# Modular forms and theta functions Part of the [Fungrim Identities](/compute-engine/reference/fungrim/) reference — **318 identities** for modular forms and theta functions. :::info[Generated reference] This page is generated from the compiled Fungrim artifact by `scripts/fungrim/gen-reference-doc.ts` (upstream snapshot `3a299164c683`, translator `grim2mathjson 0.1.0`). Do not edit it by hand. The corpus is MIT-licensed; see `data/fungrim/LICENSE`. ::: ## Contents - [Dedekind eta function](#dedekind-eta-function) (23) - [Illustrations of Eisenstein series](#illustrations-of-eisenstein-series) (41) - [Jacobi theta functions](#jacobi-theta-functions) (206) - [Modular j-invariant](#modular-j-invariant) (21) - [Modular lambda function](#modular-lambda-function) (27) ## 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** $\Im(\tau)\gt0$. Used by the Compute Engine for simplification. **Reference:** [functions.wolfram.com](http://functions.wolfram.com/EllipticFunctions/DedekindEta/13/01/0002/) [`02d14f` · Fungrim entry ↗](https://fungrim.org/entry/02d14f) --- $$\mathrm{DedekindEta}(16\imaginaryI)=\frac{\mathrm{DedekindEta}(\imaginaryI)\sqrt[4]{2^{1/4}-1}\sqrt{(1+2^{1/2})^{1/2}-2^{5/8}}}{2^{\frac{113}{64}}\sqrt[16]{1+\sqrt{2}}}$$ Used by the Compute Engine for simplification. **Reference:** [math.stackexchange.com](https://math.stackexchange.com/questions/1334684/what-is-the-exact-value-of-eta6i/1334940) [`0701dc` · Fungrim entry ↗](https://fungrim.org/entry/0701dc) --- $$\mathrm{DedekindEta}(\tau+1)=\exp(\frac{\pi\imaginaryI}{12})\mathrm{DedekindEta}(\tau)$$ **Holds when** $\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`1bae52` · Fungrim entry ↗](https://fungrim.org/entry/1bae52) --- $$\tau\mapsto\mathrm{DedekindEta}(\tau)^{\prime}(\tau)=\frac{1}{\pi}((\frac{1}{2}\imaginaryI)\mathrm{DedekindEta}(\tau)\mathrm{WeierstrassZeta}(\frac{1}{2}, \tau))$$ **Holds when** $\Im(\tau)\gt0$. **Symbols:** **WeierstrassZeta** — Weierstrass zeta function. Used by the Compute Engine for simplification. [`1c25d3` · Fungrim entry ↗](https://fungrim.org/entry/1c25d3) --- $$\mathrm{DedekindEta}(\exp(\frac{2\pi\imaginaryI}{3}))=\frac{\sqrt[8]{3}\exp(-((1/24\imaginaryI)\pi))\sqrt{\Gamma(1/3)}^{3}}{2\pi}$$ Used by the Compute Engine for simplification. [`204acd` · Fungrim entry ↗](https://fungrim.org/entry/204acd) --- $$\mathrm{DedekindEta}(4\imaginaryI)=(\mathrm{DedekindEta}(\imaginaryI))(2^{\frac{13}{16}}\sqrt[4]{1+\sqrt{2}})^{-1}$$ Used by the Compute Engine for simplification. [`3a56d8` · Fungrim entry ↗](https://fungrim.org/entry/3a56d8) --- $$\mathrm{DedekindEta}(-(\frac{1}{\tau}))=\sqrt{-(\imaginaryI\tau)}\mathrm{DedekindEta}(\tau)$$ **Holds when** $\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`3b806f` · Fungrim entry ↗](https://fungrim.org/entry/3b806f) --- $$\mathrm{DedekindEta}(6\imaginaryI)=\frac{\mathrm{DedekindEta}(\imaginaryI)\sqrt[6]{\frac{5-3^{1/2}}{2}-\frac{1}{2}(2^{1/2}\times3^{3/4})}}{6^{\frac{3}{8}}}$$ Used by the Compute Engine for simplification. **Reference:** [math.stackexchange.com](https://math.stackexchange.com/questions/1334684/what-is-the-exact-value-of-eta6i/1334940) [`62ffb3` · Fungrim entry ↗](https://fungrim.org/entry/62ffb3) --- $$\mathrm{DedekindEta}(\tau)=\exp(\frac{\pi\imaginaryI\tau}{12})\mathrm{JacobiTheta}(3, \frac{\tau+1}{2}, 3\tau)$$ **Holds when** $\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`737805` · Fungrim entry ↗](https://fungrim.org/entry/737805) --- $$\mathrm{DedekindEta}(7\imaginaryI)=\sqrt[4]{7^{1/2}-7/2+\frac{1}{2}((4\times7^{1/2}-7)^{1/2})}/\sqrt{7}\mathrm{DedekindEta}(\imaginaryI)$$ Used by the Compute Engine for simplification. **Reference:** [math.stackexchange.com](https://math.stackexchange.com/questions/1334684/what-is-the-exact-value-of-eta6i/1334940) [`7cc3d3` · Fungrim entry ↗](https://fungrim.org/entry/7cc3d3) --- $$\tau\mapsto\mathrm{DedekindEta}(\tau)^{\prime}(\tau)=\frac{1}{12}(\imaginaryI\pi\mathrm{DedekindEta}(\tau))\mathrm{EisensteinE}(2, \tau)$$ **Holds when** $\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`871996` · Fungrim entry ↗](https://fungrim.org/entry/871996) --- $$\mathrm{DedekindEta}(2\imaginaryI)=\frac{\mathrm{DedekindEta}(\imaginaryI)}{2^{\frac{3}{8}}}$$ Used by the Compute Engine for simplification. [`87e9ed` · Fungrim entry ↗](https://fungrim.org/entry/87e9ed) --- $$\mathrm{DedekindEtaEpsilon}(a, b, c, d)=\exp(\pi\imaginaryI((a+d)/(12c)-\mathrm{DedekindSum}(d, c)-\frac{1}{4}))$$ **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 ↗](https://fungrim.org/entry/921ef0) --- $$\mathrm{DedekindEta}(\imaginaryI)=\frac{\Gamma(\frac{1}{4})}{2\pi^{\frac{3}{4}}}$$ Used by the Compute Engine for simplification. [`9b8c9f` · Fungrim entry ↗](https://fungrim.org/entry/9b8c9f) --- $$\mathrm{DedekindEta}(3\imaginaryI)=(\mathrm{DedekindEta}(\imaginaryI))(3^{\frac{3}{8}}\sqrt[12]{2+\sqrt{3}})^{-1}$$ Used by the Compute Engine for simplification. [`9ce413` · Fungrim entry ↗](https://fungrim.org/entry/9ce413) --- $$\mathrm{DedekindEta}(\tau+\frac{1}{2})=\frac{\exp((1/24\imaginaryI)\pi)\mathrm{DedekindEta}(2\tau)^3}{\mathrm{DedekindEta}(\tau)\mathrm{DedekindEta}(4\tau)}$$ **Holds when** $\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`a1a3d4` · Fungrim entry ↗](https://fungrim.org/entry/a1a3d4) --- $$\mathrm{DedekindEta}(\tau+24)=\mathrm{DedekindEta}(\tau)$$ **Holds when** $\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`acee1a` · Fungrim entry ↗](https://fungrim.org/entry/acee1a) --- $$\mathrm{DedekindEta}(8\imaginaryI)=\frac{\mathrm{DedekindEta}(\imaginaryI)\sqrt{2^{1/4}-1}}{2^{\frac{41}{32}}\sqrt[8]{1+\sqrt{2}}}$$ Used by the Compute Engine for simplification. **Reference:** [math.stackexchange.com](https://math.stackexchange.com/questions/1334684/what-is-the-exact-value-of-eta6i/1334940) [`be2f32` · Fungrim entry ↗](https://fungrim.org/entry/be2f32) --- $$\mathrm{DedekindEta}(5\imaginaryI)=\frac{\mathrm{DedekindEta}(\imaginaryI)}{\sqrt{5\varphi}}$$ Used by the Compute Engine for simplification. **Reference:** [math.stackexchange.com](https://math.stackexchange.com/questions/1334684/what-is-the-exact-value-of-eta6i/1334940) [`d2900f` · Fungrim entry ↗](https://fungrim.org/entry/d2900f) --- $$(\mathrm{DedekindEta}(\tau)^2(33\tau\mapsto\mathrm{DedekindEta}(\tau)^{\doubleprime}(\tau)^2+\mathrm{DedekindEta}(\tau)\tau\mapsto\mathrm{DedekindEta}(\tau)^{(4)}(\tau))-18{\tau\mapsto\mathrm{DedekindEta}(\tau)^{\prime}(\tau)}^4+12\mathrm{DedekindEta}(\tau)\tau\mapsto\mathrm{DedekindEta}(\tau)^{\doubleprime}(\tau)\tau\mapsto\mathrm{DedekindEta}(\tau)^{\prime}(\tau)^2)-28\mathrm{DedekindEta}(\tau)^2\tau\mapsto\mathrm{DedekindEta}(\tau)^{\tripleprime}(\tau)\tau\mapsto\mathrm{DedekindEta}(\tau)^{\prime}(\tau)=0$$ **Holds when** $\Im(\tau)\gt0$. Used by the Compute Engine for simplification. **Reference:** [functions.wolfram.com](http://functions.wolfram.com/EllipticFunctions/DedekindEta/13/01/0001/) [`df5f38` · Fungrim entry ↗](https://fungrim.org/entry/df5f38) --- $$\mathrm{DedekindEta}(\sqrt{3}\imaginaryI)=\frac{\sqrt[8]{3}\sqrt{\Gamma(1/3)}^{3}}{\pi\times2^{\frac{4}{3}}}$$ Used by the Compute Engine for simplification. **Reference:** [math.stackexchange.com](https://math.stackexchange.com/questions/1334684/what-is-the-exact-value-of-eta6i/1334940) [`e3e4c5` · Fungrim entry ↗](https://fungrim.org/entry/e3e4c5) --- $$\mathrm{DedekindEtaEpsilon}(1, b, 0, 1)=\exp(\frac{\pi\imaginaryI b}{12})$$ **Symbols:** **DedekindEtaEpsilon** — Root of unity in the functional equation of the Dedekind eta function. Used by the Compute Engine for simplification. [`f04e01` · Fungrim entry ↗](https://fungrim.org/entry/f04e01) --- $$\mathrm{DedekindEta}(\tau)=\exp(\frac{\pi\imaginaryI\tau}{12})\mathrm{EulerQSeries}(\exp(2\pi\imaginaryI\tau))$$ **Holds when** $\Im(\tau)\gt0$. **Symbols:** **EulerQSeries** — Euler's q-series. Used by the Compute Engine for simplification. [`ff587a` · Fungrim entry ↗](https://fungrim.org/entry/ff587a) --- ## Illustrations of Eisenstein series $$\mathrm{EisensteinE}(2, \tau)=-(\frac{(12\imaginaryI)\tau\mapsto\mathrm{DedekindEta}(\tau)^{\prime}(\tau)}{\pi\mathrm{DedekindEta}(\tau)})$$ **Holds when** $\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`03ad5a` · Fungrim entry ↗](https://fungrim.org/entry/03ad5a) --- $$\mathrm{EisensteinE}(8, \tau)=\mathrm{EisensteinE}(4, \tau)^2$$ **Holds when** $\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`044128` · Fungrim entry ↗](https://fungrim.org/entry/044128) --- $$\mathrm{EisensteinE}(2k, \tau)=\frac{\mathrm{EisensteinG}(2k, \tau)}{2\Zeta(2k)}$$ **Holds when** $k\in\N^*\land\Im(\tau)\gt0$. **Symbols:** **EisensteinG** — Eisenstein series. Used by the Compute Engine for simplification. [`0a2120` · Fungrim entry ↗](https://fungrim.org/entry/0a2120) --- $$\mathrm{EisensteinE}(6, \tau)=\frac{\mathrm{DedekindEta}(\tau)^{24}}{\mathrm{DedekindEta}(2\tau)^{12}}-480\mathrm{DedekindEta}(2\tau)^{12}-(16\,896\mathrm{DedekindEta}(2\tau)^{12}\mathrm{DedekindEta}(4\tau)^8)/\mathrm{DedekindEta}(\tau)^8+\frac{8\,192\mathrm{DedekindEta}(4\tau)^{24}}{\mathrm{DedekindEta}(2\tau)^{12}}$$ **Holds when** $\Im(\tau)\gt0$. 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 ↗](https://fungrim.org/entry/0a5ef4) --- $$\mathrm{EisensteinG}(6, \exp(\frac{2\pi\imaginaryI}{3}))=\frac{\Gamma(1/3)^{18}}{8\,960\pi^6}$$ **Symbols:** **EisensteinG** — Eisenstein series. Used by the Compute Engine for simplification. [`0fda1b` · Fungrim entry ↗](https://fungrim.org/entry/0fda1b) --- $$\mathrm{EisensteinE}(6, \tau)=\frac{1}{2}((\mathrm{JacobiTheta}(3, 0, \tau)^{12}+\mathrm{JacobiTheta}(4, 0, \tau)^{12})-3\mathrm{JacobiTheta}(2, 0, \tau)^8(\mathrm{JacobiTheta}(3, 0, \tau)^4+\mathrm{JacobiTheta}(4, 0, \tau)^4))$$ **Holds when** $\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`10f3b2` · Fungrim entry ↗](https://fungrim.org/entry/10f3b2) --- $$\mathrm{EisensteinE}(2k, \tau)=1-\frac{4k(\sum_{n=1}^{\infty}\sum_{m=1}^{\infty}\exp(2\imaginaryI\pi mn\tau)n^{2k-1})}{\mathrm{BernoulliB}(2k)}$$ **Holds when** $k\in\N^*\land\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`15b347` · Fungrim entry ↗](https://fungrim.org/entry/15b347) --- $$\mathrm{EisensteinE}(6, \tau)=1+63(\sum_{m=1}^{\infty}\frac{2\cos(\pi m\tau)^4+11\cos(\pi m\tau)^2+2}{\sin(\pi m\tau)^6})$$ **Holds when** $\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`171724` · Fungrim entry ↗](https://fungrim.org/entry/171724) --- $$\mathrm{EisensteinE}(2, \tau)=1+6(\sum_{m=1}^{\infty}(\sin(\pi m\tau)^2)^{-1})$$ **Holds when** $\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`18a4d1` · Fungrim entry ↗](https://fungrim.org/entry/18a4d1) --- $$\mathrm{EisensteinG}(2k, \tau+n)=\mathrm{EisensteinG}(2k, \tau)$$ **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 ↗](https://fungrim.org/entry/23a5e0) --- $$\mathrm{EisensteinE}(2, \exp(\frac{2\pi\imaginaryI}{3}))=\frac{2\sqrt{3}}{\pi}$$ Used by the Compute Engine for simplification. [`30a054` · Fungrim entry ↗](https://fungrim.org/entry/30a054) --- $$\mathrm{EisensteinG}(4, \exp(\frac{2\pi\imaginaryI}{3}))=\mathrm{EisensteinE}(4, \exp(\frac{2\pi\imaginaryI}{3}))=0$$ **Symbols:** **EisensteinG** — Eisenstein series. Used by the Compute Engine for simplification. [`3102a7` · Fungrim entry ↗](https://fungrim.org/entry/3102a7) --- $$\mathrm{EisensteinE}(12, \tau)=\frac{1}{691}(441\mathrm{EisensteinE}(4, \tau)^3+250\mathrm{EisensteinE}(6, \tau)^2)$$ **Holds when** $\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`36fff2` · Fungrim entry ↗](https://fungrim.org/entry/36fff2) --- $$\mathrm{EisensteinE}(2, \tau)=(6\mathrm{WeierstrassZeta}(\frac{1}{2}, \tau))/\pi^2$$ **Holds when** $\Im(\tau)\gt0$. **Symbols:** **WeierstrassZeta** — Weierstrass zeta function. Used by the Compute Engine for simplification. [`3bf702` · Fungrim entry ↗](https://fungrim.org/entry/3bf702) --- $$\tau\mapsto\mathrm{EisensteinE}(6, \tau)^{\prime}(\tau)=\frac{1}{2}(2\pi\imaginaryI(\mathrm{EisensteinE}(2, \tau)\mathrm{EisensteinE}(6, \tau)-\mathrm{EisensteinE}(4, \tau)^2))$$ **Holds when** $\Im(\tau)\gt0$. 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](https://doi.org/10.1007/978-1-4757-3675-5_3) [`3bfced` · Fungrim entry ↗](https://fungrim.org/entry/3bfced) --- $$\mathrm{EisensteinE}(4, \tau)=\frac{\mathrm{DedekindEta}(\tau)^{16}}{\mathrm{DedekindEta}(2\tau)^8}+(256\mathrm{DedekindEta}(2\tau)^{16})/\mathrm{DedekindEta}(\tau)^8$$ **Holds when** $\Im(\tau)\gt0$. 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. [`4da2cd` · Fungrim entry ↗](https://fungrim.org/entry/4da2cd) --- $$\mathrm{EisensteinE}(4, \imaginaryI)=\frac{3\Gamma(1/4)^8}{64\pi^6}$$ Used by the Compute Engine for simplification. [`53fcdd` · Fungrim entry ↗](https://fungrim.org/entry/53fcdd) --- $$\mathrm{EisensteinG}(2, \imaginaryI)=\pi$$ **Symbols:** **EisensteinG** — Eisenstein series. Used by the Compute Engine for simplification. [`570399` · Fungrim entry ↗](https://fungrim.org/entry/570399) --- $$\mathrm{EisensteinE}(6, \exp(\frac{2\pi\imaginaryI}{3}))=\frac{27\Gamma(1/3)^{18}}{512\pi^{12}}$$ Used by the Compute Engine for simplification. [`6c71c0` · Fungrim entry ↗](https://fungrim.org/entry/6c71c0) --- $$\mathrm{EisensteinE}(8, \tau)=\frac{1}{2}(\mathrm{JacobiTheta}(2, 0, \tau)^{16}+\mathrm{JacobiTheta}(3, 0, \tau)^{16}+\mathrm{JacobiTheta}(4, 0, \tau)^{16})$$ **Holds when** $\Im(\tau)\gt0$. Used by the Compute Engine for expansion. [`6d2880` · Fungrim entry ↗](https://fungrim.org/entry/6d2880) --- $$\mathrm{EisensteinE}(2, \tau)=1-12(\sum_{m=1}^{\infty}(\cos(2\pi m\tau)-1)^{-1})$$ **Holds when** $\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`7b62e4` · Fungrim entry ↗](https://fungrim.org/entry/7b62e4) --- $$\mathrm{EisensteinE}(2k, \tau)=1-\frac{4k(\sum_{n=1}^{\infty}\mathrm{DivisorSigma}(2k-1, n)\exp(2\imaginaryI\pi n\tau))}{\mathrm{BernoulliB}(2k)}$$ **Holds when** $k\in\N^*\land\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`7c00e6` · Fungrim entry ↗](https://fungrim.org/entry/7c00e6) --- $$\tau\mapsto\mathrm{EisensteinE}(2, \tau)^{\prime}(\tau)=\frac{1}{12}(2\pi\imaginaryI(\mathrm{EisensteinE}(2, \tau)^2-\mathrm{EisensteinE}(4, \tau)))$$ **Holds when** $\Im(\tau)\gt0$. 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](https://doi.org/10.1007/978-1-4757-3675-5_3) [`7cda09` · Fungrim entry ↗](https://fungrim.org/entry/7cda09) --- $$\mathrm{EisensteinE}(2k, \tau)=1-\frac{4k(\sum_{n=1}^{\infty}(\exp(2\imaginaryI\pi n\tau)n^{2k-1})/(1-\exp(2\imaginaryI\pi n\tau)))}{\mathrm{BernoulliB}(2k)}$$ **Holds when** $k\in\N^*\land\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`848d97` · Fungrim entry ↗](https://fungrim.org/entry/848d97) --- $$\mathrm{EisensteinE}(14, \tau)=\mathrm{EisensteinE}(4, \tau)\mathrm{EisensteinE}(10, \tau)$$ **Holds when** $\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`9e1f83` · Fungrim entry ↗](https://fungrim.org/entry/9e1f83) --- $$\mathrm{EisensteinG}(2, \exp(\frac{2\pi\imaginaryI}{3}))=\frac{2\pi}{\sqrt{3}}$$ **Symbols:** **EisensteinG** — Eisenstein series. Used by the Compute Engine for simplification. [`9ea739` · Fungrim entry ↗](https://fungrim.org/entry/9ea739) --- $$\mathrm{EisensteinE}(6, \tau)^2=\frac{1}{8}({(\mathrm{JacobiTheta}(2, 0, \tau)^8+\mathrm{JacobiTheta}(3, 0, \tau)^8+\mathrm{JacobiTheta}(4, 0, \tau)^8)}^3-54(\mathrm{JacobiTheta}(2, 0, \tau)\mathrm{JacobiTheta}(3, 0, \tau)\mathrm{JacobiTheta}(4, 0, \tau))^8)$$ **Holds when** $\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`a0dff6` · Fungrim entry ↗](https://fungrim.org/entry/a0dff6) --- $$\mathrm{EisensteinG}(6, \imaginaryI)=\mathrm{EisensteinE}(6, \imaginaryI)=0$$ **Symbols:** **EisensteinG** — Eisenstein series. Used by the Compute Engine for simplification. [`a4109c` · Fungrim entry ↗](https://fungrim.org/entry/a4109c) --- $$\mathrm{EisensteinE}(2, \imaginaryI)=\frac{3}{\pi}$$ Used by the Compute Engine for simplification. [`a691b3` · Fungrim entry ↗](https://fungrim.org/entry/a691b3) --- $$\mathrm{EisensteinE}(4, \tau)=1+30(\sum_{m=1}^{\infty}\frac{\cos(\pi m\tau)^2+1}{\sin(\pi m\tau)^4})$$ **Holds when** $\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`a92c1a` · Fungrim entry ↗](https://fungrim.org/entry/a92c1a) --- $$\mathrm{EisensteinE}(10, \tau)=\mathrm{EisensteinE}(4, \tau)\mathrm{EisensteinE}(6, \tau)$$ **Holds when** $\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`adaf5a` · Fungrim entry ↗](https://fungrim.org/entry/adaf5a) --- $$\tau\mapsto\mathrm{EisensteinE}(4, \tau)^{\prime}(\tau)=\frac{1}{3}(2\pi\imaginaryI(\mathrm{EisensteinE}(2, \tau)\mathrm{EisensteinE}(4, \tau)-\mathrm{EisensteinE}(6, \tau)))$$ **Holds when** $\Im(\tau)\gt0$. 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](https://doi.org/10.1007/978-1-4757-3675-5_3) [`af2ea9` · Fungrim entry ↗](https://fungrim.org/entry/af2ea9) --- $$\mathrm{EisensteinG}(2k, \tau)=2\Zeta(2k)+2(\sum_{m=1}^{\infty}\sum_{n\in \Z}((m\tau+n)^{2k})^{-1})$$ **Holds when** $k\in\N^*\land\Im(\tau)\gt0$. **Symbols:** **EisensteinG** — Eisenstein series. Used by the Compute Engine for simplification. [`b07750` · Fungrim entry ↗](https://fungrim.org/entry/b07750) --- $$\mathrm{EisensteinG}(2, \tau)=2\mathrm{WeierstrassZeta}(\frac{1}{2}, \tau)$$ **Holds when** $\Im(\tau)\gt0$. **Symbols:** **EisensteinG** — Eisenstein series; **WeierstrassZeta** — Weierstrass zeta function. Used by the Compute Engine for simplification. [`b52b6f` · Fungrim entry ↗](https://fungrim.org/entry/b52b6f) --- $$\mathrm{EisensteinE}(4, \tau)^3-\mathrm{EisensteinE}(6, \tau)^2=\frac{1}{4}(27(\mathrm{JacobiTheta}(2, 0, \tau)\mathrm{JacobiTheta}(3, 0, \tau)\mathrm{JacobiTheta}(4, 0, \tau))^8)$$ **Holds when** $\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`bd7d8e` · Fungrim entry ↗](https://fungrim.org/entry/bd7d8e) --- $$\mathrm{EisensteinE}(4, \tau)=\frac{1}{2}(\mathrm{JacobiTheta}(2, 0, \tau)^8+\mathrm{JacobiTheta}(3, 0, \tau)^8+\mathrm{JacobiTheta}(4, 0, \tau)^8)$$ **Holds when** $\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`cc579c` · Fungrim entry ↗](https://fungrim.org/entry/cc579c) --- $$\mathrm{EisensteinE}(2k, \tau+n)=\mathrm{EisensteinE}(2k, \tau)$$ **Holds when** $k\in\N^*\land\Im(\tau)\gt0\land n\in\Z$. Used by the Compute Engine for simplification. [`d56eb6` · Fungrim entry ↗](https://fungrim.org/entry/d56eb6) --- $$\mathrm{EisensteinG}(2, \tau)=-(\frac{1}{\mathrm{DedekindEta}(\tau)}((4\imaginaryI)\pi\tau\mapsto\mathrm{DedekindEta}(\tau)^{\prime}(\tau)))$$ **Holds when** $\Im(\tau)\gt0$. **Symbols:** **EisensteinG** — Eisenstein series. Used by the Compute Engine for simplification. [`dbf388` · Fungrim entry ↗](https://fungrim.org/entry/dbf388) --- $$\mathrm{EisensteinG}(4, \imaginaryI)=\frac{\Gamma(1/4)^8}{960\pi^2}$$ **Symbols:** **EisensteinG** — Eisenstein series. Used by the Compute Engine for simplification. [`e03b7c` · Fungrim entry ↗](https://fungrim.org/entry/e03b7c) --- $$\mathrm{EisensteinE}(14, \tau)=\mathrm{EisensteinE}(4, \tau)^2\mathrm{EisensteinE}(6, \tau)$$ **Holds when** $\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`e60fd4` · Fungrim entry ↗](https://fungrim.org/entry/e60fd4) --- $$\mathrm{EisensteinE}(14, \tau)=\mathrm{EisensteinE}(6, \tau)\mathrm{EisensteinE}(8, \tau)$$ **Holds when** $\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`feb95e` · Fungrim entry ↗](https://fungrim.org/entry/feb95e) --- ## 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** $z\in\C\land\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`0096a8` · Fungrim entry ↗](https://fungrim.org/entry/0096a8) --- $$z\mapsto\frac{\mathrm{JacobiTheta}(3, z, \tau)}{\mathrm{JacobiTheta}(1, z, \tau)}^{\prime}(z)=-(\frac{\pi\mathrm{JacobiTheta}(2, z, \tau)\mathrm{JacobiTheta}(4, z, \tau)\mathrm{JacobiTheta}(3, 0, \tau)^2}{\mathrm{JacobiTheta}(1, z, \tau)^2})$$ **Holds when** $z\in\C\land\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`0373dc` · Fungrim entry ↗](https://fungrim.org/entry/0373dc) --- $$\mathrm{JacobiTheta}(4, 0, \tau, 2r+1)=0$$ **Holds when** $\Im(\tau)\gt0\land r\in\N$. Used by the Compute Engine for simplification. [`055b0a` · Fungrim entry ↗](https://fungrim.org/entry/055b0a) --- $$\mathrm{JacobiTheta}(2, z, \frac{-1}{\tau})=\sqrt{\frac{\tau}{\imaginaryI}}\exp(\pi\imaginaryI\tau z^2)\mathrm{JacobiTheta}(4, \tau z, \tau)$$ **Holds when** $z\in\C\land\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`06319a` · Fungrim entry ↗](https://fungrim.org/entry/06319a) --- $$\mathrm{JacobiTheta}(3, 0, \tau)^2=1+4(\sum_{n=1}^{\infty}\frac{\exp(\pi\imaginaryI\tau)^{n}}{1+\exp(\pi\imaginaryI\tau)^{2n}})$$ **Holds when** $\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`0650f8` · Fungrim entry ↗](https://fungrim.org/entry/0650f8) --- $$\mathrm{JacobiTheta}(3, 0, \tau)\mathrm{JacobiTheta}(4, 0, \tau)\mathrm{JacobiTheta}(3, z+w, \tau)\mathrm{JacobiTheta}(4, z-w, \tau)=\mathrm{JacobiTheta}(3, z, \tau)\mathrm{JacobiTheta}(4, z, \tau)\mathrm{JacobiTheta}(3, w, \tau)\mathrm{JacobiTheta}(4, w, \tau)-\mathrm{JacobiTheta}(1, z, \tau)\mathrm{JacobiTheta}(2, z, \tau)\mathrm{JacobiTheta}(1, w, \tau)\mathrm{JacobiTheta}(2, w, \tau)$$ **Holds when** $z\in\C\land w\in\C\land\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`077394` · Fungrim entry ↗](https://fungrim.org/entry/077394) --- $$\mathrm{JacobiTheta}(2, z, \tau)=\exp(\pi\imaginaryI(z+\frac{\tau}{4}))\mathrm{JacobiTheta}(3, z+\frac{\tau}{2}, \tau)$$ **Holds when** $z\in\C\land\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`0878a4` · Fungrim entry ↗](https://fungrim.org/entry/0878a4) --- $$\mathrm{JacobiTheta}(1, z, \tau)^4+\mathrm{JacobiTheta}(3, z, \tau)^4=\mathrm{JacobiTheta}(2, z, \tau)^4+\mathrm{JacobiTheta}(4, z, \tau)^4$$ **Holds when** $z\in\C\land\Im(\tau)\gt0$. Used by the Compute Engine for expansion. [`08822c` · Fungrim entry ↗](https://fungrim.org/entry/08822c) --- $$\mathrm{JacobiTheta}(3, 2z, 2\tau)=\frac{\mathrm{JacobiTheta}(3, 1/4-z, \tau)\mathrm{JacobiTheta}(3, \frac{1}{4}+z, \tau)}{\mathrm{JacobiTheta}(4, 0, 2\tau)}$$ **Holds when** $z\in\C\land\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`0a9ec2` · Fungrim entry ↗](https://fungrim.org/entry/0a9ec2) --- $$\mathrm{JacobiTheta}(3, 0, \tau)^2\mathrm{JacobiTheta}(2, z, \tau)^2=\mathrm{JacobiTheta}(2, 0, \tau)^2\mathrm{JacobiTheta}(3, z, \tau)^2-\mathrm{JacobiTheta}(4, 0, \tau)^2\mathrm{JacobiTheta}(1, z, \tau)^2$$ **Holds when** $z\in\C\land\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`0e2635` · Fungrim entry ↗](https://fungrim.org/entry/0e2635) --- $$\mathrm{JacobiTheta}(3, z+\tau, \tau)=\exp(-(\pi\imaginaryI(2z+\tau)))\mathrm{JacobiTheta}(3, z, \tau)$$ **Holds when** $z\in\C\land\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`103bfb` · Fungrim entry ↗](https://fungrim.org/entry/103bfb) --- $$\mathrm{JacobiTheta}(4, z, \tau)=\imaginaryI\exp(\pi\imaginaryI(z+\frac{\tau}{4}))\mathrm{JacobiTheta}(2, z+\frac{1}{2}+\frac{\tau}{2}, \tau)$$ **Holds when** $z\in\C\land\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`10ca40` · Fungrim entry ↗](https://fungrim.org/entry/10ca40) --- $$\mathrm{JacobiTheta}(3, 0, \imaginaryI)=\frac{\Gamma(\frac{1}{4})}{\sqrt{2}\pi^{\frac{3}{4}}}$$ Used by the Compute Engine for simplification. [`1403b5` · Fungrim entry ↗](https://fungrim.org/entry/1403b5) --- $$\mathrm{JacobiTheta}(1, z, \tau)\mathrm{JacobiTheta}(1, w, \tau)=\mathrm{JacobiTheta}(3, z+w, 2\tau)\mathrm{JacobiTheta}(2, z-w, 2\tau)-\mathrm{JacobiTheta}(2, z+w, 2\tau)\mathrm{JacobiTheta}(3, z-w, 2\tau)$$ **Holds when** $z\in\C\land w\in\C\land\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`1792a9` · Fungrim entry ↗](https://fungrim.org/entry/1792a9) --- $$\mathrm{JacobiTheta}(4, z, \tau+2n)=\mathrm{JacobiTheta}(4, z, \tau)$$ **Holds when** $z\in\C\land\Im(\tau)\gt0\land n\in\Z$. Used by the Compute Engine for simplification. [`19acd8` · Fungrim entry ↗](https://fungrim.org/entry/19acd8) --- $$\mathrm{JacobiTheta}(2, 0, 2\tau)(\mathrm{JacobiTheta}(1, z, \tau)^2-\mathrm{JacobiTheta}(2, z, \tau)^2)=\mathrm{JacobiTheta}(3, 0, 2\tau)(\mathrm{JacobiTheta}(4, z, \tau)^2-\mathrm{JacobiTheta}(3, z, \tau)^2)$$ **Holds when** $z\in\C\land\Im(\tau)\gt0$. Used by the Compute Engine for expansion. [`1c67c8` · Fungrim entry ↗](https://fungrim.org/entry/1c67c8) --- $$\mathrm{JacobiTheta}(4, 0, \tau)^4-\mathrm{JacobiTheta}(2, 0, \tau)^4=1-24(\sum_{n=0}^{\infty}\frac{(2n+1)\exp(\pi\imaginaryI\tau)^{2n+1}}{1+\exp(\pi\imaginaryI\tau)^{2n+1}})$$ **Holds when** $\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`1cec67` · Fungrim entry ↗](https://fungrim.org/entry/1cec67) --- $$\mathrm{JacobiTheta}(1, z, \tau+n)=\exp(\frac{\pi\imaginaryI n}{4})\mathrm{JacobiTheta}(1, z, \tau)$$ **Holds when** $z\in\C\land\Im(\tau)\gt0\land n\in\Z$. Used by the Compute Engine for simplification. [`1fa8e7` · Fungrim entry ↗](https://fungrim.org/entry/1fa8e7) --- $$\mathrm{JacobiTheta}(3, 0, \tau)^4=\mathrm{JacobiTheta}(2, 0, \tau)^4+\mathrm{JacobiTheta}(4, 0, \tau)^4$$ **Holds when** $\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`1fbc09` · Fungrim entry ↗](https://fungrim.org/entry/1fbc09) --- $$\mathrm{JacobiTheta}(2, z+w, \tau)\mathrm{JacobiTheta}(2, z-w, \tau)\mathrm{JacobiTheta}(2, 0, \tau)^2=\mathrm{JacobiTheta}(2, z, \tau)^2\mathrm{JacobiTheta}(2, w, \tau)^2-\mathrm{JacobiTheta}(1, z, \tau)^2\mathrm{JacobiTheta}(1, w, \tau)^2=\mathrm{JacobiTheta}(3, z, \tau)^2\mathrm{JacobiTheta}(3, w, \tau)^2-\mathrm{JacobiTheta}(4, z, \tau)^2\mathrm{JacobiTheta}(4, w, \tau)^2$$ **Holds when** $z\in\C\land w\in\C\land\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`1feda6` · Fungrim entry ↗](https://fungrim.org/entry/1feda6) --- $$\mathrm{JacobiTheta}(3, 2z, \tau)=\frac{\mathrm{JacobiTheta}(1, z, \tau)^4+\mathrm{JacobiTheta}(3, z, \tau)^4}{\mathrm{JacobiTheta}(3, 0, \tau)^3}$$ **Holds when** $z\in\C\land\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`20d581` · Fungrim entry ↗](https://fungrim.org/entry/20d581) --- $$2\mathrm{JacobiTheta}(2, 0, 2\tau)^2=\mathrm{JacobiTheta}(3, 0, \tau)^2-\mathrm{JacobiTheta}(4, 0, \tau)^2$$ **Holds when** $\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`21c2f7` · Fungrim entry ↗](https://fungrim.org/entry/21c2f7) --- $$\mathrm{JacobiTheta}(4, 2z, \tau)=\frac{\mathrm{JacobiTheta}(1, z, \tau)^2\mathrm{JacobiTheta}(2, z, \tau)^2+\mathrm{JacobiTheta}(3, z, \tau)^2\mathrm{JacobiTheta}(4, z, \tau)^2}{\mathrm{JacobiTheta}(4, 0, \tau)\mathrm{JacobiTheta}(3, 0, \tau)^2}$$ **Holds when** $z\in\C\land\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`21dc98` · Fungrim entry ↗](https://fungrim.org/entry/21dc98) --- $$z\mapsto\frac{\mathrm{JacobiTheta}(4, z, \tau)}{\mathrm{JacobiTheta}(3, z, \tau)}^{\prime}(z)=\frac{\pi\mathrm{JacobiTheta}(1, z, \tau)\mathrm{JacobiTheta}(2, z, \tau)\mathrm{JacobiTheta}(2, 0, \tau)^2}{\mathrm{JacobiTheta}(3, z, \tau)^2}$$ **Holds when** $z\in\C\land\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`23077c` · Fungrim entry ↗](https://fungrim.org/entry/23077c) --- $$\mathrm{JacobiTheta}(3, z, \tau)=\exp(\pi\imaginaryI(z+\frac{\tau}{4}))\mathrm{JacobiTheta}(1, z+\frac{1}{2}+\frac{\tau}{2}, \tau)$$ **Holds when** $z\in\C\land\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`235d0d` · Fungrim entry ↗](https://fungrim.org/entry/235d0d) --- $$\mathrm{JacobiTheta}(2, 0, \tau)^2\mathrm{JacobiTheta}(4, z, \tau)^2=\mathrm{JacobiTheta}(3, 0, \tau)^2\mathrm{JacobiTheta}(1, z, \tau)^2+\mathrm{JacobiTheta}(4, 0, \tau)^2\mathrm{JacobiTheta}(2, z, \tau)^2$$ **Holds when** $z\in\C\land\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`265d9c` · Fungrim entry ↗](https://fungrim.org/entry/265d9c) --- $$\frac{\mathrm{JacobiTheta}(1, 0, \tau, 3)}{\mathrm{JacobiTheta}(1, 0, \tau, 1)}=\frac{\mathrm{JacobiTheta}(2, 0, \tau, 2)}{\mathrm{JacobiTheta}(2, 0, \tau)}+\frac{\mathrm{JacobiTheta}(3, 0, \tau, 2)}{\mathrm{JacobiTheta}(3, 0, \tau)}+\frac{\mathrm{JacobiTheta}(4, 0, \tau, 2)}{\mathrm{JacobiTheta}(4, 0, \tau)}$$ **Holds when** $\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`278274` · Fungrim entry ↗](https://fungrim.org/entry/278274) --- $$\mathrm{JacobiTheta}(1, 4z, 4\tau)=\frac{\mathrm{JacobiTheta}(1, z, \tau)\mathrm{JacobiTheta}(1, 1/4-z, \tau)\mathrm{JacobiTheta}(1, \frac{1}{4}+z, \tau)\mathrm{JacobiTheta}(2, z, \tau)}{\mathrm{JacobiTheta}(3, 0, \tau)\mathrm{JacobiTheta}(4, 0, \tau)\mathrm{JacobiTheta}(3, \frac{1}{4}, \tau)}$$ **Holds when** $z\in\C\land\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`27b169` · Fungrim entry ↗](https://fungrim.org/entry/27b169) --- $$z\mapsto\frac{\mathrm{JacobiTheta}(3, z, \tau)}{\mathrm{JacobiTheta}(2, z, \tau)}^{\prime}(z)=\frac{\pi\mathrm{JacobiTheta}(1, z, \tau)\mathrm{JacobiTheta}(4, z, \tau)\mathrm{JacobiTheta}(4, 0, \tau)^2}{\mathrm{JacobiTheta}(2, z, \tau)^2}$$ **Holds when** $z\in\C\land\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`2853d4` · Fungrim entry ↗](https://fungrim.org/entry/2853d4) --- $$\mathrm{JacobiTheta}(3, z, \tau+n)=\begin{cases}\mathrm{JacobiTheta}(3, z, \tau)&\mathrm{IsEven}(n)\\\mathrm{JacobiTheta}(4, z, \tau)&\mathrm{IsOdd}(n)\end{cases}$$ **Holds when** $z\in\C\land\Im(\tau)\gt0\land n\in\Z$. Used by the Compute Engine for simplification. [`28b4c3` · Fungrim entry ↗](https://fungrim.org/entry/28b4c3) --- $$\mathrm{JacobiTheta}(3, z+\frac{\tau}{2}, \tau)=\exp(-(\pi\imaginaryI(z+\frac{\tau}{4})))\mathrm{JacobiTheta}(2, z, \tau)$$ **Holds when** $z\in\C\land\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`2d2dde` · Fungrim entry ↗](https://fungrim.org/entry/2d2dde) --- $$\mathrm{JacobiTheta}(3, z+m+n\tau, \tau)=\exp(-(\pi\imaginaryI(\tau n^2+2nz)))\mathrm{JacobiTheta}(3, z, \tau)$$ **Holds when** $z\in\C\land\Im(\tau)\gt0\land m\in\Z\land n\in\Z$. Used by the Compute Engine for expansion. [`2e4da0` · Fungrim entry ↗](https://fungrim.org/entry/2e4da0) --- $$\mathrm{JacobiTheta}(1, \frac{n}{4}, \imaginaryI)=\begin{cases}0&\mathrm{CongruentMod}(n, 0, 4)\\(-1)^{\lfloor n/4\rfloor}\mathrm{JacobiTheta}(4, 0, \imaginaryI)&\mathrm{CongruentMod}(n, 2, 4)\\(-1)^{\lfloor n/4\rfloor}\frac{\sqrt{2^{1/2}-1}}{2^{7/16}}\sqrt[4]{\sqrt{2}+1}\mathrm{JacobiTheta}(3, 0, \imaginaryI)&\top\end{cases}$$ **Holds when** $n\in\Z$. Used by the Compute Engine for simplification. [`2f3ed3` · Fungrim entry ↗](https://fungrim.org/entry/2f3ed3) --- $$\mathrm{JacobiTheta}(1, z+2n, \tau)=\mathrm{JacobiTheta}(1, z, \tau)$$ **Holds when** $z\in\C\land\Im(\tau)\gt0\land n\in\Z$. Used by the Compute Engine for simplification. [`2faeb9` · Fungrim entry ↗](https://fungrim.org/entry/2faeb9) --- $$\mathrm{JacobiTheta}(3, 0, 1+\frac{\imaginaryI}{2})=\frac{\mathrm{JacobiTheta}(3, 0, \imaginaryI){(2^{1/2}-1)}^{\frac{2}{3}}\sqrt[12]{4+3\sqrt{2}}}{2^{\frac{7}{24}}}$$ Used by the Compute Engine for simplification. **Reference:** [doi.org](https://doi.org/10.1016/j.jmaa.2003.12.009) [`324483` · Fungrim entry ↗](https://fungrim.org/entry/324483) --- $$\mathrm{JacobiTheta}(3, 2z, 2\tau)=\frac{\mathrm{JacobiTheta}(1, z, \tau)^2+\mathrm{JacobiTheta}(2, z, \tau)^2}{2\mathrm{JacobiTheta}(2, 0, 2\tau)}$$ **Holds when** $z\in\C\land\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`3479be` · Fungrim entry ↗](https://fungrim.org/entry/3479be) --- $$\mathrm{JacobiTheta}(3, 0, \tau)\mathrm{JacobiTheta}(4, 0, \tau)\mathrm{JacobiTheta}(1, z+w, \tau)\mathrm{JacobiTheta}(2, z-w, \tau)=\mathrm{JacobiTheta}(1, z, \tau)\mathrm{JacobiTheta}(2, z, \tau)\mathrm{JacobiTheta}(3, w, \tau)\mathrm{JacobiTheta}(4, w, \tau)+\mathrm{JacobiTheta}(3, z, \tau)\mathrm{JacobiTheta}(4, z, \tau)\mathrm{JacobiTheta}(1, w, \tau)\mathrm{JacobiTheta}(2, w, \tau)$$ **Holds when** $z\in\C\land w\in\C\land\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`34d1c6` · Fungrim entry ↗](https://fungrim.org/entry/34d1c6) --- $$z\mapsto\frac{\mathrm{JacobiTheta}(3, z, \tau)}{\mathrm{JacobiTheta}(4, z, \tau)}^{\prime}(z)=-(\frac{\pi\mathrm{JacobiTheta}(1, z, \tau)\mathrm{JacobiTheta}(2, z, \tau)\mathrm{JacobiTheta}(2, 0, \tau)^2}{\mathrm{JacobiTheta}(4, z, \tau)^2})$$ **Holds when** $z\in\C\land\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`378949` · Fungrim entry ↗](https://fungrim.org/entry/378949) --- $$\tau\mapsto\mathrm{JacobiTheta}(j, z, \tau, s)^{\prime}(\tau)=\frac{\mathrm{JacobiTheta}(j, z, \tau, 2r+s)}{(4\imaginaryI\pi)^{r}}$$ **Holds when** $j\in\lbrace1, 2, 3, 4\rbrace\land z\in\C\land\Im(\tau)\gt0\land r\in\N\land s\in\N$. Used by the Compute Engine for simplification. [`37e644` · Fungrim entry ↗](https://fungrim.org/entry/37e644) --- $$\mathrm{JacobiTheta}(3, -z, \tau)=\mathrm{JacobiTheta}(3, z, \tau)$$ **Holds when** $z\in\C\land\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`380076` · Fungrim entry ↗](https://fungrim.org/entry/380076) --- $$\mathrm{JacobiTheta}(3, 0, 1+10\imaginaryI)=\frac{\sqrt{5}\mathrm{JacobiTheta}(3, 0, \imaginaryI)\times2^{\frac{7}{8}}}{5(\sqrt[4]{5}-1)\sqrt{1+\sqrt{5}}}$$ Used by the Compute Engine for simplification. **Reference:** [doi.org](https://doi.org/10.1016/j.jmaa.2003.12.009) [`390158` · Fungrim entry ↗](https://fungrim.org/entry/390158) --- $$\mathrm{JacobiTheta}(2, 2z, \tau)=\frac{\mathrm{JacobiTheta}(2, z, \tau)^4-\mathrm{JacobiTheta}(1, z, \tau)^4}{\mathrm{JacobiTheta}(2, 0, \tau)^3}$$ **Holds when** $z\in\C\land\Im(\tau)\gt0$. Used by the Compute Engine for expansion. [`3a77e0` · Fungrim entry ↗](https://fungrim.org/entry/3a77e0) --- $$\mathrm{JacobiTheta}(4, z+w, \tau)\mathrm{JacobiTheta}(4, z-w, \tau)\mathrm{JacobiTheta}(3, 0, \tau)^2=\mathrm{JacobiTheta}(1, z, \tau)^2\mathrm{JacobiTheta}(2, w, \tau)^2+\mathrm{JacobiTheta}(3, z, \tau)^2\mathrm{JacobiTheta}(4, w, \tau)^2=\mathrm{JacobiTheta}(2, z, \tau)^2\mathrm{JacobiTheta}(1, w, \tau)^2+\mathrm{JacobiTheta}(4, z, \tau)^2\mathrm{JacobiTheta}(3, w, \tau)^2$$ **Holds when** $z\in\C\land w\in\C\land\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`3cac28` · Fungrim entry ↗](https://fungrim.org/entry/3cac28) --- $$\mathrm{JacobiTheta}(3, \frac{n}{4}, \imaginaryI)=\begin{cases}\mathrm{JacobiTheta}(3, 0, \imaginaryI)&\mathrm{CongruentMod}(n, 0, 4)\\\mathrm{JacobiTheta}(4, 0, \imaginaryI)&\mathrm{CongruentMod}(n, 2, 4)\\\frac{(2^{1/2}+1)^{1/4}}{2^{7/16}}\mathrm{JacobiTheta}(3, 0, \imaginaryI)&\top\end{cases}$$ **Holds when** $n\in\Z$. Used by the Compute Engine for simplification. [`3fb309` · Fungrim entry ↗](https://fungrim.org/entry/3fb309) --- $$\mathrm{JacobiTheta}(3, 0, \frac{\imaginaryI}{2})=\sqrt{\frac{\sqrt{2}+1}{2}}\sqrt[4]{2}\mathrm{JacobiTheta}(3, 0, \imaginaryI)$$ Used by the Compute Engine for simplification. **Reference:** [doi.org](https://doi.org/10.1016/j.jmaa.2003.12.009) [`4256f0` · Fungrim entry ↗](https://fungrim.org/entry/4256f0) --- $$\mathrm{JacobiTheta}(4, z+\frac{\tau}{2}, \tau)=\exp(-(\pi\imaginaryI(z+\frac{\tau}{4})))\imaginaryI\mathrm{JacobiTheta}(1, z, \tau)$$ **Holds when** $z\in\C\land\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`429093` · Fungrim entry ↗](https://fungrim.org/entry/429093) --- $$\mathrm{JacobiTheta}(1, z+m+n\tau, \tau)=(-1)^{m+n}\exp(-(\pi\imaginaryI(\tau n^2+2nz)))\mathrm{JacobiTheta}(1, z, \tau)$$ **Holds when** $z\in\C\land\Im(\tau)\gt0\land m\in\Z\land n\in\Z$. Used by the Compute Engine for simplification. [`43fa0e` · Fungrim entry ↗](https://fungrim.org/entry/43fa0e) --- $$\mathrm{JacobiTheta}(4, z+n, \tau)=\mathrm{JacobiTheta}(4, z, \tau)$$ **Holds when** $z\in\C\land\Im(\tau)\gt0\land n\in\Z$. Used by the Compute Engine for simplification. [`4448f1` · Fungrim entry ↗](https://fungrim.org/entry/4448f1) --- $$\mathrm{JacobiTheta}(1, z+w, \tau)\mathrm{JacobiTheta}(1, z-w, \tau)\mathrm{JacobiTheta}(2, 0, \tau)^2=\mathrm{JacobiTheta}(1, z, \tau)^2\mathrm{JacobiTheta}(2, w, \tau)^2-\mathrm{JacobiTheta}(2, z, \tau)^2\mathrm{JacobiTheta}(1, w, \tau)^2=\mathrm{JacobiTheta}(4, z, \tau)^2\mathrm{JacobiTheta}(3, w, \tau)^2-\mathrm{JacobiTheta}(3, z, \tau)^2\mathrm{JacobiTheta}(4, w, \tau)^2$$ **Holds when** $z\in\C\land w\in\C\land\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`45165c` · Fungrim entry ↗](https://fungrim.org/entry/45165c) --- $$\mathrm{JacobiTheta}(4, z+w, \tau)\mathrm{JacobiTheta}(4, z-w, \tau)\mathrm{JacobiTheta}(4, 0, \tau)^2=\mathrm{JacobiTheta}(3, z, \tau)^2\mathrm{JacobiTheta}(3, w, \tau)^2-\mathrm{JacobiTheta}(2, z, \tau)^2\mathrm{JacobiTheta}(2, w, \tau)^2=\mathrm{JacobiTheta}(4, z, \tau)^2\mathrm{JacobiTheta}(4, w, \tau)^2-\mathrm{JacobiTheta}(1, z, \tau)^2\mathrm{JacobiTheta}(1, w, \tau)^2$$ **Holds when** $z\in\C\land w\in\C\land\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`45a130` · Fungrim entry ↗](https://fungrim.org/entry/45a130) --- $$2\mathrm{JacobiTheta}(1, 0, 2\tau, 1)\mathrm{JacobiTheta}(4, 0, 2\tau)=\mathrm{JacobiTheta}(1, 0, \tau, 1)\mathrm{JacobiTheta}(2, 0, \tau)$$ **Holds when** $\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`46f244` · Fungrim entry ↗](https://fungrim.org/entry/46f244) --- $$\mathrm{JacobiTheta}(2, 0, \tau, 2r+1)=0$$ **Holds when** $\Im(\tau)\gt0\land r\in\N$. Used by the Compute Engine for simplification. [`474c51` · Fungrim entry ↗](https://fungrim.org/entry/474c51) --- $$\mathrm{JacobiTheta}(3, 0, \frac{\tau}{2})\mathrm{JacobiTheta}(4, 0, \frac{\tau}{2})=\mathrm{JacobiTheta}(4, 0, \tau)^2$$ **Holds when** $\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`476642` · Fungrim entry ↗](https://fungrim.org/entry/476642) --- $$\mathrm{JacobiTheta}(2, 0, \tau)\mathrm{JacobiTheta}(4, 0, \tau)\mathrm{JacobiTheta}(1, z+w, \tau)\mathrm{JacobiTheta}(3, z-w, \tau)=\mathrm{JacobiTheta}(1, z, \tau)\mathrm{JacobiTheta}(3, z, \tau)\mathrm{JacobiTheta}(2, w, \tau)\mathrm{JacobiTheta}(4, w, \tau)+\mathrm{JacobiTheta}(2, z, \tau)\mathrm{JacobiTheta}(4, z, \tau)\mathrm{JacobiTheta}(1, w, \tau)\mathrm{JacobiTheta}(3, w, \tau)$$ **Holds when** $z\in\C\land w\in\C\land\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`47e587` · Fungrim entry ↗](https://fungrim.org/entry/47e587) --- $$\mathrm{JacobiTheta}(2, 0, y\imaginaryI)=\mathrm{JacobiTheta}(3, 0, \frac{\imaginaryI}{y}+1)/\sqrt{y}$$ **Holds when** $y\in\lparen0, \infty\rparen$. Used by the Compute Engine for simplification. [`47f4ba` · Fungrim entry ↗](https://fungrim.org/entry/47f4ba) --- $$\mathrm{JacobiTheta}(2, z+\frac{1}{2}, \tau)=-\mathrm{JacobiTheta}(1, z, \tau)$$ **Holds when** $z\in\C\land\Im(\tau)\gt0$. Used by the Compute Engine for expansion. [`47f6dd` · Fungrim entry ↗](https://fungrim.org/entry/47f6dd) --- $$\mathrm{JacobiTheta}(3, 0, 5\imaginaryI)=\frac{\sqrt{5}\mathrm{JacobiTheta}(3, 0, \imaginaryI)}{5\sqrt{5^{1/2}-2}}$$ Used by the Compute Engine for simplification. **Reference:** [doi.org](https://doi.org/10.1016/j.jmaa.2003.12.009) [`483e7e` · Fungrim entry ↗](https://fungrim.org/entry/483e7e) --- $$\mathrm{JacobiTheta}(2, z+w, \tau)\mathrm{JacobiTheta}(2, z-w, \tau)\mathrm{JacobiTheta}(4, 0, \tau)^2=\mathrm{JacobiTheta}(4, z, \tau)^2\mathrm{JacobiTheta}(2, w, \tau)^2-\mathrm{JacobiTheta}(1, z, \tau)^2\mathrm{JacobiTheta}(3, w, \tau)^2=\mathrm{JacobiTheta}(2, z, \tau)^2\mathrm{JacobiTheta}(4, w, \tau)^2-\mathrm{JacobiTheta}(3, z, \tau)^2\mathrm{JacobiTheta}(1, w, \tau)^2$$ **Holds when** $z\in\C\land w\in\C\land\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`48a1c6` · Fungrim entry ↗](https://fungrim.org/entry/48a1c6) --- $$\mathrm{JacobiTheta}(1, z, \tau)=-\imaginaryI\exp(\pi\imaginaryI(z+\frac{\tau}{4}))\mathrm{JacobiTheta}(4, z+\frac{\tau}{2}, \tau)$$ **Holds when** $z\in\C\land\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`4c462b` · Fungrim entry ↗](https://fungrim.org/entry/4c462b) --- $$\mathrm{JacobiTheta}(3, 0, 1+\imaginaryI)=\frac{\mathrm{JacobiTheta}(3, 0, \imaginaryI)}{\sqrt[4]{2}}$$ Used by the Compute Engine for simplification. **Reference:** [doi.org](https://doi.org/10.1016/j.jmaa.2003.12.009) [`4c8873` · Fungrim entry ↗](https://fungrim.org/entry/4c8873) --- $$\mathrm{JacobiTheta}(2, z, \tau+4n)=(-1)^{n}\mathrm{JacobiTheta}(2, z, \tau)$$ **Holds when** $z\in\C\land\Im(\tau)\gt0\land n\in\Z$. Used by the Compute Engine for expansion. [`4cf228` · Fungrim entry ↗](https://fungrim.org/entry/4cf228) --- $$\mathrm{JacobiTheta}(4, 0, \tau)^8=1+16(\sum_{n=1}^{\infty}\frac{(-1)^{n}n^3\exp(\pi\imaginaryI\tau)^{n}}{1-\exp(\pi\imaginaryI\tau)^{n}})$$ **Holds when** $\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`4d26ec` · Fungrim entry ↗](https://fungrim.org/entry/4d26ec) --- $$\mathrm{JacobiTheta}(4, -z, \tau)=\mathrm{JacobiTheta}(4, z, \tau)$$ **Holds when** $z\in\C\land\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`4f939e` · Fungrim entry ↗](https://fungrim.org/entry/4f939e) --- $$\mathrm{JacobiTheta}(3, 0, \frac{\imaginaryI}{3})=\sqrt[4]{2\sqrt{3}+3}\mathrm{JacobiTheta}(3, 0, \imaginaryI)$$ Used by the Compute Engine for simplification. **Reference:** [doi.org](https://doi.org/10.1016/j.jmaa.2003.12.009) [`52302f` · Fungrim entry ↗](https://fungrim.org/entry/52302f) --- $$\mathrm{JacobiTheta}(3, 0, 1+6\imaginaryI)=\frac{\mathrm{JacobiTheta}(3, 0, \imaginaryI)\sqrt[3]{1+\sqrt{3}+\sqrt{2}\sqrt[4]{27}}}{2^{\frac{11}{24}}\times3^{\frac{3}{8}}\sqrt[6]{3^{1/2}-1}}$$ Used by the Compute Engine for simplification. **Reference:** [doi.org](https://doi.org/10.1016/j.jmaa.2003.12.009) [`5384f3` · Fungrim entry ↗](https://fungrim.org/entry/5384f3) --- $$\mathrm{JacobiTheta}(3, 2z, 4\tau)=\frac{1}{2}(\mathrm{JacobiTheta}(3, z, \tau)+\mathrm{JacobiTheta}(4, z, \tau))$$ **Holds when** $z\in\C\land\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`53fef4` · Fungrim entry ↗](https://fungrim.org/entry/53fef4) --- $$\mathrm{JacobiTheta}(2, 0, \tau)\mathrm{JacobiTheta}(3, 0, \tau)\mathrm{JacobiTheta}(4, 0, \tau)=2\mathrm{DedekindEta}(\tau)^3$$ **Holds when** $\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`557b19` · Fungrim entry ↗](https://fungrim.org/entry/557b19) --- $$\mathrm{JacobiTheta}(1, z+\frac{1}{2}, \tau)=\mathrm{JacobiTheta}(2, z, \tau)$$ **Holds when** $z\in\C\land\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`563d18` · Fungrim entry ↗](https://fungrim.org/entry/563d18) --- $$\mathrm{JacobiTheta}(3, z+w, \tau)\mathrm{JacobiTheta}(3, z-w, \tau)\mathrm{JacobiTheta}(4, 0, \tau)^2=\mathrm{JacobiTheta}(4, z, \tau)^2\mathrm{JacobiTheta}(3, w, \tau)^2-\mathrm{JacobiTheta}(1, z, \tau)^2\mathrm{JacobiTheta}(2, w, \tau)^2=\mathrm{JacobiTheta}(3, z, \tau)^2\mathrm{JacobiTheta}(4, w, \tau)^2-\mathrm{JacobiTheta}(2, z, \tau)^2\mathrm{JacobiTheta}(1, w, \tau)^2$$ **Holds when** $z\in\C\land w\in\C\land\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`5752b8` · Fungrim entry ↗](https://fungrim.org/entry/5752b8) --- $$\mathrm{JacobiTheta}(1, 0, \frac{\tau}{2}, 1)\mathrm{JacobiTheta}(2, 0, \frac{\tau}{2})=2\mathrm{JacobiTheta}(1, 0, \tau, 1)\mathrm{JacobiTheta}(4, 0, \tau)$$ **Holds when** $\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`59184e` · Fungrim entry ↗](https://fungrim.org/entry/59184e) --- $$\mathrm{JacobiTheta}(1, -z, \tau)=-\mathrm{JacobiTheta}(1, z, \tau)$$ **Holds when** $z\in\C\land\Im(\tau)\gt0$. Used by the Compute Engine for expansion. [`59f8e1` · Fungrim entry ↗](https://fungrim.org/entry/59f8e1) --- $$\mathrm{JacobiTheta}(2, 0, \frac{\tau}{2})^2=2\mathrm{JacobiTheta}(2, 0, \tau)\mathrm{JacobiTheta}(3, 0, \tau)$$ **Holds when** $\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`59fd23` · Fungrim entry ↗](https://fungrim.org/entry/59fd23) --- $$\mathrm{JacobiTheta}(1, z, \tau)^4-\mathrm{JacobiTheta}(2, z, \tau)^4=\mathrm{JacobiTheta}(4, z, \tau)^4-\mathrm{JacobiTheta}(3, z, \tau)^4$$ **Holds when** $z\in\C\land\Im(\tau)\gt0$. Used by the Compute Engine for expansion. [`5a3ebf` · Fungrim entry ↗](https://fungrim.org/entry/5a3ebf) --- $$\mathrm{JacobiTheta}(1, z+n, \tau)=(-1)^{n}\mathrm{JacobiTheta}(1, z, \tau)$$ **Holds when** $z\in\C\land\Im(\tau)\gt0\land n\in\Z$. Used by the Compute Engine for simplification. [`5cdae6` · Fungrim entry ↗](https://fungrim.org/entry/5cdae6) --- $$\mathrm{JacobiTheta}(4, z, \tau)=\mathrm{JacobiTheta}(3, z+\frac{1}{2}, \tau)$$ **Holds when** $z\in\C\land\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`5d41b1` · Fungrim entry ↗](https://fungrim.org/entry/5d41b1) --- $$\mathrm{JacobiTheta}(1, z, \tau)\mathrm{JacobiTheta}(2, w, \tau)=\mathrm{JacobiTheta}(1, z+w, 2\tau)\mathrm{JacobiTheta}(4, z-w, 2\tau)+\mathrm{JacobiTheta}(4, z+w, 2\tau)\mathrm{JacobiTheta}(1, z-w, 2\tau)$$ **Holds when** $z\in\C\land w\in\C\land\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`5f9e54` · Fungrim entry ↗](https://fungrim.org/entry/5f9e54) --- $$\mathrm{JacobiTheta}(1, 2z, \tau)=\frac{2\mathrm{JacobiTheta}(1, z, \tau)\mathrm{JacobiTheta}(2, z, \tau)\mathrm{JacobiTheta}(3, z, \tau)\mathrm{JacobiTheta}(4, z, \tau)}{\mathrm{JacobiTheta}(2, 0, \tau)\mathrm{JacobiTheta}(3, 0, \tau)\mathrm{JacobiTheta}(4, 0, \tau)}$$ **Holds when** $z\in\C\land\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`5fe58d` · Fungrim entry ↗](https://fungrim.org/entry/5fe58d) --- $$z\mapsto\frac{\mathrm{JacobiTheta}(2, z, \tau)}{\mathrm{JacobiTheta}(3, z, \tau)}^{\prime}(z)=-(\frac{\pi\mathrm{JacobiTheta}(1, z, \tau)\mathrm{JacobiTheta}(4, z, \tau)\mathrm{JacobiTheta}(4, 0, \tau)^2}{\mathrm{JacobiTheta}(3, z, \tau)^2})$$ **Holds when** $z\in\C\land\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`64b65d` · Fungrim entry ↗](https://fungrim.org/entry/64b65d) --- $$\mathrm{JacobiTheta}(4, z, \tau+n)=\begin{cases}\mathrm{JacobiTheta}(4, z, \tau)&\mathrm{IsEven}(n)\\\mathrm{JacobiTheta}(3, z, \tau)&\mathrm{IsOdd}(n)\end{cases}$$ **Holds when** $z\in\C\land\Im(\tau)\gt0\land n\in\Z$. Used by the Compute Engine for simplification. [`64f0a5` · Fungrim entry ↗](https://fungrim.org/entry/64f0a5) --- $$\mathrm{JacobiTheta}(1, z+w, \tau)\mathrm{JacobiTheta}(1, z-w, \tau)\mathrm{JacobiTheta}(4, 0, \tau)^2=\mathrm{JacobiTheta}(3, z, \tau)^2\mathrm{JacobiTheta}(2, w, \tau)^2-\mathrm{JacobiTheta}(2, z, \tau)^2\mathrm{JacobiTheta}(3, w, \tau)^2=\mathrm{JacobiTheta}(1, z, \tau)^2\mathrm{JacobiTheta}(4, w, \tau)^2-\mathrm{JacobiTheta}(4, z, \tau)^2\mathrm{JacobiTheta}(1, w, \tau)^2$$ **Holds when** $z\in\C\land w\in\C\land\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`663a02` · Fungrim entry ↗](https://fungrim.org/entry/663a02) --- $$\mathrm{JacobiTheta}(3, 0, 6\imaginaryI)=\frac{\mathrm{JacobiTheta}(3, 0, \imaginaryI)\sqrt[3]{-4+2\sqrt{2}\times3^{3/4}+2\sqrt{3}+3\sqrt{2}-3^{3/4}+3^{5/4}}}{2\times3^{\frac{3}{8}}\sqrt[6]{(2^{1/2}-1)(3^{1/2}-1)}}$$ Used by the Compute Engine for simplification. **Reference:** [doi.org](https://doi.org/10.1016/j.jmaa.2003.12.009) [`669765` · Fungrim entry ↗](https://fungrim.org/entry/669765) --- $$\mathrm{JacobiTheta}(1, z, \frac{\tau}{2})=\frac{2\mathrm{JacobiTheta}(1, z, \tau)\mathrm{JacobiTheta}(4, z, \tau)}{\mathrm{JacobiTheta}(2, 0, \frac{\tau}{2})}$$ **Holds when** $z\in\C\land\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`66eb8b` · Fungrim entry ↗](https://fungrim.org/entry/66eb8b) --- $$\mathrm{JacobiTheta}(3, z+w, \tau)\mathrm{JacobiTheta}(3, z-w, \tau)\mathrm{JacobiTheta}(2, 0, \tau)^2=\mathrm{JacobiTheta}(3, z, \tau)^2\mathrm{JacobiTheta}(2, w, \tau)^2+\mathrm{JacobiTheta}(4, z, \tau)^2\mathrm{JacobiTheta}(1, w, \tau)^2=\mathrm{JacobiTheta}(2, z, \tau)^2\mathrm{JacobiTheta}(3, w, \tau)^2+\mathrm{JacobiTheta}(1, z, \tau)^2\mathrm{JacobiTheta}(4, w, \tau)^2$$ **Holds when** $z\in\C\land w\in\C\land\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`66efb8` · Fungrim entry ↗](https://fungrim.org/entry/66efb8) --- $$\mathrm{JacobiTheta}(3, 0, 1+12\imaginaryI)=\frac{\frac{\mathrm{JacobiTheta}(3, 0, \imaginaryI)\sqrt[3]{2-3\times2^{1/2}+3^{5/4}+3^{3/4}}}{2^{19/48}\times3^{3/8}}}{\sqrt[3]{-1+\sqrt{2}\times3^{3/4}-\sqrt{3}}\sqrt[12]{2^{1/2}-1}\sqrt[6]{1+\sqrt{3}}}$$ Used by the Compute Engine for simplification. **Reference:** [doi.org](https://doi.org/10.1016/j.jmaa.2003.12.009) [`675f23` · Fungrim entry ↗](https://fungrim.org/entry/675f23) --- $$\mathrm{JacobiTheta}(4, 2z, 2\tau)=\frac{\mathrm{JacobiTheta}(3, z, \tau)\mathrm{JacobiTheta}(4, z, \tau)}{\mathrm{JacobiTheta}(4, 0, 2\tau)}$$ **Holds when** $z\in\C\land\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`686ce0` · Fungrim entry ↗](https://fungrim.org/entry/686ce0) --- $$\mathrm{JacobiTheta}(3, z, \frac{\tau}{2})=\frac{\mathrm{JacobiTheta}(2, z, \tau)^2+\mathrm{JacobiTheta}(3, z, \tau)^2}{\mathrm{JacobiTheta}(3, 0, \frac{\tau}{2})}$$ **Holds when** $z\in\C\land\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`69b32e` · Fungrim entry ↗](https://fungrim.org/entry/69b32e) --- $$\mathrm{JacobiTheta}(2, z, \tau)=\exp(\pi\imaginaryI(z+\frac{\tau}{4}))\mathrm{JacobiTheta}(4, z+\frac{1}{2}+\frac{\tau}{2}, \tau)$$ **Holds when** $z\in\C\land\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`6a7704` · Fungrim entry ↗](https://fungrim.org/entry/6a7704) --- $$\mathrm{JacobiTheta}(3, 0, 45\imaginaryI)=\frac{\sqrt{10}(3+\sqrt{5}+(\sqrt{3}+\sqrt{5}+\sqrt[4]{60})\sqrt[3]{2+\sqrt{3}})\mathrm{JacobiTheta}(3, 0, \imaginaryI)}{30\sqrt{1+\sqrt{5}}}$$ Used by the Compute Engine for simplification. **Reference:** [doi.org](https://doi.org/10.1016/j.jmaa.2003.12.009) [`6ade92` · Fungrim entry ↗](https://fungrim.org/entry/6ade92) --- $$\mathrm{JacobiTheta}(1, z, \tau+1)=\exp(\frac{\pi\imaginaryI}{4})\mathrm{JacobiTheta}(1, z, \tau)$$ **Holds when** $z\in\C\land\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`6b2078` · Fungrim entry ↗](https://fungrim.org/entry/6b2078) --- $$\mathrm{JacobiTheta}(3, 0, 1+4\imaginaryI)=\frac{\sqrt[4]{\sqrt{2}+1}}{2^{\frac{7}{16}}}\mathrm{JacobiTheta}(3, 0, \imaginaryI)$$ Used by the Compute Engine for simplification. **Reference:** [doi.org](https://doi.org/10.1016/j.jmaa.2003.12.009) [`6cbce8` · Fungrim entry ↗](https://fungrim.org/entry/6cbce8) --- $$\mathrm{JacobiTheta}(4, z, \tau)=-\imaginaryI\exp(\pi\imaginaryI(z+\frac{\tau}{4}))\mathrm{JacobiTheta}(1, z+\frac{\tau}{2}, \tau)$$ **Holds when** $z\in\C\land\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`6d918c` · Fungrim entry ↗](https://fungrim.org/entry/6d918c) --- $$\mathrm{JacobiTheta}(3, 0, \tau)^2\mathrm{JacobiTheta}(3, z, \tau)^2=\mathrm{JacobiTheta}(4, 0, \tau)^2\mathrm{JacobiTheta}(4, z, \tau)^2+\mathrm{JacobiTheta}(2, 0, \tau)^2\mathrm{JacobiTheta}(2, z, \tau)^2$$ **Holds when** $z\in\C\land\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`6fad93` · Fungrim entry ↗](https://fungrim.org/entry/6fad93) --- $$\mathrm{JacobiTheta}(4, 2z, \tau)=\frac{\mathrm{JacobiTheta}(3, z, \tau)^4-\mathrm{JacobiTheta}(2, z, \tau)^4}{\mathrm{JacobiTheta}(4, 0, \tau)^3}$$ **Holds when** $z\in\C\land\Im(\tau)\gt0$. Used by the Compute Engine for expansion. [`7131cd` · Fungrim entry ↗](https://fungrim.org/entry/7131cd) --- $$\mathrm{JacobiTheta}(2, 2z, 2\tau)=\frac{\mathrm{JacobiTheta}(2, z, \tau)^2-\mathrm{JacobiTheta}(1, z, \tau)^2}{2\mathrm{JacobiTheta}(3, 0, 2\tau)}$$ **Holds when** $z\in\C\land\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`7137a2` · Fungrim entry ↗](https://fungrim.org/entry/7137a2) --- $$z\mapsto\frac{\mathrm{JacobiTheta}(2, z, \tau)}{\mathrm{JacobiTheta}(1, z, \tau)}^{\prime}(z)=-(\frac{\pi\mathrm{JacobiTheta}(3, z, \tau)\mathrm{JacobiTheta}(4, z, \tau)\mathrm{JacobiTheta}(2, 0, \tau)^2}{\mathrm{JacobiTheta}(1, z, \tau)^2})$$ **Holds when** $z\in\C\land\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`713b6b` · Fungrim entry ↗](https://fungrim.org/entry/713b6b) --- $$\mathrm{JacobiTheta}(3, z, \tau)=\exp(\pi\imaginaryI(z+\frac{\tau}{4}))\mathrm{JacobiTheta}(2, z+\frac{\tau}{2}, \tau)$$ **Holds when** $z\in\C\land\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`71d5ee` · Fungrim entry ↗](https://fungrim.org/entry/71d5ee) --- $$\mathrm{JacobiTheta}(3, 0, 7\imaginaryI)=\sqrt{\frac{1}{14}((\sqrt{13+7^{1/2}}+\sqrt{7+3\times7^{1/2}})\sqrt[8]{28})}\mathrm{JacobiTheta}(3, 0, \imaginaryI)$$ Used by the Compute Engine for simplification. **Reference:** [doi.org](https://doi.org/10.1016/j.jmaa.2003.12.009) [`72f583` · Fungrim entry ↗](https://fungrim.org/entry/72f583) --- $$\mathrm{JacobiTheta}(4, z, \tau)\mathrm{JacobiTheta}(4, w, \tau)=\mathrm{JacobiTheta}(3, z+w, 2\tau)\mathrm{JacobiTheta}(3, z-w, 2\tau)-\mathrm{JacobiTheta}(2, z+w, 2\tau)\mathrm{JacobiTheta}(2, z-w, 2\tau)$$ **Holds when** $z\in\C\land w\in\C\land\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`73eb5d` · Fungrim entry ↗](https://fungrim.org/entry/73eb5d) --- $$\mathrm{JacobiTheta}(4, 0, \frac{\tau}{2})^2=\mathrm{JacobiTheta}(3, 0, \tau)^2-\mathrm{JacobiTheta}(2, 0, \tau)^2$$ **Holds when** $\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`7527f1` · Fungrim entry ↗](https://fungrim.org/entry/7527f1) --- $$\mathrm{JacobiTheta}(1, z+w, \tau)\mathrm{JacobiTheta}(1, z-w, \tau)\mathrm{JacobiTheta}(3, 0, \tau)^2=\mathrm{JacobiTheta}(1, z, \tau)^2\mathrm{JacobiTheta}(3, w, \tau)^2-\mathrm{JacobiTheta}(3, z, \tau)^2\mathrm{JacobiTheta}(1, w, \tau)^2=\mathrm{JacobiTheta}(4, z, \tau)^2\mathrm{JacobiTheta}(2, w, \tau)^2-\mathrm{JacobiTheta}(2, z, \tau)^2\mathrm{JacobiTheta}(4, w, \tau)^2$$ **Holds when** $z\in\C\land w\in\C\land\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`75cb8c` · Fungrim entry ↗](https://fungrim.org/entry/75cb8c) --- $$\mathrm{JacobiTheta}(3, z, \tau+2n)=\mathrm{JacobiTheta}(3, z, \tau)$$ **Holds when** $z\in\C\land\Im(\tau)\gt0\land n\in\Z$. Used by the Compute Engine for simplification. [`772c88` · Fungrim entry ↗](https://fungrim.org/entry/772c88) --- $$z\mapsto\frac{\mathrm{JacobiTheta}(4, z, \tau)}{\mathrm{JacobiTheta}(2, z, \tau)}^{\prime}(z)=\frac{\pi\mathrm{JacobiTheta}(1, z, \tau)\mathrm{JacobiTheta}(3, z, \tau)\mathrm{JacobiTheta}(3, 0, \tau)^2}{\mathrm{JacobiTheta}(2, z, \tau)^2}$$ **Holds when** $z\in\C\land\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`775637` · Fungrim entry ↗](https://fungrim.org/entry/775637) --- $$\mathrm{JacobiTheta}(3, 2z, \tau)=\frac{\mathrm{JacobiTheta}(2, z, \tau)^2\mathrm{JacobiTheta}(3, z, \tau)^2+\mathrm{JacobiTheta}(1, z, \tau)^2\mathrm{JacobiTheta}(4, z, \tau)^2}{\mathrm{JacobiTheta}(3, 0, \tau)\mathrm{JacobiTheta}(2, 0, \tau)^2}$$ **Holds when** $z\in\C\land\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`794106` · Fungrim entry ↗](https://fungrim.org/entry/794106) --- $$\mathrm{JacobiTheta}(3, 0, \sqrt{6}\imaginaryI)=\sqrt{\frac{1}{\pi}(2\mathrm{EllipticK}((2-3^{1/2})^2(2^{1/2}-3^{1/2})^2))}$$ Used by the Compute Engine for simplification. **Reference:** [mathworld.wolfram.com](http://mathworld.wolfram.com/PolyasRandomWalkConstants.html) [`799b5e` · Fungrim entry ↗](https://fungrim.org/entry/799b5e) --- $$\mathrm{JacobiTheta}(2, 0, \imaginaryI)=\mathrm{JacobiTheta}(4, 0, \imaginaryI)=\frac{\mathrm{JacobiTheta}(3, 0, \imaginaryI)}{\sqrt[4]{2}}$$ Used by the Compute Engine for simplification. [`7d7c65` · Fungrim entry ↗](https://fungrim.org/entry/7d7c65) --- $$\mathrm{JacobiTheta}(3, 2z, 2\tau)=\frac{\mathrm{JacobiTheta}(3, z, \tau)^2+\mathrm{JacobiTheta}(4, z, \tau)^2}{2\mathrm{JacobiTheta}(3, 0, 2\tau)}$$ **Holds when** $z\in\C\land\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`7e0002` · Fungrim entry ↗](https://fungrim.org/entry/7e0002) --- $$\mathrm{JacobiTheta}(3, 0, \frac{\imaginaryI}{4})=\frac{(1+\frac{1}{\sqrt[4]{2}})\mathrm{JacobiTheta}(3, 0, \imaginaryI)\sqrt{1+\sqrt{2}}}{\sqrt{1+\sqrt{2}}}$$ Used by the Compute Engine for simplification. **Reference:** [doi.org](https://doi.org/10.1016/j.jmaa.2003.12.009) [`7f9273` · Fungrim entry ↗](https://fungrim.org/entry/7f9273) --- $$\mathrm{JacobiTheta}(4, 0, 1+y\imaginaryI)=\mathrm{JacobiTheta}(3, 0, y\imaginaryI)$$ **Holds when** $y\in\lparen0, \infty\rparen$. Used by the Compute Engine for simplification. [`81550a` · Fungrim entry ↗](https://fungrim.org/entry/81550a) --- $$\mathrm{JacobiTheta}(3, 0, 9\imaginaryI)=\frac{1}{3}((1+\sqrt[3]{2(\sqrt{3}+1)})\mathrm{JacobiTheta}(3, 0, \imaginaryI))$$ Used by the Compute Engine for simplification. **Reference:** [doi.org](https://doi.org/10.1016/j.jmaa.2003.12.009) [`8356db` · Fungrim entry ↗](https://fungrim.org/entry/8356db) --- $$\mathrm{JacobiTheta}(3, 0, \tau)=\frac{\mathrm{DedekindEta}((\tau+1)/2)^2}{\mathrm{DedekindEta}(\tau+1)}$$ **Holds when** $\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`85b2ff` · Fungrim entry ↗](https://fungrim.org/entry/85b2ff) --- $$z\mapsto\frac{\mathrm{JacobiTheta}(2, z, \tau)}{\mathrm{JacobiTheta}(4, z, \tau)}^{\prime}(z)=-(\frac{\pi\mathrm{JacobiTheta}(1, z, \tau)\mathrm{JacobiTheta}(3, z, \tau)\mathrm{JacobiTheta}(3, 0, \tau)^2}{\mathrm{JacobiTheta}(4, z, \tau)^2})$$ **Holds when** $z\in\C\land\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`89985a` · Fungrim entry ↗](https://fungrim.org/entry/89985a) --- $$\mathrm{JacobiTheta}(2, z+w, \tau)\mathrm{JacobiTheta}(2, z-w, \tau)\mathrm{JacobiTheta}(3, 0, \tau)^2=\mathrm{JacobiTheta}(2, z, \tau)^2\mathrm{JacobiTheta}(3, w, \tau)^2-\mathrm{JacobiTheta}(4, z, \tau)^2\mathrm{JacobiTheta}(1, w, \tau)^2=\mathrm{JacobiTheta}(3, z, \tau)^2\mathrm{JacobiTheta}(2, w, \tau)^2-\mathrm{JacobiTheta}(1, z, \tau)^2\mathrm{JacobiTheta}(4, w, \tau)^2$$ **Holds when** $z\in\C\land w\in\C\land\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`89c9e4` · Fungrim entry ↗](https://fungrim.org/entry/89c9e4) --- $$\mathrm{JacobiTheta}(3, 0, \tau)^4=1+8(\sum_{n=0}^{\infty}\frac{2n\exp(\pi\imaginaryI\tau)^{2n}}{1+\exp(\pi\imaginaryI\tau)^{2n}})+8(\sum_{n=0}^{\infty}\frac{(2n+1)\exp(\pi\imaginaryI\tau)^{2n+1}}{1-\exp(\pi\imaginaryI\tau)^{2n+1}})$$ **Holds when** $\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`8a316c` · Fungrim entry ↗](https://fungrim.org/entry/8a316c) --- $$\mathrm{JacobiTheta}(4, z, \tau)=1+2(\sum_{n=1}^{\infty}(-1)^{n}\exp(\pi\imaginaryI\tau)^{n^2}\cos(2n\pi z))$$ **Holds when** $z\in\C\land\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`8a34d1` · Fungrim entry ↗](https://fungrim.org/entry/8a34d1) --- $$\mathrm{JacobiTheta}(4, 2z, \tau)=\frac{\mathrm{JacobiTheta}(4, z, \tau)^4-\mathrm{JacobiTheta}(1, z, \tau)^4}{\mathrm{JacobiTheta}(4, 0, \tau)^3}$$ **Holds when** $z\in\C\land\Im(\tau)\gt0$. Used by the Compute Engine for expansion. [`8b825c` · Fungrim entry ↗](https://fungrim.org/entry/8b825c) --- $$\mathrm{JacobiTheta}(4, \frac{n}{4}, \imaginaryI)=\begin{cases}\mathrm{JacobiTheta}(4, 0, \imaginaryI)&\mathrm{CongruentMod}(n, 0, 4)\\\mathrm{JacobiTheta}(3, 0, \imaginaryI)&\mathrm{CongruentMod}(n, 2, 4)\\\frac{(2^{1/2}+1)^{1/4}}{2^{7/16}}\mathrm{JacobiTheta}(3, 0, \imaginaryI)&\top\end{cases}$$ **Holds when** $n\in\Z$. Used by the Compute Engine for simplification. [`8c4ab4` · Fungrim entry ↗](https://fungrim.org/entry/8c4ab4) --- $$\mathrm{JacobiTheta}(4, z+m+n\tau, \tau)=(-1)^{n}\exp(-(\pi\imaginaryI(\tau n^2+2nz)))\mathrm{JacobiTheta}(4, z, \tau)$$ **Holds when** $z\in\C\land\Im(\tau)\gt0\land m\in\Z\land n\in\Z$. Used by the Compute Engine for expansion. [`8d6a1d` · Fungrim entry ↗](https://fungrim.org/entry/8d6a1d) --- $$\mathrm{JacobiTheta}(1, 0, \tau)=0$$ **Holds when** $\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`8f43ab` · Fungrim entry ↗](https://fungrim.org/entry/8f43ab) --- $$\mathrm{JacobiTheta}(4, 2z, \tau)=\frac{\mathrm{JacobiTheta}(1, z, \tau)^2\mathrm{JacobiTheta}(3, z, \tau)^2+\mathrm{JacobiTheta}(2, z, \tau)^2\mathrm{JacobiTheta}(4, z, \tau)^2}{\mathrm{JacobiTheta}(4, 0, \tau)\mathrm{JacobiTheta}(2, 0, \tau)^2}$$ **Holds when** $z\in\C\land\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`931201` · Fungrim entry ↗](https://fungrim.org/entry/931201) --- $$(30{\tau\mapsto\mathrm{JacobiTheta}(j, 0, \tau)^{\prime}(\tau)}^3-15\tau\mapsto\mathrm{JacobiTheta}(j, 0, \tau)^{(0)}(\tau)\tau\mapsto\mathrm{JacobiTheta}(j, 0, \tau)^{\prime}(\tau)\tau\mapsto\mathrm{JacobiTheta}(j, 0, \tau)^{\doubleprime}(\tau)+\tau\mapsto\mathrm{JacobiTheta}(j, 0, \tau)^{(0)}(\tau)^2\tau\mapsto\mathrm{JacobiTheta}(j, 0, \tau)^{\tripleprime}(\tau))^2+32{(\tau\mapsto\mathrm{JacobiTheta}(j, 0, \tau)^{(0)}(\tau)\tau\mapsto\mathrm{JacobiTheta}(j, 0, \tau)^{\doubleprime}(\tau)-3\tau\mapsto\mathrm{JacobiTheta}(j, 0, \tau)^{\prime}(\tau)^2)}^3+\pi^2(\tau\mapsto\mathrm{JacobiTheta}(j, 0, \tau)^{(0)}(\tau)\tau\mapsto\mathrm{JacobiTheta}(j, 0, \tau)^{\doubleprime}(\tau)-3\tau\mapsto\mathrm{JacobiTheta}(j, 0, \tau)^{\prime}(\tau)^2)^2{\tau\mapsto\mathrm{JacobiTheta}(j, 0, \tau)^{(0)}(\tau)}^{10}=0$$ **Holds when** $j\in\lbrace1, 2, 3, 4\rbrace\land\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`936694` · Fungrim entry ↗](https://fungrim.org/entry/936694) --- $$\mathrm{JacobiTheta}(4, 0, \tau)=\frac{1}{\mathrm{DedekindEta}(\tau)}(\mathrm{DedekindEta}(\tau/2)^2)$$ **Holds when** $\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`9448f2` · Fungrim entry ↗](https://fungrim.org/entry/9448f2) --- $$\mathrm{JacobiTheta}(1, z, \tau)=-\mathrm{JacobiTheta}(2, z+\frac{1}{2}, \tau)$$ **Holds when** $z\in\C\land\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`95988c` · Fungrim entry ↗](https://fungrim.org/entry/95988c) --- $$\mathrm{JacobiTheta}(4, z, \frac{\tau}{2})=\frac{\mathrm{JacobiTheta}(3, z, \tau)^2-\mathrm{JacobiTheta}(2, z, \tau)^2}{\mathrm{JacobiTheta}(4, 0, \frac{\tau}{2})}$$ **Holds when** $z\in\C\land\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`95e508` · Fungrim entry ↗](https://fungrim.org/entry/95e508) --- $$\mathrm{JacobiTheta}(3, 0, 4\imaginaryI)=\frac{1}{2}((1+2^{-(1/4)})\mathrm{JacobiTheta}(3, 0, \imaginaryI))$$ Used by the Compute Engine for simplification. **Reference:** [doi.org](https://doi.org/10.1016/j.jmaa.2003.12.009) [`95e9e4` · Fungrim entry ↗](https://fungrim.org/entry/95e9e4) --- $$\mathrm{JacobiTheta}(2, 0, \tau)\mathrm{JacobiTheta}(4, 0, \tau)\mathrm{JacobiTheta}(2, z+w, \tau)\mathrm{JacobiTheta}(4, z-w, \tau)=\mathrm{JacobiTheta}(2, z, \tau)\mathrm{JacobiTheta}(4, z, \tau)\mathrm{JacobiTheta}(2, w, \tau)\mathrm{JacobiTheta}(4, w, \tau)-\mathrm{JacobiTheta}(1, z, \tau)\mathrm{JacobiTheta}(3, z, \tau)\mathrm{JacobiTheta}(1, w, \tau)\mathrm{JacobiTheta}(3, w, \tau)$$ **Holds when** $z\in\C\land w\in\C\land\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`9973ef` · Fungrim entry ↗](https://fungrim.org/entry/9973ef) --- $$2\mathrm{JacobiTheta}(2, 0, 2\tau)\mathrm{JacobiTheta}(3, 0, 2\tau)=\mathrm{JacobiTheta}(2, 0, \tau)^2$$ **Holds when** $\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`9a2054` · Fungrim entry ↗](https://fungrim.org/entry/9a2054) --- $$\mathrm{JacobiTheta}(2, z, \tau)\mathrm{JacobiTheta}(2, w, \tau)=\mathrm{JacobiTheta}(2, z+w, 2\tau)\mathrm{JacobiTheta}(3, z-w, 2\tau)+\mathrm{JacobiTheta}(3, z+w, 2\tau)\mathrm{JacobiTheta}(2, z-w, 2\tau)$$ **Holds when** $z\in\C\land w\in\C\land\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`9a9487` · Fungrim entry ↗](https://fungrim.org/entry/9a9487) --- $$\mathrm{JacobiTheta}(3, z+w, \tau)\mathrm{JacobiTheta}(3, z-w, \tau)\mathrm{JacobiTheta}(3, 0, \tau)^2=\mathrm{JacobiTheta}(1, z, \tau)^2\mathrm{JacobiTheta}(1, w, \tau)^2+\mathrm{JacobiTheta}(3, z, \tau)^2\mathrm{JacobiTheta}(3, w, \tau)^2=\mathrm{JacobiTheta}(2, z, \tau)^2\mathrm{JacobiTheta}(2, w, \tau)^2+\mathrm{JacobiTheta}(4, z, \tau)^2\mathrm{JacobiTheta}(4, w, \tau)^2$$ **Holds when** $z\in\C\land w\in\C\land\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`9aa437` · Fungrim entry ↗](https://fungrim.org/entry/9aa437) --- $$\mathrm{JacobiTheta}(3, z, \tau+1)=\mathrm{JacobiTheta}(4, z, \tau)$$ **Holds when** $z\in\C\land\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`9c1e9a` · Fungrim entry ↗](https://fungrim.org/entry/9c1e9a) --- $$z\mapsto\frac{\mathrm{JacobiTheta}(4, z, \tau)}{\mathrm{JacobiTheta}(1, z, \tau)}^{\prime}(z)=-(\frac{\pi\mathrm{JacobiTheta}(2, z, \tau)\mathrm{JacobiTheta}(3, z, \tau)\mathrm{JacobiTheta}(4, 0, \tau)^2}{\mathrm{JacobiTheta}(1, z, \tau)^2})$$ **Holds when** $z\in\C\land\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`a0552b` · Fungrim entry ↗](https://fungrim.org/entry/a0552b) --- $$\mathrm{JacobiTheta}(2, 2z, 4\tau)=\frac{1}{2}(\mathrm{JacobiTheta}(3, z, \tau)-\mathrm{JacobiTheta}(4, z, \tau))$$ **Holds when** $z\in\C\land\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`a0a1ee` · Fungrim entry ↗](https://fungrim.org/entry/a0a1ee) --- $$\mathrm{JacobiTheta}(3, 0, \tau, 2r+1)=0$$ **Holds when** $\Im(\tau)\gt0\land r\in\N$. Used by the Compute Engine for simplification. [`a19141` · Fungrim entry ↗](https://fungrim.org/entry/a19141) --- $$z\mapsto\mathrm{JacobiTheta}(j, z, \tau)^{\prime}(z)=\mathrm{JacobiTheta}(j, z, \tau, r)$$ **Holds when** $j\in\lbrace1, 2, 3, 4\rbrace\land z\in\C\land\Im(\tau)\gt0\land r\in\N$. Used by the Compute Engine for simplification. [`a222ed` · Fungrim entry ↗](https://fungrim.org/entry/a222ed) --- $$\mathrm{JacobiTheta}(2, 4z, 4\tau)=\frac{\mathrm{JacobiTheta}(2, 1/8-z, \tau)\mathrm{JacobiTheta}(2, \frac{1}{8}+z, \tau)\mathrm{JacobiTheta}(2, 3/8-z, \tau)\mathrm{JacobiTheta}(2, \frac{3}{8}+z, \tau)}{\mathrm{JacobiTheta}(3, 0, \tau)\mathrm{JacobiTheta}(4, 0, \tau)\mathrm{JacobiTheta}(3, \frac{1}{4}, \tau)}$$ **Holds when** $z\in\C\land\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`a255e1` · Fungrim entry ↗](https://fungrim.org/entry/a255e1) --- $$z\mapsto\frac{\mathrm{JacobiTheta}(1, z, \tau)}{\mathrm{JacobiTheta}(4, z, \tau)}^{\prime}(z)=\frac{\pi\mathrm{JacobiTheta}(2, z, \tau)\mathrm{JacobiTheta}(3, z, \tau)\mathrm{JacobiTheta}(4, 0, \tau)^2}{\mathrm{JacobiTheta}(4, z, \tau)^2}$$ **Holds when** $z\in\C\land\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`a4eecf` · Fungrim entry ↗](https://fungrim.org/entry/a4eecf) --- $$\mathrm{JacobiTheta}(4, z, \tau+1)=\mathrm{JacobiTheta}(3, z, \tau)$$ **Holds when** $z\in\C\land\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`a5c258` · Fungrim entry ↗](https://fungrim.org/entry/a5c258) --- $$\mathrm{JacobiTheta}(j, z^\star, \tau)=\mathrm{JacobiTheta}(j, z, -\tau^\star)^\star$$ **Holds when** $j\in\lbrace1, 2, 3, 4\rbrace\land z\in\C\land\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`a891da` · Fungrim entry ↗](https://fungrim.org/entry/a891da) --- $$\mathrm{JacobiTheta}(3, 2z, \tau)=\frac{\mathrm{JacobiTheta}(3, z, \tau)^2\mathrm{JacobiTheta}(4, z, \tau)^2-\mathrm{JacobiTheta}(1, z, \tau)^2\mathrm{JacobiTheta}(2, z, \tau)^2}{\mathrm{JacobiTheta}(3, 0, \tau)\mathrm{JacobiTheta}(4, 0, \tau)^2}$$ **Holds when** $z\in\C\land\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`a94b43` · Fungrim entry ↗](https://fungrim.org/entry/a94b43) --- $$\mathrm{JacobiTheta}(2, 0, \tau)=\frac{1}{\mathrm{DedekindEta}(\tau)}(2\mathrm{DedekindEta}(2\tau)^2)$$ **Holds when** $\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`a9c825` · Fungrim entry ↗](https://fungrim.org/entry/a9c825) --- $$\mathrm{JacobiTheta}(2, z, \frac{\tau}{2})=\frac{2\mathrm{JacobiTheta}(2, z, \tau)\mathrm{JacobiTheta}(3, z, \tau)}{\mathrm{JacobiTheta}(2, 0, \frac{\tau}{2})}$$ **Holds when** $z\in\C\land\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`a9cdda` · Fungrim entry ↗](https://fungrim.org/entry/a9cdda) --- $$\mathrm{JacobiTheta}(2, 2z, \tau)=\frac{\mathrm{JacobiTheta}(2, z, \tau)^2\mathrm{JacobiTheta}(3, z, \tau)^2-\mathrm{JacobiTheta}(1, z, \tau)^2\mathrm{JacobiTheta}(4, z, \tau)^2}{\mathrm{JacobiTheta}(2, 0, \tau)\mathrm{JacobiTheta}(3, 0, \tau)^2}$$ **Holds when** $z\in\C\land\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`aaa582` · Fungrim entry ↗](https://fungrim.org/entry/aaa582) --- $$\mathrm{JacobiTheta}(3, 0, \tau)^2\mathrm{JacobiTheta}(4, z, \tau)^2=\mathrm{JacobiTheta}(2, 0, \tau)^2\mathrm{JacobiTheta}(1, z, \tau)^2+\mathrm{JacobiTheta}(4, 0, \tau)^2\mathrm{JacobiTheta}(3, z, \tau)^2$$ **Holds when** $z\in\C\land\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`abbe42` · Fungrim entry ↗](https://fungrim.org/entry/abbe42) --- $$\mathrm{JacobiTheta}(1, z, \tau+8n)=\mathrm{JacobiTheta}(1, z, \tau)$$ **Holds when** $z\in\C\land\Im(\tau)\gt0\land n\in\Z$. Used by the Compute Engine for simplification. [`abc1e7` · Fungrim entry ↗](https://fungrim.org/entry/abc1e7) --- $$\mathrm{JacobiTheta}(2, 2z, \tau)=\frac{\mathrm{JacobiTheta}(2, z, \tau)^2\mathrm{JacobiTheta}(4, z, \tau)^2-\mathrm{JacobiTheta}(1, z, \tau)^2\mathrm{JacobiTheta}(3, z, \tau)^2}{\mathrm{JacobiTheta}(2, 0, \tau)\mathrm{JacobiTheta}(4, 0, \tau)^2}$$ **Holds when** $z\in\C\land\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`b1d07b` · Fungrim entry ↗](https://fungrim.org/entry/b1d07b) --- $$\mathrm{JacobiTheta}(1, 0, \tau, 2r)=0$$ **Holds when** $\Im(\tau)\gt0\land r\in\N$. Used by the Compute Engine for simplification. [`b3c440` · Fungrim entry ↗](https://fungrim.org/entry/b3c440) --- $$\mathrm{JacobiTheta}(3, z, \tau)=\mathrm{JacobiTheta}(4, z+\frac{1}{2}, \tau)$$ **Holds when** $z\in\C\land\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`b3fc6d` · Fungrim entry ↗](https://fungrim.org/entry/b3fc6d) --- $$\mathrm{JacobiTheta}(2, z+2n, \tau)=\mathrm{JacobiTheta}(2, z, \tau)$$ **Holds when** $z\in\C\land\Im(\tau)\gt0\land n\in\Z$. Used by the Compute Engine for simplification. [`b46534` · Fungrim entry ↗](https://fungrim.org/entry/b46534) --- $$\mathrm{JacobiTheta}(3, 0, 1+2\imaginaryI)=\frac{\mathrm{JacobiTheta}(3, 0, \imaginaryI)}{\sqrt[8]{2}}$$ Used by the Compute Engine for simplification. **Reference:** [doi.org](https://doi.org/10.1016/j.jmaa.2003.12.009) [`b58070` · Fungrim entry ↗](https://fungrim.org/entry/b58070) --- $$\mathrm{JacobiTheta}(4, z+\tau, \tau)=-\exp(-(\pi\imaginaryI(2z+\tau)))\mathrm{JacobiTheta}(4, z, \tau)$$ **Holds when** $z\in\C\land\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`b83f63` · Fungrim entry ↗](https://fungrim.org/entry/b83f63) --- $$\mathrm{JacobiTheta}(1, z, \tau+2n)=\imaginaryI^{n}\mathrm{JacobiTheta}(1, z, \tau)$$ **Holds when** $z\in\C\land\Im(\tau)\gt0\land n\in\Z$. Used by the Compute Engine for expansion. [`b978f0` · Fungrim entry ↗](https://fungrim.org/entry/b978f0) --- $$\mathrm{JacobiTheta}(1, z, \tau+4n)=(-1)^{n}\mathrm{JacobiTheta}(1, z, \tau)$$ **Holds when** $z\in\C\land\Im(\tau)\gt0\land n\in\Z$. Used by the Compute Engine for expansion. [`b9c650` · Fungrim entry ↗](https://fungrim.org/entry/b9c650) --- $$2\mathrm{JacobiTheta}(3, 0, 2\tau)^2=\mathrm{JacobiTheta}(3, 0, \tau)^2+\mathrm{JacobiTheta}(4, 0, \tau)^2$$ **Holds when** $\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`c3d8c2` · Fungrim entry ↗](https://fungrim.org/entry/c3d8c2) --- $$\mathrm{JacobiTheta}(3, z, \frac{-1}{\tau})=\sqrt{\frac{\tau}{\imaginaryI}}\exp(\pi\imaginaryI\tau z^2)\mathrm{JacobiTheta}(3, \tau z, \tau)$$ **Holds when** $z\in\C\land\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`c4b16c` · Fungrim entry ↗](https://fungrim.org/entry/c4b16c) --- $$\mathrm{JacobiTheta}(3, 0, \sqrt{6}\imaginaryI)=\sqrt[4]{\frac{\sqrt{6}\Gamma(1/24)\Gamma(5/24)\Gamma(7/24)\Gamma(11/24)}{96(18-10\sqrt{3}-7\sqrt{6}+12\sqrt{2})\pi^3}}$$ Used by the Compute Engine for simplification. **Reference:** [mathworld.wolfram.com](http://mathworld.wolfram.com/PolyasRandomWalkConstants.html) [`c60033` · Fungrim entry ↗](https://fungrim.org/entry/c60033) --- $$\mathrm{JacobiTheta}(2, 0, \tau)^4=8(\sum_{n=0}^{\infty}\frac{(2n+1)\exp(\pi\imaginaryI\tau)^{2n+1}}{1+\exp(\pi\imaginaryI\tau)^{2n+1}})+8(\sum_{n=0}^{\infty}\frac{(2n+1)\exp(\pi\imaginaryI\tau)^{2n+1}}{1-\exp(\pi\imaginaryI\tau)^{2n+1}})$$ **Holds when** $\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`c743eb` · Fungrim entry ↗](https://fungrim.org/entry/c743eb) --- $$\frac{\mathrm{JacobiTheta}(2, z, \tau, 1)}{\pi\mathrm{JacobiTheta}(2, z, \tau)}=4(\sum_{n=1}^{\infty}\frac{\sin(2\pi nz)\times(-1)^{n}\exp(2\imaginaryI\pi n\tau)}{1-\exp(2\imaginaryI\pi n\tau)})-\tan(\pi z)$$ **Holds when** $z\in\C\land\Im(\tau)\gt0\land\vert\Im(z)\vert\lt\vert\Im(\tau)\vert\land\cos(\pi z)\ne0$. Used by the Compute Engine for simplification. [`c7f7a5` · Fungrim entry ↗](https://fungrim.org/entry/c7f7a5) --- $$\mathrm{JacobiTheta}(4, z+w, \tau)\mathrm{JacobiTheta}(4, z-w, \tau)\mathrm{JacobiTheta}(2, 0, \tau)^2=\mathrm{JacobiTheta}(4, z, \tau)^2\mathrm{JacobiTheta}(2, w, \tau)^2+\mathrm{JacobiTheta}(3, z, \tau)^2\mathrm{JacobiTheta}(1, w, \tau)^2=\mathrm{JacobiTheta}(1, z, \tau)^2\mathrm{JacobiTheta}(3, w, \tau)^2+\mathrm{JacobiTheta}(2, z, \tau)^2\mathrm{JacobiTheta}(4, w, \tau)^2$$ **Holds when** $z\in\C\land w\in\C\land\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`c891a1` · Fungrim entry ↗](https://fungrim.org/entry/c891a1) --- $$\mathrm{JacobiTheta}(4, z, \frac{\tau}{2})=\frac{\mathrm{JacobiTheta}(4, z, \tau)^2+\mathrm{JacobiTheta}(1, z, \tau)^2}{\mathrm{JacobiTheta}(3, 0, \frac{\tau}{2})}$$ **Holds when** $z\in\C\land\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`c92a6f` · Fungrim entry ↗](https://fungrim.org/entry/c92a6f) --- $$z\mapsto\frac{\mathrm{JacobiTheta}(1, z, \tau)}{\mathrm{JacobiTheta}(2, z, \tau)}^{\prime}(z)=\frac{\pi\mathrm{JacobiTheta}(3, z, \tau)\mathrm{JacobiTheta}(4, z, \tau)\mathrm{JacobiTheta}(2, 0, \tau)^2}{\mathrm{JacobiTheta}(2, z, \tau)^2}$$ **Holds when** $z\in\C\land\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`cb493d` · Fungrim entry ↗](https://fungrim.org/entry/cb493d) --- $$\mathrm{JacobiTheta}(3, 0, 5\imaginaryI)=\frac{\mathrm{JacobiTheta}(3, 0, \imaginaryI)\sqrt{5+2\sqrt{5}}}{5^{\frac{3}{4}}}$$ Used by the Compute Engine for simplification. **Reference:** [doi.org](https://doi.org/10.1016/j.jmaa.2003.12.009) [`cb6c9c` · Fungrim entry ↗](https://fungrim.org/entry/cb6c9c) --- $$\mathrm{JacobiTheta}(2, z+\frac{\tau}{2}, \tau)=\exp(-(\pi\imaginaryI(z+\frac{\tau}{4})))\mathrm{JacobiTheta}(3, z, \tau)$$ **Holds when** $z\in\C\land\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`cc6d21` · Fungrim entry ↗](https://fungrim.org/entry/cc6d21) --- $$\mathrm{JacobiTheta}(2, z+\tau, \tau)=\exp(-(\pi\imaginaryI(2z+\tau)))\mathrm{JacobiTheta}(2, z, \tau)$$ **Holds when** $z\in\C\land\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`cd5f45` · Fungrim entry ↗](https://fungrim.org/entry/cd5f45) --- $$\mathrm{JacobiTheta}(2, z, \tau+1)=\exp(\frac{\pi\imaginaryI}{4})\mathrm{JacobiTheta}(2, z, \tau)$$ **Holds when** $z\in\C\land\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`cde93e` · Fungrim entry ↗](https://fungrim.org/entry/cde93e) --- $$\mathrm{JacobiTheta}(3, 0, 2\imaginaryI)=\frac{1}{2}(\sqrt{\sqrt{2}+2}\mathrm{JacobiTheta}(3, 0, \imaginaryI))$$ Used by the Compute Engine for simplification. **Reference:** [doi.org](https://doi.org/10.1016/j.jmaa.2003.12.009) [`cf3c8e` · Fungrim entry ↗](https://fungrim.org/entry/cf3c8e) --- $$\mathrm{JacobiTheta}(4, 0, y\imaginaryI)=\mathrm{JacobiTheta}(3, 0, 1+y\imaginaryI)$$ **Holds when** $y\in\lparen0, \infty\rparen$. Used by the Compute Engine for simplification. [`cf7ee3` · Fungrim entry ↗](https://fungrim.org/entry/cf7ee3) --- $$\mathrm{JacobiTheta}(2, z, \tau+n)=\exp(\frac{\pi\imaginaryI n}{4})\mathrm{JacobiTheta}(2, z, \tau)$$ **Holds when** $z\in\C\land\Im(\tau)\gt0\land n\in\Z$. Used by the Compute Engine for simplification. [`d0dfba` · Fungrim entry ↗](https://fungrim.org/entry/d0dfba) --- $$\mathrm{JacobiTheta}(2, z, \tau+2n)=\imaginaryI^{n}\mathrm{JacobiTheta}(2, z, \tau)$$ **Holds when** $z\in\C\land\Im(\tau)\gt0\land n\in\Z$. Used by the Compute Engine for expansion. [`d11b7f` · Fungrim entry ↗](https://fungrim.org/entry/d11b7f) --- $$\mathrm{JacobiTheta}(3, 0, \imaginaryI)=\frac{\sqrt[4]{\pi}}{\Gamma(\frac{3}{4})}$$ Used by the Compute Engine for simplification. [`d15f11` · Fungrim entry ↗](https://fungrim.org/entry/d15f11) --- $$\mathrm{JacobiTheta}(2, z+m+n\tau, \tau)=(-1)^{m}\exp(-(\pi\imaginaryI(\tau n^2+2nz)))\mathrm{JacobiTheta}(2, z, \tau)$$ **Holds when** $z\in\C\land\Im(\tau)\gt0\land m\in\Z\land n\in\Z$. Used by the Compute Engine for simplification. [`d29148` · Fungrim entry ↗](https://fungrim.org/entry/d29148) --- $$\mathrm{JacobiTheta}(3, z, \tau)\mathrm{JacobiTheta}(4, w, \tau)=\mathrm{JacobiTheta}(4, z+w, 2\tau)\mathrm{JacobiTheta}(4, z-w, 2\tau)-\mathrm{JacobiTheta}(1, z+w, 2\tau)\mathrm{JacobiTheta}(1, z-w, 2\tau)$$ **Holds when** $z\in\C\land w\in\C\land\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`d36e97` · Fungrim entry ↗](https://fungrim.org/entry/d36e97) --- $$z\mapsto\frac{\mathrm{JacobiTheta}(1, z, \tau)}{\mathrm{JacobiTheta}(3, z, \tau)}^{\prime}(z)=\frac{\pi\mathrm{JacobiTheta}(2, z, \tau)\mathrm{JacobiTheta}(4, z, \tau)\mathrm{JacobiTheta}(3, 0, \tau)^2}{\mathrm{JacobiTheta}(3, z, \tau)^2}$$ **Holds when** $z\in\C\land\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`d41a95` · Fungrim entry ↗](https://fungrim.org/entry/d41a95) --- $$\mathrm{JacobiTheta}(1, z+\frac{\tau}{2}, \tau)=\exp(-(\pi\imaginaryI(z+\frac{\tau}{4})))\imaginaryI\mathrm{JacobiTheta}(4, z, \tau)$$ **Holds when** $z\in\C\land\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`d5a29e` · Fungrim entry ↗](https://fungrim.org/entry/d5a29e) --- $$\mathrm{JacobiTheta}(1, z+\tau, \tau)=-\exp(-(\pi\imaginaryI(2z+\tau)))\mathrm{JacobiTheta}(1, z, \tau)$$ **Holds when** $z\in\C\land\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`d989cd` · Fungrim entry ↗](https://fungrim.org/entry/d989cd) --- $$\mathrm{JacobiTheta}(2, 2z, 2\tau)=\frac{\mathrm{JacobiTheta}(3, z, \tau)^2-\mathrm{JacobiTheta}(4, z, \tau)^2}{2\mathrm{JacobiTheta}(2, 0, 2\tau)}$$ **Holds when** $z\in\C\land\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`db4e29` · Fungrim entry ↗](https://fungrim.org/entry/db4e29) --- $$\mathrm{JacobiTheta}(4, 0, \tau)^4=(1+8(\sum_{n=0}^{\infty}\frac{2n\exp(\pi\imaginaryI\tau)^{2n}}{1+\exp(\pi\imaginaryI\tau)^{2n}}))-8(\sum_{n=0}^{\infty}\frac{(2n+1)\exp(\pi\imaginaryI\tau)^{2n+1}}{1+\exp(\pi\imaginaryI\tau)^{2n+1}})$$ **Holds when** $\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`dc7c83` · Fungrim entry ↗](https://fungrim.org/entry/dc7c83) --- $$\mathrm{JacobiTheta}(2, \frac{n}{4}, \imaginaryI)=\begin{cases}(-1)^{\lfloor(n+1)/4\rfloor}\mathrm{JacobiTheta}(4, 0, \imaginaryI)&\mathrm{CongruentMod}(n, 0, 4)\\0&\mathrm{CongruentMod}(n, 2, 4)\\(-1)^{\lfloor(n+1)/4\rfloor}\frac{\sqrt{2^{1/2}-1}}{2^{7/16}}\sqrt[4]{\sqrt{2}+1}\mathrm{JacobiTheta}(3, 0, \imaginaryI)&\top\end{cases}$$ **Holds when** $n\in\Z$. Used by the Compute Engine for simplification. [`dd5f43` · Fungrim entry ↗](https://fungrim.org/entry/dd5f43) --- $$\mathrm{JacobiTheta}(3, 0, \frac{\tau}{2})^2=\mathrm{JacobiTheta}(2, 0, \tau)^2+\mathrm{JacobiTheta}(3, 0, \tau)^2$$ **Holds when** $\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`de7918` · Fungrim entry ↗](https://fungrim.org/entry/de7918) --- $$\frac{\mathrm{JacobiTheta}(1, z, \tau, 1)}{\pi\mathrm{JacobiTheta}(1, z, \tau)}=\cot(\pi z)+4(\sum_{n=1}^{\infty}\frac{\sin(2\pi nz)\exp(2\imaginaryI\pi n\tau)}{1-\exp(2\imaginaryI\pi n\tau)})$$ **Holds when** $z\in\C\land\Im(\tau)\gt0\land\vert\Im(z)\vert\lt\vert\Im(\tau)\vert\land\sin(\pi z)\ne0$. Used by the Compute Engine for simplification. [`dfbddd` · Fungrim entry ↗](https://fungrim.org/entry/dfbddd) --- $$\mathrm{JacobiTheta}(2, 0, \tau)\mathrm{JacobiTheta}(3, 0, \tau)\mathrm{JacobiTheta}(2, z+w, \tau)\mathrm{JacobiTheta}(3, z-w, \tau)=\mathrm{JacobiTheta}(2, z, \tau)\mathrm{JacobiTheta}(3, z, \tau)\mathrm{JacobiTheta}(2, w, \tau)\mathrm{JacobiTheta}(3, w, \tau)-\mathrm{JacobiTheta}(1, z, \tau)\mathrm{JacobiTheta}(4, z, \tau)\mathrm{JacobiTheta}(1, w, \tau)\mathrm{JacobiTheta}(4, w, \tau)$$ **Holds when** $z\in\C\land w\in\C\land\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`dfea7d` · Fungrim entry ↗](https://fungrim.org/entry/dfea7d) --- $$\mathrm{JacobiTheta}(1, z, \tau)^4-\mathrm{JacobiTheta}(4, z, \tau)^4=\mathrm{JacobiTheta}(2, z, \tau)^4-\mathrm{JacobiTheta}(3, z, \tau)^4$$ **Holds when** $z\in\C\land\Im(\tau)\gt0$. Used by the Compute Engine for expansion. [`e08bb4` · Fungrim entry ↗](https://fungrim.org/entry/e08bb4) --- $$\mathrm{JacobiTheta}(1, 2z, 2\tau)=\frac{\mathrm{JacobiTheta}(1, z, \tau)\mathrm{JacobiTheta}(2, z, \tau)}{\mathrm{JacobiTheta}(4, 0, 2\tau)}$$ **Holds when** $z\in\C\land\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`e13fe9` · Fungrim entry ↗](https://fungrim.org/entry/e13fe9) --- $$\mathrm{JacobiTheta}(2, 0, 1+y\imaginaryI)=\frac{\sqrt{2}(1+\imaginaryI)\mathrm{JacobiTheta}(3, 0, \frac{\imaginaryI}{y}+1)}{2\sqrt{y}}$$ **Holds when** $y\in\lparen0, \infty\rparen$. Used by the Compute Engine for simplification. [`e2288d` · Fungrim entry ↗](https://fungrim.org/entry/e2288d) --- $$\mathrm{JacobiTheta}(3, 0, 1+8\imaginaryI)=\frac{\sqrt[8]{16+15\sqrt[4]{2}+12\sqrt{2}+9\sqrt[4]{8}}}{2^{\frac{7}{8}}}\mathrm{JacobiTheta}(3, 0, \imaginaryI)$$ Used by the Compute Engine for simplification. **Reference:** [doi.org](https://doi.org/10.1016/j.jmaa.2003.12.009) [`e2bc80` · Fungrim entry ↗](https://fungrim.org/entry/e2bc80) --- $$\mathrm{JacobiTheta}(3, 0, \tau)=1+2(\sum_{n=1}^{\infty}\frac{\mathrm{LiouvilleLambda}(n)\exp(\pi\imaginaryI\tau)^{n}}{1-\exp(\pi\imaginaryI\tau)^{n}})$$ **Holds when** $\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`e4e707` · Fungrim entry ↗](https://fungrim.org/entry/e4e707) --- $$\mathrm{JacobiTheta}(3, z+n, \tau)=\mathrm{JacobiTheta}(3, z, \tau)$$ **Holds when** $z\in\C\land\Im(\tau)\gt0\land n\in\Z$. Used by the Compute Engine for simplification. [`e56f77` · Fungrim entry ↗](https://fungrim.org/entry/e56f77) --- $$\mathrm{JacobiTheta}(3, z, \frac{\tau}{2})=\frac{\mathrm{JacobiTheta}(4, z, \tau)^2-\mathrm{JacobiTheta}(1, z, \tau)^2}{\mathrm{JacobiTheta}(4, 0, \frac{\tau}{2})}$$ **Holds when** $z\in\C\land\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`e6d333` · Fungrim entry ↗](https://fungrim.org/entry/e6d333) --- $$\mathrm{JacobiTheta}(2, 2z, \tau)=\frac{\mathrm{JacobiTheta}(3, z, \tau)^4-\mathrm{JacobiTheta}(4, z, \tau)^4}{\mathrm{JacobiTheta}(2, 0, \tau)^3}$$ **Holds when** $z\in\C\land\Im(\tau)\gt0$. Used by the Compute Engine for expansion. [`e6dc09` · Fungrim entry ↗](https://fungrim.org/entry/e6dc09) --- $$\mathrm{JacobiTheta}(1, z, \frac{-1}{\tau})=-\imaginaryI\sqrt{\frac{\tau}{\imaginaryI}}\exp(\pi\imaginaryI\tau z^2)\mathrm{JacobiTheta}(1, \tau z, \tau)$$ **Holds when** $z\in\C\land\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`e8ce0b` · Fungrim entry ↗](https://fungrim.org/entry/e8ce0b) --- $$\mathrm{JacobiTheta}(j, z, \tau, 2)-4\pi\imaginaryI\tau\mapsto\mathrm{JacobiTheta}(j, z, \tau)^{\prime}(\tau)=0$$ **Holds when** $j\in\lbrace1, 2, 3, 4\rbrace\land z\in\C\land\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`ebc673` · Fungrim entry ↗](https://fungrim.org/entry/ebc673) --- $$\mathrm{JacobiTheta}(1, z, \tau)=-\imaginaryI\exp(\pi\imaginaryI(z+\frac{\tau}{4}))\mathrm{JacobiTheta}(3, z+\frac{1}{2}+\frac{\tau}{2}, \tau)$$ **Holds when** $z\in\C\land\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`ed0756` · Fungrim entry ↗](https://fungrim.org/entry/ed0756) --- $$\mathrm{JacobiTheta}(3, 2z, \tau)=\frac{\mathrm{JacobiTheta}(2, z, \tau)^4+\mathrm{JacobiTheta}(4, z, \tau)^4}{\mathrm{JacobiTheta}(3, 0, \tau)^3}$$ **Holds when** $z\in\C\land\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`ed3ff9` · Fungrim entry ↗](https://fungrim.org/entry/ed3ff9) --- $$\mathrm{JacobiTheta}(4, z, \frac{-1}{\tau})=\sqrt{\frac{\tau}{\imaginaryI}}\exp(\pi\imaginaryI\tau z^2)\mathrm{JacobiTheta}(2, \tau z, \tau)$$ **Holds when** $z\in\C\land\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`ed8ba7` · Fungrim entry ↗](https://fungrim.org/entry/ed8ba7) --- $$\mathrm{JacobiTheta}(2, 0, \tau)\mathrm{JacobiTheta}(3, 0, \tau)\mathrm{JacobiTheta}(1, z+w, \tau)\mathrm{JacobiTheta}(4, z-w, \tau)=\mathrm{JacobiTheta}(1, z, \tau)\mathrm{JacobiTheta}(4, z, \tau)\mathrm{JacobiTheta}(2, w, \tau)\mathrm{JacobiTheta}(3, w, \tau)+\mathrm{JacobiTheta}(2, z, \tau)\mathrm{JacobiTheta}(3, z, \tau)\mathrm{JacobiTheta}(1, w, \tau)\mathrm{JacobiTheta}(4, w, \tau)$$ **Holds when** $z\in\C\land w\in\C\land\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`ee8617` · Fungrim entry ↗](https://fungrim.org/entry/ee8617) --- $$\mathrm{JacobiTheta}(2, 2z, 2\tau)=\frac{\mathrm{JacobiTheta}(1, 1/4-z, \tau)\mathrm{JacobiTheta}(1, \frac{1}{4}+z, \tau)}{\mathrm{JacobiTheta}(4, 0, 2\tau)}$$ **Holds when** $z\in\C\land\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`f12569` · Fungrim entry ↗](https://fungrim.org/entry/f12569) --- $$\mathrm{JacobiTheta}(3, 0, 3\imaginaryI)=\frac{\mathrm{JacobiTheta}(3, 0, \imaginaryI)\sqrt{1+\sqrt{3}}}{\sqrt[4]{2}\times3^{\frac{3}{8}}}$$ Used by the Compute Engine for simplification. **Reference:** [doi.org](https://doi.org/10.1016/j.jmaa.2003.12.009) [`f12e20` · Fungrim entry ↗](https://fungrim.org/entry/f12e20) --- $$\mathrm{JacobiTheta}(4, 0, 2\tau)^2=\mathrm{JacobiTheta}(3, 0, \tau)\mathrm{JacobiTheta}(4, 0, \tau)$$ **Holds when** $\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`f14471` · Fungrim entry ↗](https://fungrim.org/entry/f14471) --- $$\mathrm{JacobiTheta}(1, 0, \tau, 1)=\pi\mathrm{JacobiTheta}(2, 0, \tau)\mathrm{JacobiTheta}(3, 0, \tau)\mathrm{JacobiTheta}(4, 0, \tau)$$ **Holds when** $\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`f2e28a` · Fungrim entry ↗](https://fungrim.org/entry/f2e28a) --- $$\mathrm{JacobiTheta}(3, z, \tau)=1+2(\sum_{n=1}^{\infty}\exp(\pi\imaginaryI\tau)^{n^2}\cos(2n\pi z))$$ **Holds when** $z\in\C\land\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`f3e75c` · Fungrim entry ↗](https://fungrim.org/entry/f3e75c) --- $$\mathrm{JacobiTheta}(3, z, \tau)\mathrm{JacobiTheta}(3, w, \tau)=\mathrm{JacobiTheta}(3, z+w, 2\tau)\mathrm{JacobiTheta}(3, z-w, 2\tau)+\mathrm{JacobiTheta}(2, z+w, 2\tau)\mathrm{JacobiTheta}(2, z-w, 2\tau)$$ **Holds when** $z\in\C\land w\in\C\land\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`f4554f` · Fungrim entry ↗](https://fungrim.org/entry/f4554f) --- $$\mathrm{JacobiTheta}(2, z+n, \tau)=(-1)^{n}\mathrm{JacobiTheta}(2, z, \tau)$$ **Holds when** $z\in\C\land\Im(\tau)\gt0\land n\in\Z$. Used by the Compute Engine for simplification. [`f697d5` · Fungrim entry ↗](https://fungrim.org/entry/f697d5) --- $$\mathrm{JacobiTheta}(3, 0, \tau)^2=1+2(\sum_{n=1}^{\infty}(\cos(\pi\tau n))^{-1})$$ **Holds when** $\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`f8cd8f` · Fungrim entry ↗](https://fungrim.org/entry/f8cd8f) --- $$\mathrm{JacobiTheta}(2, 0, \tau)^2\mathrm{JacobiTheta}(3, z, \tau)^2=\mathrm{JacobiTheta}(4, 0, \tau)^2\mathrm{JacobiTheta}(1, z, \tau)^2+\mathrm{JacobiTheta}(3, 0, \tau)^2\mathrm{JacobiTheta}(2, z, \tau)^2$$ **Holds when** $z\in\C\land\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`fa7251` · Fungrim entry ↗](https://fungrim.org/entry/fa7251) --- $$\mathrm{JacobiTheta}(2, z, \tau+8n)=\mathrm{JacobiTheta}(2, z, \tau)$$ **Holds when** $z\in\C\land\Im(\tau)\gt0\land n\in\Z$. Used by the Compute Engine for simplification. [`fb4b1b` · Fungrim entry ↗](https://fungrim.org/entry/fb4b1b) --- $$\mathrm{JacobiTheta}(2, -z, \tau)=\mathrm{JacobiTheta}(2, z, \tau)$$ **Holds when** $z\in\C\land\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`fb55cb` · Fungrim entry ↗](https://fungrim.org/entry/fb55cb) --- $$\mathrm{JacobiTheta}(4, 4z, 4\tau)=\frac{\mathrm{JacobiTheta}(4, z, \tau)\mathrm{JacobiTheta}(4, 1/4-z, \tau)\mathrm{JacobiTheta}(4, \frac{1}{4}+z, \tau)\mathrm{JacobiTheta}(3, z, \tau)}{\mathrm{JacobiTheta}(3, 0, \tau)\mathrm{JacobiTheta}(4, 0, \tau)\mathrm{JacobiTheta}(3, \frac{1}{4}, \tau)}$$ **Holds when** $z\in\C\land\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`fc3c44` · Fungrim entry ↗](https://fungrim.org/entry/fc3c44) --- $$\mathrm{JacobiTheta}(j, z, -\tau^\star)=\mathrm{JacobiTheta}(j, z^\star, \tau)^\star$$ **Holds when** $j\in\lbrace1, 2, 3, 4\rbrace\land z\in\C\land\Im(\tau)\gt0$. Used by the Compute Engine for simplification. [`fe1b96` · Fungrim entry ↗](https://fungrim.org/entry/fe1b96) --- ## 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 ↗](https://fungrim.org/entry/1356e4) --- $$\mathrm{ModularJ}(\frac{1}{2}(1+\sqrt{163}\imaginaryI))=-640\,320^3$$ **Symbols:** **ModularJ** — Modular j-invariant. Used by the Compute Engine for simplification. [`1cb24e` · Fungrim entry ↗](https://fungrim.org/entry/1cb24e) --- $$\mathrm{ModularJ}(2\imaginaryI)=66^3=287\,496$$ **Symbols:** **ModularJ** — Modular j-invariant. Used by the Compute Engine for simplification. [`229c97` · Fungrim entry ↗](https://fungrim.org/entry/229c97) --- $$\mathrm{ModularJ}(\frac{1}{2}(1+\sqrt{7}\imaginaryI))=-15^3$$ **Symbols:** **ModularJ** — Modular j-invariant. Used by the Compute Engine for simplification. [`29c095` · Fungrim entry ↗](https://fungrim.org/entry/29c095) --- $$\mathrm{ModularJ}(4\imaginaryI)=27(724+513\sqrt{2})^3$$ **Symbols:** **ModularJ** — Modular j-invariant. Used by the Compute Engine for simplification. [`3189b9` · Fungrim entry ↗](https://fungrim.org/entry/3189b9) --- $$\tau\mapsto\mathrm{ModularJ}(\tau)^{\prime}(\tau)=\frac{(-2\imaginaryI)\pi\mathrm{EisensteinE}(6, \tau)\mathrm{ModularJ}(\tau)}{\mathrm{EisensteinE}(4, \tau)}$$ **Holds when** $\Im(\tau)\gt0\land\mathrm{EisensteinE}(4, \tau)\ne0$. **Symbols:** **ModularJ** — Modular j-invariant. Used by the Compute Engine for simplification. [`348b26` · Fungrim entry ↗](https://fungrim.org/entry/348b26) --- $$\mathrm{ModularJ}(\frac{1}{2}(1+\sqrt{19}\imaginaryI))=-96^3$$ **Symbols:** **ModularJ** — Modular j-invariant. Used by the Compute Engine for simplification. [`3ee358` · Fungrim entry ↗](https://fungrim.org/entry/3ee358) --- $$\mathrm{ModularJ}(-(\frac{1}{\tau}))=\mathrm{ModularJ}(\tau)$$ **Holds when** $\Im(\tau)\gt0$. **Symbols:** **ModularJ** — Modular j-invariant. Used by the Compute Engine for simplification. [`42a909` · Fungrim entry ↗](https://fungrim.org/entry/42a909) --- $$\mathrm{Count}(\mathrm{Solutions}(\tau\mapsto\mathrm{ModularJ}(\tau)=z, \mathrm{ModularGroupFundamentalDomain}))=1$$ **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 ↗](https://fungrim.org/entry/441301) --- $$\mathrm{ModularJ}(\frac{1}{2}(1+\sqrt{43}\imaginaryI))=-960^3$$ **Symbols:** **ModularJ** — Modular j-invariant. Used by the Compute Engine for simplification. [`5b108e` · Fungrim entry ↗](https://fungrim.org/entry/5b108e) --- $$\mathrm{ModularJ}(\tau)={(\frac{\mathrm{DedekindEta}(\tau)}{\mathrm{DedekindEta}(2\tau)}^8+(256\mathrm{DedekindEta}(2\tau)^{16})/\mathrm{DedekindEta}(\tau)^{16})}^3$$ **Holds when** $\Im(\tau)\gt0$. **Symbols:** **ModularJ** — Modular j-invariant. Used by the Compute Engine for expansion. [`664b4c` · Fungrim entry ↗](https://fungrim.org/entry/664b4c) --- $$\mathrm{ModularJ}(3\imaginaryI)=64(2+\sqrt{3})^2(21+20\sqrt{3})^3$$ **Symbols:** **ModularJ** — Modular j-invariant. Used by the Compute Engine for simplification. [`8be46c` · Fungrim entry ↗](https://fungrim.org/entry/8be46c) --- $$\mathrm{ModularJ}(\frac{1}{2}(1+\sqrt{67}\imaginaryI))=-5\,280^3$$ **Symbols:** **ModularJ** — Modular j-invariant. Used by the Compute Engine for simplification. [`951017` · Fungrim entry ↗](https://fungrim.org/entry/951017) --- $$\mathrm{ModularJ}(\exp(\frac{\pi\imaginaryI}{3}))=0$$ **Symbols:** **ModularJ** — Modular j-invariant. Used by the Compute Engine for simplification. [`9aa62c` · Fungrim entry ↗](https://fungrim.org/entry/9aa62c) --- $$\mathrm{ModularJ}(\frac{1}{2}(1+\sqrt{11}\imaginaryI))=-32^3$$ **Symbols:** **ModularJ** — Modular j-invariant. Used by the Compute Engine for simplification. [`a498dd` · Fungrim entry ↗](https://fungrim.org/entry/a498dd) --- $$\mathrm{ModularJ}(\tau+1)=\mathrm{ModularJ}(\tau)$$ **Holds when** $\Im(\tau)\gt0$. **Symbols:** **ModularJ** — Modular j-invariant. Used by the Compute Engine for simplification. [`a997f2` · Fungrim entry ↗](https://fungrim.org/entry/a997f2) --- $$\mathrm{ModularJ}(\imaginaryI)=1\,728$$ **Symbols:** **ModularJ** — Modular j-invariant. Used by the Compute Engine for simplification. [`ad228f` · Fungrim entry ↗](https://fungrim.org/entry/ad228f) --- $$\mathrm{ModularJ}(\tau)=\frac{32{(\mathrm{JacobiTheta}(2, 0, \tau)^8+\mathrm{JacobiTheta}(3, 0, \tau)^8+\mathrm{JacobiTheta}(4, 0, \tau)^8)}^3}{(\mathrm{JacobiTheta}(2, 0, \tau)\mathrm{JacobiTheta}(3, 0, \tau)\mathrm{JacobiTheta}(4, 0, \tau))^8}$$ **Holds when** $\Im(\tau)\gt0$. **Symbols:** **ModularJ** — Modular j-invariant. Used by the Compute Engine for simplification. [`cedcfc` · Fungrim entry ↗](https://fungrim.org/entry/cedcfc) --- $$\mathrm{ModularJ}(\tau)=\frac{\mathrm{EisensteinE}(4, \tau)^3}{\mathrm{DedekindEta}(\tau)^{24}}$$ **Holds when** $\Im(\tau)\gt0$. **Symbols:** **ModularJ** — Modular j-invariant. Used by the Compute Engine for expansion. [`dc8251` · Fungrim entry ↗](https://fungrim.org/entry/dc8251) --- $$\mathrm{Map}(\mathrm{ModularGroupFundamentalDomain}, \tau\mapsto\mathrm{ModularJ}(\tau))=\C$$ **Symbols:** **ModularGroupFundamentalDomain** — Fundamental domain for action of the modular group; **ModularJ** — Modular j-invariant. Used by the Compute Engine for simplification. [`dcc8b1` · Fungrim entry ↗](https://fungrim.org/entry/dcc8b1) --- $$\tau\mapsto\mathrm{ModularJ}(\tau)^{\prime}(\tau)=((-2\imaginaryI)\pi\mathrm{EisensteinE}(14, \tau))/\mathrm{DedekindEta}(\tau)^{24}$$ **Holds when** $\Im(\tau)\gt0$. **Symbols:** **ModularJ** — Modular j-invariant. Used by the Compute Engine for simplification. [`f0f53b` · Fungrim entry ↗](https://fungrim.org/entry/f0f53b) --- ## 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:** **ModularLambda** — Modular lambda function. Used by the Compute Engine for simplification. [`033d39` · Fungrim entry ↗](https://fungrim.org/entry/033d39) --- $$1-\mathrm{ModularLambda}(\tau)=\frac{\mathrm{JacobiTheta}(4, 0, \tau)^4}{\mathrm{JacobiTheta}(3, 0, \tau)^4}$$ **Holds when** $\Im(\tau)\gt0$. **Symbols:** **ModularLambda** — Modular lambda function. Used by the Compute Engine for simplification. [`04d3a6` · Fungrim entry ↗](https://fungrim.org/entry/04d3a6) --- $$\mathrm{ModularLambda}(\frac{1+\imaginaryI}{2})=2$$ **Symbols:** **ModularLambda** — Modular lambda function. Used by the Compute Engine for simplification. [`078869` · Fungrim entry ↗](https://fungrim.org/entry/078869) --- $$\mathrm{ModularLambda}(-(\frac{1}{\tau}))=1-\mathrm{ModularLambda}(\tau)$$ **Holds when** $\Im(\tau)\gt0$. **Symbols:** **ModularLambda** — Modular lambda function. Used by the Compute Engine for simplification. [`07bf27` · Fungrim entry ↗](https://fungrim.org/entry/07bf27) --- $$\mathrm{ModularLambda}(\tau)=\frac{\mathrm{WeierstrassP}((1+\tau)/2, \tau)-\mathrm{WeierstrassP}(\tau/2, \tau)}{\mathrm{WeierstrassP}(1/2, \tau)-\mathrm{WeierstrassP}(\tau/2, \tau)}$$ **Holds when** $\Im(\tau)\gt0$. **Symbols:** **ModularLambda** — Modular lambda function; **WeierstrassP** — Weierstrass elliptic function. Used by the Compute Engine for simplification. [`166402` · Fungrim entry ↗](https://fungrim.org/entry/166402) --- $$\tau\mapsto\mathrm{ModularLambda}(\tau)^{\prime}(\tau)=\frac{1}{3}(\pi\imaginaryI((\mathrm{EisensteinE}(2, \tau/2)+8\mathrm{EisensteinE}(2, 2\tau))-6\mathrm{EisensteinE}(2, \tau)))\mathrm{ModularLambda}(\tau)$$ **Holds when** $\Im(\tau)\gt0$. **Symbols:** **ModularLambda** — Modular lambda function. Used by the Compute Engine for simplification. [`27b2c7` · Fungrim entry ↗](https://fungrim.org/entry/27b2c7) --- $$\mathrm{ModularLambda}(\frac{1}{1-\tau})=\frac{1}{1-\mathrm{ModularLambda}(\tau)}$$ **Holds when** $\Im(\tau)\gt0$. **Symbols:** **ModularLambda** — Modular lambda function. Used by the Compute Engine for expansion. [`2ba627` · Fungrim entry ↗](https://fungrim.org/entry/2ba627) --- $$\mathrm{ModularLambda}(2\imaginaryI)=17-12\sqrt{2}$$ **Symbols:** **ModularLambda** — Modular lambda function. Used by the Compute Engine for simplification. [`35c85f` · Fungrim entry ↗](https://fungrim.org/entry/35c85f) --- $$\mathrm{ModularLambda}(\frac{\tau-1}{\tau})=\frac{\mathrm{ModularLambda}(\tau)-1}{\mathrm{ModularLambda}(\tau)}$$ **Holds when** $\Im(\tau)\gt0$. **Symbols:** **ModularLambda** — Modular lambda function. Used by the Compute Engine for simplification. [`3a7a0b` · Fungrim entry ↗](https://fungrim.org/entry/3a7a0b) --- $$\mathrm{ModularJ}(\tau)=\frac{256{(-\mathrm{ModularLambda}(\tau)+\mathrm{ModularLambda}(\tau)^2+1)}^3}{(1-\mathrm{ModularLambda}(\tau))^2\mathrm{ModularLambda}(\tau)^2}$$ **Holds when** $\Im(\tau)\gt0$. **Symbols:** **ModularJ** — Modular j-invariant; **ModularLambda** — Modular lambda function. Used by the Compute Engine for simplification. [`44a529` · Fungrim entry ↗](https://fungrim.org/entry/44a529) --- $$\mathrm{ModularLambda}(\frac{\imaginaryI}{2})=12\sqrt{2}-16$$ **Symbols:** **ModularLambda** — Modular lambda function. Used by the Compute Engine for simplification. [`4877f2` · Fungrim entry ↗](https://fungrim.org/entry/4877f2) --- $$\mathrm{Map}(\mathrm{Filter}(\mathrm{HH}, \tau\mapsto\Re(\tau)=-1), \tau\mapsto\mathrm{ModularLambda}(\tau))=\lparen-\infty, 0\rparen$$ **Symbols:** **HH** — Upper complex half-plane; **ModularLambda** — 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. 118. [`4b20ab` · Fungrim entry ↗](https://fungrim.org/entry/4b20ab) --- $$\mathrm{ModularLambda}(\tau)=\frac{\mathrm{JacobiTheta}(2, 0, \tau)^4}{\mathrm{JacobiTheta}(3, 0, \tau)^4}$$ **Holds when** $\Im(\tau)\gt0$. **Symbols:** **ModularLambda** — Modular lambda function. Used by the Compute Engine for expansion. [`5b9c02` · Fungrim entry ↗](https://fungrim.org/entry/5b9c02) --- $$\mathrm{ModularLambda}(\tau)=(16\mathrm{DedekindEta}(2\tau)^{16}\mathrm{DedekindEta}(\tau/2)^8)/\mathrm{DedekindEta}(\tau)^{24}$$ **Holds when** $\Im(\tau)\gt0$. **Symbols:** **ModularLambda** — Modular lambda function. Used by the Compute Engine for simplification. [`5dd24a` · Fungrim entry ↗](https://fungrim.org/entry/5dd24a) --- $$\mathrm{ModularLambda}(\tau+2)=\mathrm{ModularLambda}(\tau)$$ **Holds when** $\Im(\tau)\gt0$. **Symbols:** **ModularLambda** — Modular lambda function. Used by the Compute Engine for simplification. [`6678af` · Fungrim entry ↗](https://fungrim.org/entry/6678af) --- $$\mathrm{ModularLambdaFundamentalDomain}=\mathrm{Filter}(\mathrm{HH}, \tau\mapsto\Re(\tau)\in\lparen-1, 1\rparen\land\min(\vert\tau-1/2\vert, \vert z+1/2\vert)\gt1/2\lor\Re(\tau)=-1\lor\vert\tau+1/2\vert=1/2)$$ **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 ↗](https://fungrim.org/entry/737f2b) --- $$\mathrm{Map}(\mathrm{Interior}(\mathrm{ModularLambdaFundamentalDomain}), \tau\mapsto\mathrm{ModularLambda}(\tau))=\C\setminus(\lparen-\infty, 0\rbrack\cup\lbrack1, \infty\rparen)$$ **Symbols:** **ModularLambda** — Modular lambda function; **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. 118. [`830dd4` · Fungrim entry ↗](https://fungrim.org/entry/830dd4) --- $$\frac{\mathrm{ModularLambda}(\tau)}{\mathrm{ModularLambda}(\tau)-1}=-(\frac{\mathrm{JacobiTheta}(2, 0, \tau)^4}{\mathrm{JacobiTheta}(4, 0, \tau)^4})$$ **Holds when** $\Im(\tau)\gt0$. **Symbols:** **ModularLambda** — Modular lambda function. Used by the Compute Engine for simplification. [`903962` · Fungrim entry ↗](https://fungrim.org/entry/903962) --- $$\mathrm{Map}(\mathrm{HH}, \tau\mapsto\mathrm{ModularLambda}(\tau))=\mathrm{Map}(\mathrm{ModularLambdaFundamentalDomain}, \tau\mapsto\mathrm{ModularLambda}(\tau))=\C\setminus\lbrace0, 1\rbrace$$ **Symbols:** **HH** — Upper complex half-plane; **ModularLambda** — Modular lambda function; **ModularLambdaFundamentalDomain** — Fundamental domain of the modular lambda function. Used by the Compute Engine for simplification. [`90b419` · Fungrim entry ↗](https://fungrim.org/entry/90b419) --- $$\mathrm{ModularLambda}(\imaginaryI)=\frac{1}{2}$$ **Symbols:** **ModularLambda** — Modular lambda function. Used by the Compute Engine for simplification. [`a35b3c` · Fungrim entry ↗](https://fungrim.org/entry/a35b3c) --- $$\mathrm{ModularLambda}(\exp(\frac{2\pi\imaginaryI}{3}))=-\exp(\frac{2\pi\imaginaryI}{3})$$ **Symbols:** **ModularLambda** — Modular lambda function. Used by the Compute Engine for simplification. [`b0e1cb` · Fungrim entry ↗](https://fungrim.org/entry/b0e1cb) --- $$\mathrm{ModularLambda}(\tau+1)=\frac{\mathrm{ModularLambda}(\tau)}{\mathrm{ModularLambda}(\tau)-1}$$ **Holds when** $\Im(\tau)\gt0$. **Symbols:** **ModularLambda** — Modular lambda function. Used by the Compute Engine for simplification. [`bbfb6c` · Fungrim entry ↗](https://fungrim.org/entry/bbfb6c) --- $$\tau\mapsto\mathrm{ModularLambda}(\tau)^{\prime}(\tau)=\frac{1}{\pi}((2\imaginaryI)(-6\mathrm{WeierstrassZeta}(\frac{1}{2}, \tau)+8\mathrm{WeierstrassZeta}(\frac{1}{2}, 2\tau)+\mathrm{WeierstrassZeta}(\frac{1}{2}, \frac{\tau}{2}))\mathrm{ModularLambda}(\tau))$$ **Holds when** $\Im(\tau)\gt0$. **Symbols:** **ModularLambda** — Modular lambda function; **WeierstrassZeta** — Weierstrass zeta function. Used by the Compute Engine for simplification. [`c18c95` · Fungrim entry ↗](https://fungrim.org/entry/c18c95) --- $$\mathrm{Map}(\mathrm{Filter}(\mathrm{HH}, \tau\mapsto\vert\tau+1/2\vert=1/2), \tau\mapsto\mathrm{ModularLambda}(\tau))=\lparen1, \infty\rparen$$ **Symbols:** **HH** — Upper complex half-plane; **ModularLambda** — 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. 118. [`e4315f` · Fungrim entry ↗](https://fungrim.org/entry/e4315f) --- $$\mathrm{ModularLambda}(\frac{\tau}{1-\tau})=\frac{1}{\mathrm{ModularLambda}(\tau)}$$ **Holds when** $\Im(\tau)\gt0$. **Symbols:** **ModularLambda** — Modular lambda function. Used by the Compute Engine for simplification. [`e9f0c8` · Fungrim entry ↗](https://fungrim.org/entry/e9f0c8) --- $$\mathrm{ModularLambda}(\frac{\tau}{2\tau+1})=\mathrm{ModularLambda}(\tau)$$ **Holds when** $\Im(\tau)\gt0$. **Symbols:** **ModularLambda** — Modular lambda function. Used by the Compute Engine for simplification. [`ec5a44` · Fungrim entry ↗](https://fungrim.org/entry/ec5a44) --- $$\mathrm{ModularLambda}(1+\imaginaryI)=-1$$ **Symbols:** **ModularLambda** — Modular lambda function. Used by the Compute Engine for simplification. [`fe2627` · Fungrim entry ↗](https://fungrim.org/entry/fe2627) ---