(\tau)\gt0$ . Symbols: ModularLambda — Modular lambda function. Used by the Compute Engine for simplification. bbfb6c · Fungrim entry ↗

$\tau\mapsto\mathrm{ModularLambda}(\tau)^{\prime}(\tau)=\frac{1}{\pi}((2\imaginaryI)(-6\mathrm{WeierstrassZeta}(\frac{1}{2}, \tau)+8\mathrm{WeierstrassZeta}(\frac{1}{2}, 2\tau)+\mathrm{WeierstrassZeta}(\frac{1}{2}, \frac{\tau}{2}))\mathrm{ModularLambda}(\tau))$

Holds when $\Im(\tau)\gt0$ . Symbols: ModularLambda — Modular lambda function; WeierstrassZeta — Weierstrass zeta function. Used by the Compute Engine for simplification. c18c95 · Fungrim entry ↗

$\mathrm{ModularLambda}(\frac{\tau}{1-\tau})=\frac{1}{\mathrm{ModularLambda}(\tau)}$

Holds when $\Im(\tau)\gt0$ . Symbols: ModularLambda — Modular lambda function. Used by the Compute Engine for simplification. e9f0c8 · Fungrim entry ↗

$\mathrm{ModularLambda}(\frac{\tau}{2\tau+1})=\mathrm{ModularLambda}(\tau)$

Holds when $\Im(\tau)\gt0$ . Symbols: ModularLambda — Modular lambda function. Used by the Compute Engine for simplification. ec5a44 · Fungrim entry ↗

$\mathrm{ModularLambda}(1+\imaginaryI)=-1$

Symbols: ModularLambda — Modular lambda function. Used by the Compute Engine for simplification. fe2627 · Fungrim entry ↗

Contents​

Dedekind eta function​

Illustrations of Eisenstein series​

Jacobi theta functions​

Modular j-invariant​

Modular lambda function​

Contents

Dedekind eta function

Illustrations of Eisenstein series

Jacobi theta functions

Modular j-invariant

Modular lambda function