LaTeX to CAS translator

This mockup demonstrates the concept of TeX to Computer Algebra System (CAS) conversion.

The demo-application converts LaTeX functions which directly translate to CAS counterparts.

Functions without explicit CAS support are available for translation via a DRMF package (under development).

The following LaTeX input ...

{\displaystyle \begin{align} \vartheta_{01}(z\mid q) &= \prod_{m=1}^\infty \left( 1 - q^{2m}\right) \left( 1 - 2 \cos(2 \pi z)q^{2m-1}+q^{4m-2}\right),\\[3pt] \vartheta_{10}(z\mid q) &= 2 q^\frac14\cos(\pi z)\prod_{m=1}^\infty \left( 1 - q^{2m}\right) \left( 1 + 2 \cos(2 \pi z)q^{2m}+q^{4m}\right),\\[3pt] \vartheta_{11}(z\mid q) &= -2 q^\frac14\sin(\pi z)\prod_{m=1}^\infty \left( 1 - q^{2m}\right)\left( 1 - 2 \cos(2 \pi z)q^{2m}+q^{4m}\right). \end{align}}

${\displaystyle {\begin{aligned}\vartheta _{01}(z\mid q)&=\prod _{m=1}^{\infty }\left(1-q^{2m}\right)\left(1-2\cos(2\pi z)q^{2m-1}+q^{4m-2}\right),\\[3pt]\vartheta _{10}(z\mid q)&=2q^{\frac {1}{4}}\cos(\pi z)\prod _{m=1}^{\infty }\left(1-q^{2m}\right)\left(1+2\cos(2\pi z)q^{2m}+q^{4m}\right),\\[3pt]\vartheta _{11}(z\mid q)&=-2q^{\frac {1}{4}}\sin(\pi z)\prod _{m=1}^{\infty }\left(1-q^{2m}\right)\left(1-2\cos(2\pi z)q^{2m}+q^{4m}\right).\end{aligned}}}$

... is translated to the CAS output ...

Semantic latex: \begin{align}\vartheta_{01}(z\mid q) &= \prod_{m=1}^\infty(1 - q^{2m})(1 - 2 \cos(2 \cpi z) q^{2m-1} + q^{4m-2}) , \\ \vartheta_{10}(z\mid q) &= 2 q^\frac14 \cos(\cpi z) \prod_{m=1}^\infty(1 - q^{2m})(1 + 2 \cos(2 \cpi z) q^{2m} + q^{4m}) , \\ \vartheta_{11}(z\mid q) &= - 2 q^\frac14 \sin(\cpi z) \prod_{m=1}^\infty(1 - q^{2m})(1 - 2 \cos(2 \cpi z) q^{2m} + q^{4m}) .\end{align}

Confidence: 0

Mathematica

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(LaTeX -> SymPy) No translation possible for given token: Unknown relation. Cannot translate: \mid [\mid]

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(LaTeX -> Maple) No translation possible for given token: Unknown relation. Cannot translate: \mid [\mid]

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${\begin{aligned}\vartheta _{01}(z\mid q)&=\prod _{m=1}^{\infty }\left(1-q^{2m}\right)\left(1-2\cos(2\pi z)q^{2m-1}+q^{4m-2}\right),\\[3pt]\vartheta _{10}(z\mid q)&=2q^{\frac {1}{4}}\cos(\pi z)\prod _{m=1}^{\infty }\left(1-q^{2m}\right)\left(1+2\cos(2\pi z)q^{2m}+q^{4m}\right),\\[3pt]\vartheta _{11}(z\mid q)&=-2q^{\frac {1}{4}}\sin(\pi z)\prod _{m=1}^{\infty }\left(1-q^{2m}\right)\left(1-2\cos(2\pi z)q^{2m}+q^{4m}\right).\end{aligned}}$

Is part of

${\begin{aligned}\vartheta _{01}(z\mid q)&=\prod _{m=1}^{\infty }\left(1-q^{2m}\right)\left(1-2\cos(2\pi z)q^{2m-1}+q^{4m-2}\right),\\[3pt]\vartheta _{10}(z\mid q)&=2q^{\frac {1}{4}}\cos(\pi z)\prod _{m=1}^{\infty }\left(1-q^{2m}\right)\left(1+2\cos(2\pi z)q^{2m}+q^{4m}\right),\\[3pt]\vartheta _{11}(z\mid q)&=-2q^{\frac {1}{4}}\sin(\pi z)\prod _{m=1}^{\infty }\left(1-q^{2m}\right)\left(1-2\cos(2\pi z)q^{2m}+q^{4m}\right).\end{aligned}}$

Complete translation information:

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Specify your own input

Formula input
Wikitext context	{{short description\|Special functions of several complex variables}} {{for\|other θ functions\|Theta function (disambiguation)}} [[Image:Complex theta minus0point1times e i pi 0point1.jpg\|400px\|thumb\|upright=1.2\|Jacobi's original theta function {{math\|''θ''<sub>1</sub>}} with {{math\|''u'' {{=}} ''i''π''z''}} and with nome {{math\|''q'' {{=}} ''e''<sup>''i''π''τ''</sup> {{=}} 0.1''e''<sup>0.1''i''π</sup>}}. Conventions are (Mathematica): <math>\begin{align} \theta_1(u;q) &= 2 q^\frac14 \sum_{n=0}^\infty (-1)^n q^{n(n+1)} \sin(2n+1)u \\ &= \sum_{n=-\infty}^\infty (-1)^{n-\frac12} q^{\left(n+\frac12\right)^2} e^{(2n+1)i u} \end{align}</math>]] In [[mathematics]], '''theta functions''' are [[special function]]s of [[several complex variables]]. They are important in many areas, including the theories of [[Abelian variety\|Abelian varieties]] and [[moduli space]]s, and of [[quadratic form]]s. They have also been applied to [[soliton]] theory. When generalized to a [[Grassmann algebra]], they also appear in [[quantum field theory]].<ref name="Tyurin2002">{{cite arxiv\|last1=Tyurin\|first1=Andrey N.\|title=Quantization, Classical and Quantum Field Theory and Theta-Functions\|eprint=math/0210466v1\|date=30 October 2002}}</ref> The most common form of theta function is that occurring in the theory of [[elliptic function]]s. With respect to one of the complex variables (conventionally called {{mvar\|z}}), a theta function has a property expressing its behavior with respect to the addition of a period of the associated elliptic functions, making it a [[quasiperiodic function]]. In the abstract theory this comes from a [[line bundle]] condition of [[descent (category theory)\|descent]]. ==Jacobi theta function== [[File:Jacobi theta 1.png\|thumb\|Jacobi theta 1]] [[File:Jacobi theta 2.png\|thumb\|Jacobi theta 2]] [[File:Jacobi theta 3.png\|thumb\|Jacobi theta 3]] [[File:Jacobi theta 4.png\|thumb\|Jacobi theta 4]] There are several closely related functions called Jacobi theta functions, and many different and incompatible systems of notation for them. One '''Jacobi theta function''' (named after [[Carl Gustav Jacob Jacobi]]) is a function defined for two complex variables {{mvar\|z}} and {{mvar\|τ}}, where {{mvar\|z}} can be any complex number and {{mvar\|τ}} is the [[half-period ratio]], confined to the [[upper half-plane]], which means it has positive imaginary part. It is given by the formula :<math>\begin{align} \vartheta(z; \tau) &= \sum_{n=-\infty}^\infty \exp \left(\pi i n^2 \tau + 2 \pi i n z\right) \\ &= 1 + 2 \sum_{n=1}^\infty \left(e^{\pi i\tau}\right)^{n^2} \cos(2\pi n z) \\ &= \sum_{n=-\infty}^\infty q^{n^2}\eta^n \end{align}</math> where {{math\|''q'' {{=}} exp(π''iτ'')}} is the [[nome (mathematics)\|nome]] and {{math\|''η'' {{=}} exp(2π''iz'')}}. It is a [[Jacobi form]]. At fixed {{mvar\|τ}}, this is a [[Fourier series]] for a 1-periodic [[entire function]] of {{mvar\|z}}. Accordingly, the theta function is 1-periodic in {{mvar\|z}}: :<math>\vartheta(z+1; \tau) = \vartheta(z; \tau).</math> It also turns out to be {{mvar\|τ}}-quasiperiodic in {{mvar\|z}}, with :<math>\vartheta(z+\tau;\tau) = \exp[-\pi i (\tau + 2 z)] \vartheta(z;\tau).</math> Thus, in general, :<math>\vartheta(z+a+b\tau;\tau) = \exp\left(-\pi i b^2 \tau -2 \pi i b z\right) \vartheta(z;\tau)</math> for any integers {{mvar\|a}} and {{mvar\|b}}. [[Image:Complex theta animated1.gif\|500px\|thumb\|center\|Theta function {{math\|''θ''<sub>1</sub>}} with different nome {{math\|''q'' {{=}} ''e''<sup>''i''π''τ''</sup>}}. The black dot in the right-hand picture indicates how {{mvar\|q}} changes with {{mvar\|τ}}.]] [[Image:Complex theta animated2.gif\|500px\|thumb\|center\|Theta function {{math\|''θ''<sub>1</sub>}} with different nome {{math\|''q'' {{=}} ''e''<sup>''i''π''τ''</sup>}}. The black dot in the right-hand picture indicates how {{mvar\|q}} changes with {{mvar\|τ}}.]] ==Auxiliary functions== The Jacobi theta function defined above is sometimes considered along with three auxiliary theta functions, in which case it is written with a double 0 subscript: :<math>\vartheta_{00}(z;\tau) = \vartheta(z;\tau)</math> The auxiliary (or half-period) functions are defined by :<math>\begin{align} \vartheta_{01}(z;\tau)& = \vartheta \left(z+\tfrac12;\tau\right)\\[3pt] \vartheta_{10}(z;\tau)& = \exp\left(\tfrac14\pi i \tau + \pi i z\right)\vartheta\left(z + \tfrac12\tau;\tau\right)\\[3pt] \vartheta_{11}(z;\tau)& = \exp\left(\tfrac14\pi i \tau + \pi i\left(z+\tfrac12\right)\right)\vartheta\left(z+\tfrac12\tau + \tfrac12;\tau\right). \end{align}</math> This notation follows [[Bernhard Riemann\|Riemann]] and [[David Mumford\|Mumford]]; [[Carl Gustav Jacobi\|Jacobi]]'s original formulation was in terms of the [[nome (mathematics)\|nome]] {{math\|''q'' {{=}} ''e''<sup>''i''π''τ''</sup>}} rather than {{mvar\|τ}}. In Jacobi's notation the {{mvar\|θ}}-functions are written: :<math>\begin{align} \theta_1(z;q) &= -\vartheta_{11}(z;\tau)\\ \theta_2(z;q) &= \vartheta_{10}(z;\tau)\\ \theta_3(z;q) &= \vartheta_{00}(z;\tau)\\ \theta_4(z;q) &= \vartheta_{01}(z;\tau) \end{align}</math> The above definitions of the Jacobi theta functions are by no means unique. See [[Jacobi theta functions (notational variations)]] for further discussion. If we set {{math\|''z'' {{=}} 0}} in the above theta functions, we obtain four functions of {{mvar\|τ}} only, defined on the upper half-plane (sometimes called theta constants.) These can be used to define a variety of [[modular forms]], and to parametrize certain curves; in particular, the '''Jacobi identity''' is :<math>\vartheta_{00}(0;\tau)^4 = \vartheta_{01}(0;\tau)^4 + \vartheta_{10}(0;\tau)^4</math> which is the [[Fermat curve]] of degree four. ==Jacobi identities== Jacobi's identities describe how theta functions transform under the [[modular group]], which is generated by {{math\|''τ'' ↦ ''τ'' + 1}} and {{math\|''τ'' ↦ −{{sfrac\|1\|''τ''}}}}. Equations for the first transform are easily found since adding one to {{mvar\|τ}} in the exponent has the same effect as adding {{sfrac\|1\|2}} to {{mvar\|z}} ({{math\|''n'' ≡ ''n''<sup>2</sup> [[Modular arithmetic\|mod]] 2}}). For the second, let :<math>\alpha = (-i \tau)^\frac12 \exp\left(\frac{\pi}{\tau} i z^2 \right).</math> Then :<math>\begin{align} \vartheta_{00}\!\left(\frac{z}{\tau}; \frac{-1}{\tau}\right)& = \alpha\,\vartheta_{00}(z; \tau)\quad& \vartheta_{01}\!\left(\frac{z}{\tau}; \frac{-1}{\tau}\right)& = \alpha\,\vartheta_{10}(z; \tau)\\[3pt] \vartheta_{10}\!\left(\frac{z}{\tau}; \frac{-1}{\tau}\right)& = \alpha\,\vartheta_{01}(z; \tau)\quad& \vartheta_{11}\!\left(\frac{z}{\tau}; \frac{-1}{\tau}\right)& = -i\alpha\,\vartheta_{11}(z; \tau). \end{align}</math> ==Theta functions in terms of the nome== Instead of expressing the Theta functions in terms of {{mvar\|z}} and {{mvar\|τ}}, we may express them in terms of arguments {{mvar\|w}} and the [[nome (mathematics)\|nome]] {{mvar\|q}}, where {{math\|''w'' {{=}} ''e''<sup>π''iz''</sup>}} and {{math\|''q'' {{=}} ''e''<sup>π''iτ''</sup>}}. In this form, the functions become :<math>\begin{align} \vartheta_{00}(w, q)& = \sum_{n=-\infty}^\infty (w^2)^n q^{n^2}\quad& \vartheta_{01}(w, q)& = \sum_{n=-\infty}^\infty (-1)^n (w^2)^n q^{n^2}\\[3pt] \vartheta_{10}(w, q)& = \sum_{n=-\infty}^\infty (w^2)^{n+\frac12} q^{\left(n + \frac12\right)^2}\quad& \vartheta_{11}(w, q)& = i \sum_{n=-\infty}^\infty (-1)^n (w^2)^{n+\frac12} q^{\left(n + \frac12\right)^2}. \end{align}</math> We see that the theta functions can also be defined in terms of {{mvar\|w}} and {{mvar\|q}}, without a direct reference to the exponential function. These formulas can, therefore, be used to define the Theta functions over other [[field (mathematics)\|fields]] where the exponential function might not be everywhere defined, such as fields of [[p-adic number\|{{mvar\|p}}-adic numbers]]. ==Product representations== The [[Jacobi triple product]] (a special case of the [[Macdonald identities]]) tells us that for complex numbers {{mvar\|w}} and {{mvar\|q}} with {{math\|{{abs\|''q''}} < 1}} and {{math\|''w'' ≠ 0}} we have :<math>\prod_{m=1}^\infty \left( 1 - q^{2m}\right) \left( 1 + w^2 q^{2m-1}\right) \left( 1 + w^{-2}q^{2m-1}\right) = \sum_{n=-\infty}^\infty w^{2n}q^{n^2}. </math> It can be proven by elementary means, as for instance in Hardy and Wright's ''[[An Introduction to the Theory of Numbers]]''. If we express the theta function in terms of the nome {{math\|''q'' {{=}} ''e''<sup>π''iτ''</sup>}} (noting some authors instead set {{math\|''q'' {{=}} ''e''<sup>2π''iτ''</sup>}}) and take {{math\|''w'' {{=}} ''e''<sup>π''iz''</sup>}} then :<math>\vartheta(z; \tau) = \sum_{n=-\infty}^\infty \exp(\pi i \tau n^2) \exp(2\pi i z n) = \sum_{n=-\infty}^\infty w^{2n}q^{n^2}. </math> We therefore obtain a product formula for the theta function in the form :<math>\vartheta(z; \tau) = \prod_{m=1}^\infty \big( 1 - \exp(2m \pi i \tau)\big) \Big( 1 + \exp\big((2m-1) \pi i \tau + 2 \pi i z\big)\Big) \Big( 1 + \exp\big((2m-1) \pi i \tau - 2 \pi i z\big)\Big). </math> In terms of {{mvar\|w}} and {{mvar\|q}}: :<math>\begin{align} \vartheta(z; \tau) &= \prod_{m=1}^\infty \left( 1 - q^{2m}\right) \left( 1 + q^{2m-1}w^2\right) \left( 1 + \frac{q^{2m-1}}{w^2}\right) \\ &= \left(q^2;q^2\right)_\infty\,\left(-w^2q;q^2\right)_\infty\,\left(-\frac{q}{w^2};q^2\right)_\infty \\ &= \left(q^2;q^2\right)_\infty\,\theta\left(-w^2q;q^2\right) \end{align}</math> where {{math\|(  ;  )<sub>∞</sub>}} is the [[q-Pochhammer symbol\|{{mvar\|q}}-Pochhammer symbol]] and {{math\|''θ''(  ;  )}} is the [[q-theta function\|{{mvar\|q}}-theta function]]. Expanding terms out, the Jacobi triple product can also be written :<math>\prod_{m=1}^\infty \left( 1 - q^{2m}\right) \Big( 1 + \left(w^2+w^{-2}\right)q^{2m-1}+q^{4m-2}\Big),</math> which we may also write as :<math>\vartheta(z\mid q) = \prod_{m=1}^\infty \left( 1 - q^{2m}\right) \left( 1 + 2 \cos(2 \pi z)q^{2m-1}+q^{4m-2}\right).</math> This form is valid in general but clearly is of particular interest when {{mvar\|z}} is real. Similar product formulas for the auxiliary theta functions are :<math>\begin{align} \vartheta_{01}(z\mid q) &= \prod_{m=1}^\infty \left( 1 - q^{2m}\right) \left( 1 - 2 \cos(2 \pi z)q^{2m-1}+q^{4m-2}\right),\\[3pt] \vartheta_{10}(z\mid q) &= 2 q^\frac14\cos(\pi z)\prod_{m=1}^\infty \left( 1 - q^{2m}\right) \left( 1 + 2 \cos(2 \pi z)q^{2m}+q^{4m}\right),\\[3pt] \vartheta_{11}(z\mid q) &= -2 q^\frac14\sin(\pi z)\prod_{m=1}^\infty \left( 1 - q^{2m}\right)\left( 1 - 2 \cos(2 \pi z)q^{2m}+q^{4m}\right). \end{align}</math> ==Integral representations== The Jacobi theta functions have the following integral representations: :<math>\begin{align} \vartheta_{00} (z; \tau) &= -i\int_{i-\infty}^{i+\infty} e^{i \pi \tau u^2} \frac{\cos(2uz + \pi u)}{\sin(\pi u)} \mathrm{d}u; \\[6pt] \vartheta_{01} (z; \tau) &= -i\int_{i-\infty}^{i+\infty} e^{i \pi \tau u^2} \frac{\cos (2uz)}{\sin (\pi u)} \mathrm{d}u; \\[6pt] \vartheta_{10} (z; \tau) &= -ie^{iz + \frac14 i \pi\tau} \int_{i-\infty}^{i+\infty} e^{i\pi\tau u^2} \frac{\cos(2uz + \pi u + \pi \tau u)}{\sin (\pi u)} \mathrm{d}u; \\[6pt] \vartheta_{11} (z; \tau) &= e^{iz + \frac14 i \pi \tau} \int_{i-\infty}^{i+\infty} e^{i\pi\tau u^2} \frac{\cos(2uz + \pi \tau u)}{\sin(\pi u)} \mathrm{d}u. \end{align}</math> ==Explicit values== See Yi (2004).<ref name="Yi">{{Cite journal \| first = Jinhee \| last = Yi \| title = Theta-function identities and the explicit formulas for theta-function and their applications \| journal = Journal of Mathematical Analysis and Applications \| pages = 381–400 \| volume = 292 \| issue = 2 \| doi=10.1016/j.jmaa.2003.12.009 \| year = 2004\| doi-access = free }}</ref><ref>Proper credit for these results goes to Ramanujan. See [[Ramanujan's lost notebook]] and a relevant reference at [[Euler function]]. The Ramanujan results quoted at [[Euler function]] plus a few elementary operations give the results below, so the results below are either in Ramanujan's lost notebook or follow immediately from it.</ref> :<math>\begin{align} \varphi(e^{-\pi x}) &= \vartheta(0; ix) = \theta_3(0;e^{-\pi x}) = \sum_{n=-\infty}^\infty e^{-x \pi n^2}\\[8pt] \varphi\left(e^{-\pi} \right) &= \frac{\sqrt[4]{\pi}}{\Gamma\left(\frac34\right)} \\[8pt] \varphi\left(e^{-2\pi}\right) &= \frac{\sqrt[4]{\pi}}{\Gamma\left(\frac34\right)} \frac{\sqrt[4]{6+4\sqrt2}}{2}\\[8pt] \varphi\left(e^{-3\pi}\right) &= \frac{\sqrt[4]{\pi}}{\Gamma\left(\frac34\right)} \frac{\sqrt[4]{27+18\sqrt3}}{3}\\[8pt] \varphi\left(e^{-4\pi}\right) &= \frac{\sqrt[4]{\pi}}{\Gamma\left(\frac34\right)} \frac{\sqrt[4]{8}+2}{4}\\[8pt] \varphi\left(e^{-5\pi}\right) &= \frac{\sqrt[4]{\pi}}{\Gamma\left(\frac34\right)} \frac{\sqrt[4]{225+ 100\sqrt5}}{5}\\[8pt] \varphi\left(e^{-6\pi}\right) &= \frac{\sqrt[3]{3\sqrt{2}+3\sqrt[4]{3}+2\sqrt{3}-\sqrt[4]{27}+\sqrt[4]{1728}-4}\cdot\sqrt[8]{243{\pi}^2}}{6\sqrt[6]{1+\sqrt6-\sqrt2-\sqrt3}{\Gamma\left(\frac34\right)}} = \frac{\sqrt[4]{\pi}}{\Gamma\left(\frac34\right)} \frac{\sqrt{\sqrt[4]{1}+\sqrt[4]{3}+\sqrt[4]{4}+\sqrt[4]{9}}}{\sqrt[8]{1728}} \\[8pt] \varphi\left(e^{-7\pi}\right) &= \frac{\sqrt[4]{\pi}}{\Gamma\left(\frac34\right)} \sqrt{\frac{\sqrt{13+\sqrt{7}}+\sqrt{7+3\sqrt{7}}}{14}\cdot \sqrt[8]{28}} = \frac{\sqrt[4]{\pi}}{\Gamma\left(\frac34\right)} \frac{\sqrt[4]{7+4\sqrt{7}+5\sqrt[4]{28}+\sqrt[4]{1372}}}{\sqrt{7}}\\[8pt] \varphi\left(e^{-8\pi}\right) &= \frac{\sqrt[4]{\pi}}{\Gamma\left(\frac34\right)} \frac{\sqrt[8]{128}+\sqrt{2+\sqrt{2}}}{4} \\[8pt] \varphi\left(e^{-9\pi}\right) &= \frac{\sqrt[4]{\pi}}{\Gamma\left(\frac34\right)} \frac{\left(1+\left(1+\sqrt{3}\right)\sqrt[3]{2-\sqrt{3}}\right)}{3} \\[8pt] \varphi\left(e^{-10\pi}\right) &= \frac{\sqrt[4]{\pi}}{\Gamma\left(\frac34\right)} \frac{\sqrt{20+\sqrt{450}+\sqrt{500}+10\sqrt[4]{20}}}{10} \\[8pt] \varphi\left(e^{-12\pi}\right) &= \frac{\sqrt[4]{\pi}}{\Gamma\left(\frac34\right)} \frac{\sqrt{\sqrt[4]{1}+\sqrt[4]{2}+\sqrt[4]{3}+\sqrt[4]{4}+\sqrt[4]{9}+\sqrt[4]{18}+\sqrt[4]{24}}}{2\sqrt[8]{108}} \\[8pt] \varphi\left(e^{-16\pi}\right) &= \frac{\sqrt[4]{\pi}}{\Gamma\left(\frac34\right)} \frac{\left(4+ \sqrt[4]{128}+ \sqrt[4]{1024\sqrt[4]{8} + 1024\sqrt[4]{2}} \right)}{16} \end{align}</math> ==Some series identities== The next two series identities were proved by [[István Mező]]:<ref name="Mezo2">{{Citation \| last = Mező \| first = István \| title = Duplication formulae involving Jacobi theta functions and Gosper's {{mvar\|q}}-trigonometric functions \| journal = Proceedings of the American Mathematical Society \| pages = 2401–2410 \| volume = 141 \| issue = 7 \| year = 2013 \| doi=10.1090/s0002-9939-2013-11576-5 \| doi-access = free }}</ref> :<math>\begin{align} \vartheta_4^2(q)&=iq^{\frac14}\sum_{k=-\infty}^\infty q^{2k^2-k}\vartheta_1\left(\frac{2k-1}{2i}\ln q,q\right),\\[6pt] \vartheta_4^2(q)&=\sum_{k=-\infty}^\infty q^{2k^2}\vartheta_4\left(\frac{k\ln q}{i},q\right). \end{align}</math> These relations hold for all {{math\|0 < ''q'' < 1}}. Specializing the values of {{mvar\|q}}, we have the next parameter free sums :<math>\begin{align} \sqrt{\frac{\pi\sqrt{e^\pi}}{2}}\cdot\frac{1}{\Gamma^2\left(\frac34\right)}&=i\sum_{k=-\infty}^\infty e^{\pi\left(k-2k^2\right)} \vartheta_1 \left(\frac{i\pi}{2}(2k-1),e^{-\pi}\right),\\[6pt] \sqrt{\frac{\pi}{2}}\cdot\frac{1}{\Gamma^2\left(\frac34\right)}&=\sum_{k=-\infty}^\infty\frac{\vartheta_4\left(ik\pi,e^{-\pi}\right)}{e^{2\pi k^2}} \end{align}</math> ==Zeros of the Jacobi theta functions== All zeros of the Jacobi theta functions are simple zeros and are given by the following: :<math>\begin{align} \vartheta(z,\tau) = \vartheta_3(z,\tau) &= 0 \quad &\Longleftrightarrow&& \quad z &= m + n \tau + \frac{1}{2} + \frac{\tau}{2} \\[3pt] \vartheta_1(z,\tau) &= 0 \quad &\Longleftrightarrow&& \quad z &= m + n \tau \\[3pt] \vartheta_2(z,\tau) &= 0 \quad &\Longleftrightarrow&& \quad z &= m + n \tau + \frac{1}{2} \\[3pt] \vartheta_4(z,\tau) &= 0 \quad &\Longleftrightarrow&& \quad z &= m + n \tau + \frac{\tau}{2} \end{align}</math> where {{mvar\|m}}, {{mvar\|n}} are arbitrary integers. ==Relation to the Riemann zeta function== The relation :<math>\vartheta\left(0;-\frac{1}{\tau}\right)=(-i\tau)^\frac12 \vartheta(0;\tau)</math> was used by [[Riemann]] to prove the functional equation for the [[Riemann zeta function]], by means of the [[Mellin transform]] :<math>\Gamma\left(\frac{s}{2}\right) \pi^{-\frac{s}{2}} \zeta(s) = \frac{1}{2}\int_0^\infty(\vartheta(0;it)-1)t^\frac{s}{2}\frac{\mathrm{d}t}{t}</math> which can be shown to be invariant under substitution of {{mvar\|s}} by {{math\|1 − ''s''}}. The corresponding integral for {{math\|''z'' ≠ 0}} is given in the article on the [[Hurwitz zeta function]]. ==Relation to the Weierstrass elliptic function== The theta function was used by Jacobi to construct (in a form adapted to easy calculation) [[Jacobi's elliptic functions\|his elliptic functions]] as the quotients of the above four theta functions, and could have been used by him to construct [[Weierstrass's elliptic functions]] also, since :<math>\wp(z;\tau) = -\big(\log \vartheta_{11}(z;\tau)\big)'' + c</math> where the second derivative is with respect to {{mvar\|z}} and the constant {{mvar\|c}} is defined so that the [[Laurent expansion]] of {{math\|℘(''z'')}} at {{math\|''z'' {{=}} 0}} has zero constant term. ==Relation to the ''q''-gamma function== The fourth theta function – and thus the others too – is intimately connected to the [[q-gamma function\|Jackson {{mvar\|q}}-gamma function]] via the relation<ref name = 'Mezo'>{{Cite journal\| last1=Mező \| first1=István \| title=A {{mvar\|q}}-Raabe formula and an integral of the fourth Jacobi theta function \| year=2012 \| journal=Journal of Number Theory \| volume=133 \| issue=2 \| pages=692–704 \| doi = 10.1016/j.jnt.2012.08.025 \| doi-access=free }}</ref> :<math>\left(\Gamma_{q^2}(x)\Gamma_{q^2}(1-x)\right)^{-1}=\frac{q^{2x(1-x)}}{\left(q^{-2};q^{-2}\right)^3_\infty\left(q^2-1\right)} \vartheta_4\left(\frac{1}{2i}(1-2x)\log q,\frac{1}{q}\right). </math> ==Relations to Dedekind eta function== Let {{math\|''η''(''τ'')}} be the [[Dedekind eta function]], and the argument of the theta function as the [[Nome (mathematics)\|nome]] {{math\|''q'' {{=}} ''e''<sup>π''iτ''</sup>}}. Then, :<math>\begin{align} \theta_2(0,q) = \vartheta_{10}(0;\tau) &= \frac{2\eta^2(2\tau)}{\eta(\tau)},\\[3pt] \theta_3(0,q) = \vartheta_{00}(0;\tau) &= \frac{\eta^5(\tau)}{\eta^2\left(\frac{1}{2}\tau\right)\eta^2(2\tau)} = \frac{\eta^2\left(\frac{1}{2}(\tau+1)\right)}{\eta(\tau+1)}, \\[3pt] \theta_4(0,q) = \vartheta_{01}(0;\tau) &= \frac{\eta^2\left(\frac{1}{2}\tau\right)}{\eta(\tau)}, \end{align}</math> and, :<math>\theta_2(0,q)\,\theta_3(0,q)\,\theta_4(0,q) = 2\eta^3(\tau).</math> See also the [[Weber modular function]]s. ==Elliptic modulus== The [[elliptic modulus]] is :<math>k(\tau) = \frac{\vartheta_{10}(0,\tau)^2 }{\vartheta_{00}(0,\tau)^2} </math> and the complementary elliptic modulus is :<math>k'(\tau) = \frac{\vartheta_{01}(0,\tau)^2 }{\vartheta_{00}(0,\tau)^2} </math> ==A solution to the heat equation== The Jacobi theta function is the [[fundamental solution]] of the one-dimensional [[heat equation]] with spatially periodic boundary conditions.<ref>{{Cite journal\|last=Ohyama\|first=Yousuke\|date=1995\|title=Differential relations of theta functions\|url=https://projecteuclid.org/euclid.ojm/1200786061\|journal=Osaka Journal of Mathematics\|language=EN\|volume=32\|issue=2\|pages=431–450\|issn=0030-6126}}</ref> Taking {{math\|''z'' {{=}} ''x''}} to be real and {{math\|''τ'' {{=}} ''it''}} with {{mvar\|t}} real and positive, we can write :<math>\vartheta (x,it)=1+2\sum_{n=1}^\infty \exp\left(-\pi n^2 t\right) \cos(2\pi nx)</math> which solves the heat equation :<math>\frac{\partial}{\partial t} \vartheta(x,it)=\frac{1}{4\pi} \frac{\partial^2}{\partial x^2} \vartheta(x,it).</math> This theta-function solution is 1-periodic in {{mvar\|x}}, and as {{math\|''t'' → 0}} it approaches the periodic [[Dirac delta function\|delta function]], or [[Dirac comb]], in the sense of [[Distribution (mathematics)\|distributions]] :<math>\lim_{t\to 0} \vartheta(x,it)=\sum_{n=-\infty}^\infty \delta(x-n)</math>. General solutions of the spatially periodic initial value problem for the heat equation may be obtained by convolving the initial data at {{math\|''t'' {{=}} 0}} with the theta function. ==Relation to the Heisenberg group== The Jacobi theta function is invariant under the action of a discrete subgroup of the [[Heisenberg group]]. This invariance is presented in the article on the [[theta representation]] of the Heisenberg group. ==Generalizations== If {{mvar\|F}} is a [[quadratic form]] in {{mvar\|n}} variables, then the theta function associated with {{mvar\|F}} is :<math>\theta_F (z)= \sum_{m\in \Z^n} e^{2\pi izF(m)}</math> with the sum extending over the [[lattice (group)\|lattice]] of integers <math>\Z^n.</math> This theta function is a [[modular form]] of weight {{math\|{{sfrac\|''n''\|2}}}} (on an appropriately defined subgroup) of the [[modular group]]. In the Fourier expansion, :<math>\hat{\theta}_F (z) = \sum_{k=0}^\infty R_F(k) e^{2\pi ikz},</math> the numbers {{math\|''R<sub>F</sub>''(''k'')}} are called the ''representation numbers'' of the form. ===Theta series of a Dirichlet character=== For <math>\chi</math> a primitive [[Dirichlet character]] modulo <math>q</math> and <math>\nu = \frac{1-\chi(-1)}{2}</math> then :<math>\theta_\chi(z) = \frac12\sum_{n=-\infty}^\infty \chi(n) n^\nu e^{2i \pi n^2 z}</math> is a weight <math>\frac12+\nu</math> modular form of level <math>4q^2</math> and character <math>\chi(d) \left(\frac{-1}{d}\right)^\nu</math>, which means :<math>\theta_\chi\left(\frac{az+b}{cz+d}\right) = \chi(d) \left(\frac{-1}{d}\right)^\nu \left(\frac{\theta_1\left(\frac{az+b}{cz+d}\right)}{\theta_1(z)}\right)^{1+2\nu}\theta_\chi(z)</math> whenever :<math>a,b,c,d\in \Z^4, ad-bc=1,c \equiv 0 \bmod 4 q^2.</math><ref>Shimura, On modular forms of half integral weight</ref> ===Ramanujan theta function=== {{further\|Ramanujan theta function\|mock theta function}} ===Riemann theta function=== Let :<math>\mathbb{H}_n=\left\{F\in M(n,\Complex) \,\big\|\, F=F^\mathsf{T} \,,\, \operatorname{Im} F >0 \right\}</math> the set of [[symmetric]] square [[matrix (mathematics)\|matrices]] whose imaginary part is [[Positive-definite matrix\|positive definite]]. <math>\mathbb{H}_n</math> is called the [[Siegel upper half-space]] and is the multi-dimensional analog of the [[upper half-plane]]. The {{mvar\|n}}-dimensional analogue of the [[modular group]] is the [[symplectic group]] <math>\operatorname{Sp}(2n,\Z);</math> for {{math\|''n'' {{=}} 1}}, <math>\operatorname{Sp}(2,\Z) = \operatorname{SL}(2,\Z).</math> The {{mvar\|n}}-dimensional analogue of the [[congruence subgroup]]s is played by :<math>\ker \big\{\operatorname{Sp}(2n,\Z)\to \operatorname{Sp}(2n,\Z/k\Z) \big\}.</math> Then, given <math>\tau \in \mathbb{H}_n,</math> the '''Riemann theta function''' is defined as :<math>\theta (z,\tau)=\sum_{m\in \Z^n} \exp\left(2\pi i \left(\tfrac12 m^\mathsf{T} \tau m +m^\mathsf{T} z \right)\right). </math> Here, <math>z \in\Complex^n</math> is an {{mvar\|n}}-dimensional complex vector, and the superscript '''T''' denotes the [[transpose]]. The Jacobi theta function is then a special case, with {{math\|''n'' {{=}} 1}} and <math>\tau \in \mathbb{H}</math> where <math>\mathbb{H}</math> is the [[upper half-plane]]. One major application of the Riemann theta function is that it allows one to give explicit formulas for meromorphic functions on compact Riemann surfaces, as well as other auxiliary objects that figure prominently in their function theory, by taking <math>\tau</math> to be the period matrix with respect to a canonical basis for its first [[Homology (mathematics)\|homology group]]. The Riemann theta converges absolutely and uniformly on compact subsets of <math>\Complex^n \times \mathbb{H}_n.</math> The functional equation is :<math>\theta (z+a+\tau b, \tau) = \exp 2\pi i \left(-b^\mathsf{T}z-\tfrac12 b^\mathsf{T}\tau b\right) \theta (z,\tau)</math> which holds for all vectors <math>a,b\in\Z^n,</math> and for all <math>z \in\Complex^n</math> and <math>\tau \in \mathbb{H}_n.</math> ===Poincaré series=== The [[Poincaré series (modular form)\|Poincaré series]] generalizes the theta series to automorphic forms with respect to arbitrary [[Fuchsian group]]s. == Notes == {{Reflist}} ==References== {{Cite book \|first=Milton \|last=Abramowitz \|authorlink=Milton Abramowitz \|first2=Irene A. \|last2=Stegun \|authorlink2=Irene Stegun \|title=Handbook of Mathematical Functions \|year=1964 \|publisher=Dover Publications \|location=New York \|isbn=978-0-486-61272-0 \|at=[https://archive.org/details/handbookofmathe000abra/page/ sec. 16.27ff.] \|title-link=Abramowitz and Stegun }} {{Cite book\|first=Naum Illyich \|last=Akhiezer \|authorlink=Naum Akhiezer \|title=Elements of the Theory of Elliptic Functions \|origyear=1970 \|series=AMS Translations of Mathematical Monographs \|volume=79 \|year=1990 \|publisher=AMS \|location=Providence, RI \|isbn=978-0-8218-4532-5 }} {{Cite book\|first=Hershel M. \|last=Farkas \|authorlink=Hershel M. Farkas \|first2=Irwin \|last2=Kra \|authorlink2=Irwin Kra \|title=Riemann Surfaces \|year=1980 \|publisher=Springer-Verlag \|location=New York \|isbn=978-0-387-90465-8 \|at=ch. 6}}. ''(for treatment of the Riemann theta)'' {{Cite book\|first=G. H. \|last=Hardy \|authorlink=G. H. Hardy \|first2=E. M. \|last2=Wright \|authorlink2=E. M. Wright \|title=An Introduction to the Theory of Numbers \|edition=4th \|year=1959 \|publisher=Clarendon Press \|location=Oxford \|isbn= }} {{Cite book\|first=David \|last=Mumford \|authorlink=David Mumford \|title=Tata Lectures on Theta I \|year=1983 \|publisher=Birkhauser \|location=Boston \|isbn=978-3-7643-3109-2 }} {{Cite book\|authorlink=James Pierpont (mathematician) \|first=James \|last=Pierpont \|title=Functions of a Complex Variable \|location=New York \|publisher=Dover Publications \|year=1959 \|isbn= }} {{Cite book\|first=Harry E. \|last=Rauch \|authorlink=Harry Rauch \|first2=Hershel M. \|last2=Farkas \|title=Theta Functions with Applications to Riemann Surfaces \|year=1974 \|publisher=Williams & Wilkins \|location=Baltimore \|isbn=978-0-683-07196-2 }} {{dlmf\|first=William P. \|last=Reinhardt\|first2=Peter L. \|last2=Walker\|id=20\|title=Theta Functions}} {{Cite book\|authorlink=E. T. Whittaker \|first=E. T. \|last=Whittaker \|first2=G. N. \|last2=Watson \|authorlink2=G. N. Watson \|title=A Course in Modern Analysis \|edition=4th \|publisher=Cambridge University Press \|location=Cambridge \|year=1927 \|at=ch. 21}} ''(history of Jacobi's {{mvar\|θ}} functions)'' ==Further reading== {{cite book\| editor1-first=Krishnaswami \| editor1-last=Alladi\|editor1-link= Krishnaswami Alladi \| title=Surveys in Number Theory \| series= Developments in Mathematics \| volume=17 \| publisher=[[Springer-Verlag]] \| year=2008 \| isbn=978-0-387-78509-7 \| first=Hershel M. \|last=Farkas \| chapter=Theta functions in complex analysis and number theory \| pages=57–87 \| zbl=1206.11055 }} {{cite book\| first=Bruno \| last=Schoeneberg \| authorlink= Bruno Schoeneberg \| title=Elliptic modular functions \| series=Die Grundlehren der mathematischen Wissenschaften \| volume=203 \| publisher=[[Springer-Verlag]] \| year=1974 \| isbn=978-3-540-06382-7 \| chapter=IX. Theta series \| pages=203–226 }} Ackerman, M. Math. Ann. (1979) 244: 75. [https://doi.org/10.1007/BF01420339 "On the Generating Functions of Certain Eisenstein Series]" Springer-Verlag Harry Rauch with Hershel M. Farkas: Theta functions with applications to Riemann Surfaces, Williams and Wilkins, Baltimore MD 1974, {{ISBN\|0-683-07196-3}}. ==External links== *{{cite web\|url=https://moiseevigor.github.io/elliptic/ \|title=Elliptic functions for Matlab and Octave\|author=Moiseev Igor}} {{PlanetMath attribution\|id=6262\|title=Integral representations of Jacobi theta functions}} {{Authority control}} [[Category:Theta functions\| ]] [[Category:Elliptic functions]] [[Category:Riemann surfaces]] [[Category:Analytic functions]] [[Category:Several complex variables]]
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