View source for Theta function

{{Short description|Special functions of several complex variables}}
{{for|other θ functions|Theta function (disambiguation)}}

[[File:Cplot of Jacobi theta 1.svg|400px|thumb|upright=1.2|Jacobi's theta function {{math|''θ''<sub>1</sub>}} with nome {{math|''q'' {{=}} ''e''<sup>''i''π''τ''</sup> {{=}} 0.1''e''<sup>0.1''i''π</sup>}}:
<math>\begin{align}
\theta_1(z,q) &= 2 q^\frac14 \sum_{n=0}^\infty (-1)^n q^{n(n+1)} \sin(2n+1)z \\
&= \sum_{n=-\infty}^\infty (-1)^{n-\frac12} q^{\left(n+\frac12\right)^2} e^{(2n+1)i z} 
.\end{align}</math>]]

In [[mathematics]], '''theta functions''' are [[special function]]s of [[several complex variables]]. They show up in many topics, including [[Abelian variety|Abelian varieties]], [[moduli space]]s, [[quadratic form]]s, and [[soliton]]s. Theta functions are parametrized by points in a [[tube domain]] inside a complex [[Lagrangian Grassmannian]],<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> namely the [[Siegel upper half space]].

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 quasiperiodicity comes from the [[Group cohomology|cohomology class]] of a [[Complex torus|line bundle on a complex torus]], a condition of [[descent (category theory)|descent]].

One interpretation of theta functions when dealing with the heat equation is that "a theta function is a special function that describes the evolution of temperature on a segment domain subject to certain boundary conditions".<ref>{{Cite book |last=Chang |first=Der-Chen |title=Heat Kernels for Elliptic and Sub-elliptic Operators |publisher=Birkhäuser |year=2011 |pages=7}}</ref>

Throughout this article, <math>(e^{\pi i\tau})^{\alpha}</math> should be interpreted as <math>e^{\alpha \pi i\tau}</math> (in order to resolve issues of choice of [[Branch point#Branch cuts|branch]]).<ref group="note">See e.g. https://dlmf.nist.gov/20.1. Note that this is, in general, not equivalent to the usual interpretation <math>(e^z)^\alpha=e^{\alpha \operatorname{Log} e^z}</math> when <math>z</math> is outside the strip <math>-\pi<\operatorname{Im}z\le\pi</math>. Here, <math>\operatorname{Log}</math> denotes the principal branch of the [[Complex logarithm#Properties|complex logarithm]].</ref>

==Jacobi theta function==

There are several closely related functions called Jacobi theta functions, and [[Jacobi theta functions (notational variations)|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 a 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 q^{n^2} \cos(2\pi n z) \\
&= \sum_{n=-\infty}^\infty q^{n^2}\eta^n 
\end{align}</math>

where {{math|''q'' {{=}} exp(''πiτ'')}} is the [[nome (mathematics)|nome]] and {{math|''η'' {{=}} exp(2''πiz'')}}. It is a [[Jacobi form]]. The restriction ensures that it is an absolutely convergent series. 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>

By [[completing the square]], it is also {{mvar|τ}}-quasiperiodic in {{mvar|z}}, with

:<math>\vartheta(z+\tau;\tau) = \exp\bigl(-\pi i (\tau + 2 z)\bigr) \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}}.

For any fixed <math>\tau </math>, the function is an entire function on the complex plane, so by [[Liouville's theorem (complex analysis)|Liouville's theorem]], it cannot be doubly periodic in <math>1, \tau </math> unless it is constant, and so the best we can do is to make it periodic in <math>1 </math> and quasi-periodic in <math>\tau </math>. Indeed, since <math display="block">\left|\frac{\vartheta(z+a+b\tau;\tau)}{\vartheta(z;\tau)}\right| = \exp\left(\pi (b^2 \Im(\tau) + 2b \Im(z)) \right) </math>and <math>\Im(\tau)> 0 </math>, the function <math>\vartheta(z, \tau) </math> is unbounded, as required by Liouville's theorem.

It is in fact the most general entire function with 2 quasi-periods, in the following sense:<ref>{{Cite book |url=http://link.springer.com/10.1007/978-0-8176-4577-9 |title=Tata Lectures on Theta I |date=2007 |publisher=Birkhäuser Boston |isbn=978-0-8176-4572-4 |series=Modern Birkhäuser Classics |location=Boston, MA |pages=4 |language=en |doi=10.1007/978-0-8176-4577-9 }}</ref>
{{Math theorem
| math_statement = If <math>f: \mathbb C \to \mathbb C</math> is entire and nonconstant, and satisfies the functional equations
<math>
\begin{cases}
f(z+1) = f(z)\\
f(z + \tau) = e^{az + 2\pi i b} f(z)
\end{cases}
</math>
for some constant <math>a, b\in \mathbb C</math>.

If <math>a = 0</math>, then <math>b = \tau</math> and <math>f(z) = e^{2\pi i z}</math>. If <math>a = -2\pi i</math>, then <math>f(z) = C \vartheta(z + \frac 1 2 \tau + b, \tau)</math> for some nonzero <math>C\in \mathbb C</math>.
}}[[Image:Complex theta animated1.gif|500px|thumb|center|Theta function {{math|''θ''<sub>1</sub>}} with different nome {{math|''q'' {{=}} ''e''<sup>''iπτ''</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>''iπτ''</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>''iπτ''</sup>}} rather than {{mvar|τ}}. In Jacobi's notation the {{mvar|θ}}-functions are written:

:<math>\begin{align}
\theta_1(z;q) &=\theta_1(\pi z,q)= -\vartheta_{11}(z;\tau)\\
\theta_2(z;q) &=\theta_2(\pi z,q)= \vartheta_{10}(z;\tau)\\
\theta_3(z;q) &=\theta_3(\pi z,q)= \vartheta_{00}(z;\tau)\\
\theta_4(z;q) &=\theta_4(\pi z,q)= \vartheta_{01}(z;\tau)
\end{align}</math>

[[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]]

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. These functions are called ''Theta Nullwert'' functions, based on the German term for ''zero value'' because of the annullation of the left entry in the theta function expression. Alternatively, we obtain four functions of {{mvar|q}} only, defined on the unit disk <math>|q|<1</math>. They are sometimes called [[theta constant]]s:<ref group="note"><math>\theta_1(q)=0</math> for all <math>q\in\mathbb{C}</math> with <math>|q|<1</math>.</ref>

:<math>\begin{align}
\vartheta_{11}(0;\tau)&=-\theta_1(q)=-\sum_{n=-\infty}^\infty (-1)^{n-1/2}q^{(n+1/2)^2} \\
\vartheta_{10}(0;\tau)&=\theta_2(q)=\sum_{n=-\infty}^\infty q^{(n+1/2)^2}\\
\vartheta_{00}(0;\tau)&=\theta_3(q)=\sum_{n=-\infty}^\infty q^{n^2}\\
\vartheta_{01}(0;\tau)&=\theta_4(q)=\sum_{n=-\infty}^\infty (-1)^n q^{n^2}
\end{align}</math>

with the [[Nome (mathematics)|nome]] {{math|''q'' {{=}} ''e''<sup>''iπτ''</sup>}}. 
Observe that <math> \theta_1(q)=0 </math>. 
These can be used to define a variety of [[modular forms]], and to parametrize certain curves; in particular, the '''Jacobi identity''' is

:<math>\theta_2(q)^4 + \theta_4(q)^4 = \theta_3(q)^4</math>

or equivalently,

:<math>\vartheta_{01}(0;\tau)^4 + \vartheta_{10}(0;\tau)^4 = \vartheta_{00}(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>''πiz''</sup>}} and {{math|''q'' {{=}} ''e''<sup>''πiτ''</sup>}}. In this form, the functions become

:<math>\begin{align}
\vartheta_{00}(w, q)& = \sum_{n=-\infty}^\infty  \left(w^2\right)^n q^{n^2}\quad&
\vartheta_{01}(w, q)& = \sum_{n=-\infty}^\infty (-1)^n \left(w^2\right)^n q^{n^2}\\[3pt]
\vartheta_{10}(w, q)& = \sum_{n=-\infty}^\infty \left(w^2\right)^{n+\frac12} q^{\left(n + \frac12\right)^2}\quad&
\vartheta_{11}(w, q)& = i \sum_{n=-\infty}^\infty (-1)^n \left(w^2\right)^{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>''πiτ''</sup>}} (noting some authors instead set {{math|''q'' {{=}} ''e''<sup>2''πiτ''</sup>}})  and take {{math|''w'' {{=}} ''e''<sup>''πiz''</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|(&nbsp;&nbsp;;&nbsp;&nbsp;)<sub>∞</sub>}} is the [[q-Pochhammer symbol|{{mvar|q}}-Pochhammer symbol]] and {{math|''θ''(&nbsp;&nbsp;;&nbsp;&nbsp;)}} 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>
In particular, <math display="block">\lim_{q\to 0}\frac{\vartheta_{10}(z\mid q)}{2 q^{\frac14}} = \cos(\pi z),\quad \lim_{q\to 0}\frac{-\vartheta_{11}(z\mid q)}{2 q^{-\frac14}} = \sin(\pi z)</math>so we may interpret them as one-parameter deformations of the periodic functions <math>\sin, \cos</math>, again validating the interpretation of the theta function as the most general 2 quasi-period function.

==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(2\pi uz + \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 (2\pi uz)}{\sin (\pi u)} \mathrm{d}u; \\[6pt]
\vartheta_{10} (z; \tau) &= -ie^{i \pi z + \frac14 i \pi\tau} \int_{i-\infty}^{i+\infty} e^{i\pi\tau u^2} \frac{\cos(2\pi uz + \pi u + \pi \tau u)}{\sin (\pi u)} \mathrm{d}u; \\[6pt]
\vartheta_{11} (z; \tau) &= e^{i \pi z + \frac14 i \pi \tau} \int_{i-\infty}^{i+\infty} e^{i\pi\tau u^2} \frac{\cos(2\pi uz + \pi \tau u)}{\sin(\pi u)} \mathrm{d}u.
\end{align}</math>

The Theta Nullwert function <math> \theta_{3}(q) </math> as this integral identity:

:<math>\theta_{3}(q) = 1 + \frac{4q\sqrt{\ln(1/q)}}{\sqrt{\pi}} \int_{0}^{\infty} \frac{\exp[-\ln(1/q)\,x^2]\{1 - q^2\cos[2\ln(1/q)\,x]\}}{1 - 2q^2\cos[2\ln(1/q)\,x] + q^4} \,\mathrm{d}x </math>

This formula was discussed in the essay ''Square series generating function transformations'' by the mathematician Maxie Schmidt from Georgia in Atlanta.

Based on this formula following three eminent examples are given:

:<math>\biggl[\frac{2}{\pi}K\bigl(\frac{1}{2}\sqrt{2}\bigr)\biggr]^{1/2} = \theta_{3}\bigl[\exp(-\pi)\bigr] = 1 + 4\exp(-\pi) \int_{0}^{\infty} \frac{\exp(-\pi x^2)[1 - \exp(-2\pi)\cos(2\pi x)]}{1 - 2\exp(-2\pi)\cos(2\pi x) + \exp(-4\pi)} \,\mathrm{d}x </math>

:<math>\biggl[\frac{2}{\pi}K(\sqrt{2} - 1)\biggr]^{1/2} = \theta_{3}\bigl[\exp(-\sqrt{2}\,\pi)\bigr] = 1 + 4\,\sqrt[4]{2}\exp(-\sqrt{2}\,\pi) \int_{0}^{\infty} \frac{\exp(-\sqrt{2}\,\pi x^2)[1 - \exp(-2\sqrt{2}\,\pi)\cos(2\sqrt{2}\,\pi x)]}{1 - 2\exp(-2\sqrt{2}\,\pi)\cos(2\sqrt{2}\,\pi x) + \exp(-4\sqrt{2}\,\pi)} \,\mathrm{d}x </math>

:<math>\biggl\{\frac{2}{\pi}K\bigl[\sin\bigl(\frac{\pi}{12}\bigr)\bigr]\biggr\}^{1/2} = \theta_{3}\bigl[\exp(-\sqrt{3}\,\pi)\bigr] = 1 + 4\,\sqrt[4]{3}\exp(-\sqrt{3}\,\pi) \int_{0}^{\infty} \frac{\exp(-\sqrt{3}\,\pi x^2)[1 - \exp(-2\sqrt{3}\,\pi)\cos(2\sqrt{3}\,\pi x)]}{1 - 2\exp(-2\sqrt{3}\,\pi)\cos(2\sqrt{3}\,\pi x) + \exp(-4\sqrt{3}\,\pi)} \,\mathrm{d}x </math>

Furthermore, the theta examples <math> \theta_{3}(\tfrac{1}{2}) </math> and <math> \theta_{3}(\tfrac{1}{3}) </math> shall be displayed:

:<math>\theta_{3}\left(\frac{1}{2}\right) = 1+2\sum_{n = 1}^{\infty} \frac{1}{2^{n^2}} = 1 + 2\pi^{-1/2}\sqrt{\ln(2)} \int_{0}^{\infty} \frac{\exp[-\ln(2)\,x^2]\{16 - 4\cos[2\ln(2)\,x]\}}{17 - 8\cos[2\ln(2)\,x]} \,\mathrm{d}x </math>

:<math>\theta_{3}\left(\frac{1}{2}\right) = 2.128936827211877158669\ldots </math>

:<math>\theta_{3}\left(\frac{1}{3}\right) = 1+2\sum_{n = 1}^{\infty} \frac{1}{3^{n^2}} = 1 + \frac{4}{3}\pi^{-1/2}\sqrt{\ln(3)} \int_{0}^{\infty} \frac{\exp[-\ln(3)\,x^2]\{81 - 9\cos[2\ln(3)\,x]\}}{82 - 18\cos[2\ln(3)\,x]} \,\mathrm{d}x </math>

:<math>\theta_{3}\left(\frac{1}{3}\right) = 1.691459681681715341348\ldots </math>

==Some interesting relations==

If <math>|q|<1</math> and <math>a>0</math>, then the following theta functions

:<math>\theta_3(a,b;q)=\sum^{\infty}_{n=-\infty}q^{an^2+bn}</math>
:<math>\theta_4(a,b;q)=\sum^{\infty}_{n=-\infty}(-1)^nq^{an^2+bn}</math>

have interesting arithmetical and modular properties. When <math>a,b,p</math> are positive integers, then <ref>{{cite arXiv|last1=Bagis|first1=Nikolaos D |title=On the Complete Evaluation of Jacobi Theta Functions|eprint=1503.01141|date=19 December 2022|class=math.GM }}</ref><ref>{{cite arXiv|last1=Bagis|first1=Nikolaos D |title=Evaluations of Series Related to Jacobi Elliptic Functions|eprint=1803.09445|date=31 July 2019|class=math.NT }}</ref>

:<math>\log\left(\frac{\theta_3\left(\frac{p}{2},\frac{p}{2}-a;q\right)}{\theta_3\left(\frac{p}{2},\frac{p}{2}-b;q\right)}\right)=-\sum^{\infty}_{n=1}q^n\left(\sum_{\begin{array}{cc}d|n\\n/d\equiv\pm a(p)\end{array}}\frac{(-1)^d}{d}-\sum_{\begin{array}{cc}d|n\\n/d\equiv\pm b(p)\end{array}}\frac{(-1)^d}{d}\right)</math>

:<math>\log\left(\frac{\theta_4\left(\frac{p}{2},\frac{p}{2}-a;q\right)}{\theta_4\left(\frac{p}{2},\frac{p}{2}-b;q\right)}\right)=-\sum^{\infty}_{n=1}q^n\left(\sum_{\begin{array}{cc}d|n\\n/d\equiv\pm a(p)\end{array}}\frac{1}{d}-\sum_{\begin{array}{cc}d|n\\n/d\equiv\pm b(p)\end{array}}\frac{1}{d}\right)</math>

Also if <math>q=e^{\pi i z}</math>, <math>Im(z)>0</math>, the functions with :

:<math>\vartheta_{+}(z)=\theta_{+}(a,p;z)=q^{p/8+a^2/(2p)-a/2}\theta_3\left(\frac{p}{2},\frac{p}{2}-a;q\right)</math>

and

:<math>\vartheta_{-}(z)=\theta_{-}(a,p;z)=q^{p/8+a^2/(2p)-a/2}\theta_4\left(\frac{p}{2},\frac{p}{2}-a;q\right)</math>

are modular forms with weight <math>1/2</math> in <math>\Gamma(2p)</math> i.e. If <math>a_1,b_1,c_1,d_1</math> are integers such that <math>a_1,d_1\equiv 1(2p)</math>, <math>b_1,c_1\equiv0(2p)</math> and <math>a_1d_1-b_1c_1=1</math> there exists  <math>\epsilon_{\pm}=\epsilon_{\pm}(a_1,b_1,c_1,d_1)</math>, <math>(\epsilon_{\pm})^{24}=1</math>, such that for all complex numbers <math>z</math> with   <math>Im(z)>0</math>, we have

:<math>\vartheta_{\pm}\left(\frac{a_1z+b_1}{c_1z+d_1}\right)=\epsilon_{\pm}\sqrt{c_1z+d_1}\vartheta_{\pm}(z)</math>

==Explicit values==
=== [[:Wiktionary:lemniscatic|Lemniscatic]] values ===

Proper credit for most of 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 they are either in Ramanujan's lost notebook or follow immediately from it. See also 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> Define,

:<math>\quad \varphi(q) =\vartheta_{00}(0;\tau) =\theta_3(0;q)=\sum_{n=-\infty}^\infty q^{n^2}</math>

with the nome <math>q =e^{\pi i \tau},</math> <math>\tau = n\sqrt{-1},</math> and [[Dedekind eta function]] <math>\eta(\tau).</math> Then for <math>n = 1,2,3,\dots</math>

:<math>\begin{align}
\varphi\left(e^{-\pi} \right) &= \frac{\sqrt[4]{\pi}}{\Gamma\left(\frac34\right)} = \sqrt2\,\eta\left(\sqrt{-1}\right)\\
\varphi\left(e^{-2\pi}\right) &= \frac{\sqrt[4]{\pi}}{\Gamma\left(\frac34\right)} \frac{\sqrt{2+\sqrt2}}{2}\\
\varphi\left(e^{-3\pi}\right) &= \frac{\sqrt[4]{\pi}}{\Gamma\left(\frac34\right)} \frac{\sqrt{1+\sqrt3}}{\sqrt[8]{108}}\\
\varphi\left(e^{-4\pi}\right) &= \frac{\sqrt[4]{\pi}}{\Gamma\left(\frac34\right)} \frac{2+\sqrt[4]{8}}{4}\\
\varphi\left(e^{-5\pi}\right) &= \frac{\sqrt[4]{\pi}}{\Gamma\left(\frac34\right)} \sqrt{\frac{2+\sqrt5}{5}}\\
\varphi\left(e^{-6\pi}\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]{12^3}}\\
\varphi\left(e^{-7\pi}\right) &= \frac{\sqrt[4]{\pi}}{\Gamma\left(\frac34\right)} \frac{\sqrt{\sqrt{13+\sqrt{7}}+\sqrt{7+3\sqrt{7}}}}{\sqrt[8]{14^3}\cdot\sqrt[16]{7}}\\
\varphi\left(e^{-8\pi}\right) &= \frac{\sqrt[4]{\pi}}{\Gamma\left(\frac34\right)} \frac{\sqrt{2+\sqrt{2}}+\sqrt[8]{128}}{4}\\
\varphi\left(e^{-9\pi}\right) &= \frac{\sqrt[4]{\pi}}{\Gamma\left(\frac34\right)} \frac{1+\sqrt[3]{2+2\sqrt{3}}}{3}\\
\varphi\left(e^{-10\pi}\right) &= \frac{\sqrt[4]{\pi}}{\Gamma\left(\frac34\right)} \frac{\sqrt{\sqrt[4]{64}+\sqrt[4]{80}+\sqrt[4]{81}+\sqrt[4]{100}}}{\sqrt[4]{200}}\\
\varphi\left(e^{-11\pi}\right) &= \frac{\sqrt[4]{\pi}}{\Gamma\left(\frac34\right)} \frac{\sqrt{11+\sqrt{11}+(5+3\sqrt{3}+\sqrt{11}+\sqrt{33})\sqrt[3]{-44+33\sqrt{3}}+(-5+3\sqrt{3}-\sqrt{11}+\sqrt{33})\sqrt[3]{44+33\sqrt{3}}}}{\sqrt[8]{52180524}}\\
\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}}\\
\varphi\left(e^{-13\pi}\right) &= \frac{\sqrt[4]{\pi}}{\Gamma\left(\frac34\right)} \frac{\sqrt{13+8\sqrt{13}+(11-6\sqrt{3}+\sqrt{13})\sqrt[3]{143+78\sqrt{3}}+(11+6\sqrt{3}+\sqrt{13})\sqrt[3]{143-78\sqrt{3}}}}{\sqrt[4]{19773}}\\
\varphi\left(e^{-14\pi}\right) &= \frac{\sqrt[4]{\pi}}{\Gamma\left(\frac34\right)} \frac{\sqrt{\sqrt{13+\sqrt{7}}+\sqrt{7+3\sqrt{7}}+\sqrt{10+2\sqrt{7}}+\sqrt[8]{28}\sqrt{4+\sqrt{7}}}}{\sqrt[16]{28^7}}\\
\varphi\left(e^{-15\pi}\right) &= \frac{\sqrt[4]{\pi}}{\Gamma\left(\frac34\right)} \frac{\sqrt{7+3\sqrt{3}+\sqrt{5}+\sqrt{15}+\sqrt[4]{60}+\sqrt[4]{1500}}}{\sqrt[8]{12^3}\cdot\sqrt{5}}\\
2\varphi\left(e^{-16\pi}\right) &= \varphi\left(e^{-4\pi}\right) + \frac{\sqrt[4]{\pi}}{\Gamma\left(\frac34\right)} \frac{\sqrt[4]{1+\sqrt{2}}}{\sqrt[16]{128}}\\
\varphi\left(e^{-17\pi}\right) &= \frac{\sqrt[4]{\pi}}{\Gamma\left(\frac34\right)} \frac{\sqrt{2}(1+\sqrt[4]{17})+\sqrt[8]{17}\sqrt{5+\sqrt{17}}}{\sqrt{17+17\sqrt{17}}}\\
2\varphi\left(e^{-20\pi}\right) &= \varphi\left(e^{-5\pi}\right) + \frac{\sqrt[4]{\pi}}{\Gamma\left(\frac34\right)} \sqrt{\frac{3+2\sqrt[4]{5}}{5\sqrt2}}\\
6\varphi\left(e^{-36\pi}\right) &= 3\varphi\left(e^{-9\pi}\right) + 2\varphi\left(e^{-4\pi}\right) - \varphi\left(e^{-\pi}\right) + \frac{\sqrt[4]{\pi}}{\Gamma\left(\frac34\right)} \sqrt[3]{\sqrt[4]{2}+\sqrt[4]{18}+\sqrt[4]{216}}
\end{align}</math>

If the reciprocal of the [[Gelfond constant]] is raised to the power of the reciprocal of an odd number, then the corresponding  <math> \vartheta_{00} </math> values or <math> \phi </math> values can be represented in a simplified way by using the [[Lemniscate elliptic functions|hyperbolic lemniscatic sine]]:

:<math> \varphi\bigl[\exp(-\tfrac{1}{5}\pi)\bigr] = \sqrt[4]{\pi}\,{\Gamma\left(\tfrac{3}{4}\right)}^{-1} \operatorname{slh}\bigl(\tfrac{1}{5}\sqrt{2}\,\varpi\bigr)\operatorname{slh}\bigl(\tfrac{2}{5}\sqrt{2}\,\varpi\bigr) </math>

:<math> \varphi\bigl[\exp(-\tfrac{1}{7}\pi)\bigr] = \sqrt[4]{\pi}\,{\Gamma\left(\tfrac{3}{4}\right)}^{-1} \operatorname{slh}\bigl(\tfrac{1}{7}\sqrt{2}\,\varpi\bigr)\operatorname{slh}\bigl(\tfrac{2}{7}\sqrt{2}\,\varpi\bigr) \operatorname{slh}\bigl(\tfrac{3}{7}\sqrt{2}\,\varpi\bigr) </math>

:<math> \varphi\bigl[\exp(-\tfrac{1}{9}\pi)\bigr] = \sqrt[4]{\pi}\,{\Gamma\left(\tfrac{3}{4}\right)}^{-1} \operatorname{slh}\bigl(\tfrac{1}{9}\sqrt{2}\,\varpi\bigr)\operatorname{slh}\bigl(\tfrac{2}{9}\sqrt{2}\,\varpi\bigr) \operatorname{slh}\bigl(\tfrac{3}{9}\sqrt{2}\,\varpi\bigr) \operatorname{slh} \bigl(\tfrac{4}{9}\sqrt{2}\,\varpi\bigr) </math>

:<math> \varphi\bigl[\exp(-\tfrac{1}{11}\pi)\bigr] = \sqrt[4]{\pi}\,{\Gamma\left(\tfrac{3}{4}\right)}^{-1} \operatorname{slh}\bigl(\tfrac{1}{11}\sqrt{2}\,\varpi\bigr)\operatorname{slh}\bigl(\tfrac{2}{11}\sqrt{2}\,\varpi\bigr) \operatorname{slh}\bigl(\tfrac{3}{11}\sqrt{2}\,\varpi\bigr) \operatorname{slh} \bigl(\tfrac{4}{11}\sqrt{2}\,\varpi\bigr) \operatorname{slh}\bigl(\tfrac{5}{11}\sqrt{2}\,\varpi\bigr) </math>

With the letter <math> \varpi </math> the [[Lemniscate constant]] is represented.

Note that the following modular identities hold:

:<math>\begin{align}
2\varphi\left(q^4\right) &= \varphi(q)+\sqrt{2\varphi^2\left(q^2\right)-\varphi^2(q)}\\
3\varphi\left(q^9\right) &= \varphi(q)+\sqrt[3]{9\frac{\varphi^4\left(q^3\right)}{\varphi(q)}-\varphi^3(q)}\\
\sqrt{5}\varphi\left(q^{25}\right) &= \varphi\left(q^5\right)\cot\left(\frac{1}{2}\arctan\left(\frac{2}{\sqrt{5}}\frac{\varphi(q)\varphi\left(q^5\right)}{\varphi^2(q)-\varphi^2\left(q^5\right)}\frac{1+s(q)-s^2(q)}{s(q)}\right)\right)
\end{align}</math>

where <math>s(q)=s\left(e^{\pi i\tau}\right)=-R\left(-e^{-\pi i/(5\tau)}\right)</math> is the [[Rogers–Ramanujan continued fraction]]:

:<math>\begin{align}
s(q) &= \sqrt[5]{\tan\left(\frac{1}{2}\arctan\left(\frac{5}{2}\frac{\varphi^2\left(q^5\right)}{\varphi^2(q)}-\frac{1}{2}\right)\right)\cot^2\left(\frac{1}{2}\operatorname{arccot}\left(\frac{5}{2}\frac{\varphi^2\left(q^5\right)}{\varphi^2(q)}-\frac{1}{2}\right)\right)}\\
&= \cfrac{e^{-\pi i/(25\tau)}}{1-\cfrac{e^{-\pi i/(5\tau)}}{1+\cfrac{e^{-2\pi i/(5\tau)}}{1-\ddots}}}
\end{align}</math>

=== [[Equianharmonic]] values ===
The mathematician [[Bruce Berndt]] found out further values<ref>{{cite journal |last1=Berndt |first1=Bruce C |last2=Rebák |first2=Örs |title=Explicit Values for Ramanujan's Theta Function {{not a typo|ϕ}}(q) |journal=Hardy-Ramanujan Journal |date=9 January 2022 |volume=44 |pages=8923 |doi=10.46298/hrj.2022.8923 |s2cid=245851672 |doi-access=free |arxiv=2112.11882 }}</ref> of the theta function:

:<math>\begin{array}{lll}
\varphi\left(\exp( -\sqrt{3}\,\pi)\right) &=& \pi^{-1}{\Gamma\left(\tfrac{4}{3}\right)}^{3/2}2^{-2/3}3^{13/8} \\
\varphi\left(\exp(-2\sqrt{3}\,\pi)\right) &=& \pi^{-1}{\Gamma\left(\tfrac{4}{3}\right)}^{3/2}2^{-2/3}3^{13/8}\cos(\tfrac{1}{24}\pi) \\
\varphi\left(\exp(-3\sqrt{3}\,\pi)\right) &=& \pi^{-1}{\Gamma\left(\tfrac{4}{3}\right)}^{3/2}2^{-2/3}3^{7/8}(\sqrt[3]{2}+1) \\
\varphi\left(\exp(-4\sqrt{3}\,\pi)\right) &=& \pi^{-1}{\Gamma\left(\tfrac{4}{3}\right)}^{3/2}2^{-5/3}3^{13/8}\Bigl(1+\sqrt{\cos(\tfrac{1}{12}\pi)}\Bigr) \\
\varphi\left(\exp(-5\sqrt{3}\,\pi)\right) &=& \pi^{-1}{\Gamma\left(\tfrac{4}{3}\right)}^{3/2}2^{-2/3}3^{5/8}\sin(\tfrac{1}{5}\pi)(\tfrac{2}{5}\sqrt[3]{100}+\tfrac{2}{5}\sqrt[3]{10}+\tfrac{3}{5}\sqrt{5}+1)
\end{array}</math>

=== Further values ===
Many values of the theta function<ref>{{cite journal |last1=Yi |first1=Jinhee |title=Theta-function identities and the explicit formulas for theta-function and their applications |journal=Journal of Mathematical Analysis and Applications |date=15 April 2004 |volume=292 |issue=2 |pages=381–400 |doi=10.1016/j.jmaa.2003.12.009 |doi-access=free }}</ref> and especially of the shown phi function can be represented in terms of the gamma function:

:<math>\begin{array}{lll}
\varphi\left(\exp( -\sqrt{2}\,\pi)\right) &=& \pi^{-1/2}\Gamma\left(\tfrac{9}{8}\right){\Gamma\left(\tfrac{5}{4}\right)}^{-1/2}2^{7/8} \\
\varphi\left(\exp(-2\sqrt{2}\,\pi)\right) &=& \pi^{-1/2}\Gamma\left(\tfrac{9}{8}\right){\Gamma\left(\tfrac{5}{4}\right)}^{-1/2}2^{1/8}\Bigl(1+\sqrt{\sqrt{2}-1}\Bigr) \\
\varphi\left(\exp(-3\sqrt{2}\,\pi)\right) &=& \pi^{-1/2}\Gamma\left(\tfrac{9}{8}\right){\Gamma\left(\tfrac{5}{4}\right)}^{-1/2}2^{3/8}3^{-1/2}(\sqrt{3}+1)\sqrt{\tan(\tfrac{5}{24}\pi)} \\
\varphi\left(\exp(-4\sqrt{2}\,\pi)\right) &=& \pi^{-1/2}\Gamma\left(\tfrac{9}{8}\right){\Gamma\left(\tfrac{5}{4}\right)}^{-1/2}2^{-1/8}\Bigl(1+\sqrt[4]{2\sqrt{2}-2}\Bigr) \\
\varphi\left(\exp(-5\sqrt{2}\,\pi)\right) &=& \pi^{-1/2}\Gamma\left(\tfrac{9}{8}\right){\Gamma\left(\tfrac{5}{4}\right)}^{-1/2} \frac{1}{15}\,2^{3/8} \times \\
 && \times \biggl[\sqrt[3]{5}\,\sqrt{10+2\sqrt{5}}\biggl(\sqrt[3]{5+\sqrt{2}+3\sqrt{3}}+\sqrt[3]{5+\sqrt{2}-3\sqrt{3}}\,\biggr)-\bigl(2-\sqrt{2}\,\bigr)\sqrt{25-10\sqrt{5}}\,\biggr] \\
\varphi\left(\exp( -\sqrt{6}\,\pi)\right) &=& \pi^{-1/2}\Gamma\left(\tfrac{5}{24}\right){\Gamma\left(\tfrac{5}{12}\right)}^{-1/2}2^{-13/24}3^{-1/8}\sqrt{\sin(\tfrac{5}{12}\pi)} \\
\varphi\left(\exp(-\tfrac{1}{2}\sqrt{6}\,\pi)\right) &=& \pi^{-1/2}\Gamma\left(\tfrac{5}{24}\right){\Gamma\left(\tfrac{5}{12}\right)}^{-1/2}2^{5/24}3^{-1/8}\sin(\tfrac{5}{24}\pi)
\end{array}</math>

==Nome power theorems==

===Direct power theorems===

For the transformation of the nome<ref>Andreas Dieckmann: ''[http://www-elsa.physik.uni-bonn.de/~dieckman/InfProd/InfProd.html#InfinitexProducts Table of Infinite Products Infinite Sums Infinite Series, Elliptic Theta.]'' Physikalisches Institut Universität Bonn, Abruf am 1. Oktober 2021.</ref> in the theta functions these formulas can be used:

:<math>\theta_{2}(q^2) = \tfrac{1}{2}\sqrt{2[\theta_{3}(q)^2 - \theta_{4}(q)^2]}</math>
:<math>\theta_{3}(q^2) = \tfrac{1}{2}\sqrt{2[\theta_{3}(q)^2 + \theta_{4}(q)^2]}</math>
:<math>\theta_{4}(q^2) = \sqrt{\theta_{4}(q)\theta_{3}(q)}</math>

The squares of the three theta zero-value functions with the square function as the inner function are also formed in the pattern of the [[Pythagorean triple]]s according to the Jacobi Identity. Furthermore, those transformations are valid:

:<math>\theta_{3}(q^4) = \tfrac{1}{2}\theta_{3}(q) + \tfrac{1}{2}\theta_{4}(q)</math>

These formulas can be used to compute the theta values of the cube of the nome:

:<math>27\,\theta_{3}(q^3)^8 - 18\,\theta_{3}(q^3)^4\theta_{3}(q)^4 - \,\theta_{3}(q)^8 = 8\,\theta_{3}(q^3)^2\theta_{3}(q)^2[2\,\theta_{4}(q)^4 - \theta_{3}(q)^4]</math>

:<math>27\,\theta_{4}(q^3)^8 - 18\,\theta_{4}(q^3)^4\theta_{4}(q)^4 - \,\theta_{4}(q)^8 = 8\,\theta_{4}(q^3)^2\theta_{4}(q)^2[2\,\theta_{3}(q)^4 - \theta_{4}(q)^4]</math>

And the following formulas can be used to compute the theta values of the fifth power of the nome:

:<math>[\theta_{3}(q)^2 - \theta_{3}(q^5)^2][5\,\theta_{3}(q^5)^2 - \theta_{3}(q)^2]^5 = 256\,\theta_{3}(q^5)^2\theta_{3}(q)^2\theta_{4}(q)^4 [\theta_{3}(q)^4 - \theta_{4}(q)^4]</math>

:<math>[\theta_{4}(q^5)^2 - \theta_{4}(q)^2][5\,\theta_{4}(q^5)^2 - \theta_{4}(q)^2]^5 = 256\,\theta_{4}(q^5)^2\theta_{4}(q)^2\theta_{3}(q)^4 [\theta_{3}(q)^4 - \theta_{4}(q)^4]</math>

=== Transformation at the cube root of the nome ===
The formulas for the theta Nullwert function values from the cube root of the elliptic nome are obtained by contrasting the two real solutions of the corresponding quartic equations:

: <math>\biggl[\frac{\theta_{3}(q^{1/3})^2}{\theta_{3}(q)^2} - \frac{3\,\theta_{3}(q^{3})^2}{\theta_{3}(q)^2}\biggr]^2 = 4 - 4\biggl[\frac{2\,\theta_{2}(q)^2 \theta_{4}(q)^2}{\theta_{3}(q)^4}\biggr]^{2/3} </math>

: <math>\biggl[\frac{3\,\theta_{4}(q^{3})^2}{\theta_{4}(q)^2} - \frac{\theta_{4}(q^{1/3})^2}{\theta_{4}(q)^2}\biggr]^2 = 4 + 4\biggl[\frac{2\,\theta_{2}(q)^2 \theta_{3}(q)^2}{\theta_{4}(q)^4}\biggr]^{2/3} </math>

=== Transformation at the fifth root of the nome ===
The [[Rogers-Ramanujan continued fraction]] can be defined in terms of the '''Jacobi theta function''' in the following way:

: <math>R(q) = \tan\biggl\{\frac{1}{2}\arctan\biggl[\frac{1}{2} - \frac{\theta _{4}(q)^2}{2\,\theta_{4}(q^5)^2}\biggr]\biggr\}^{1/5} \tan\biggl\{\frac{1}{2}\arccot\biggl[\frac{1}{2} - \frac{\theta_{4}(q)^2}{2\,\theta_{4}(q^5)^2}\biggr]\biggr\}^{2/5} </math>

: <math>R(q^2) = \tan\biggl\{\frac{1}{2}\arctan\biggl[\frac{1}{2} - \frac{\theta_{4}(q)^2}{2\,\theta_{4}(q^5)^2}\biggr]\biggr\}^{2/5} \cot\biggl\{\frac{1}{2}\arccot\biggl[\frac{1}{2} - \frac{\theta_{4}(q)^2}{2\,\theta_{4}(q^5)^2}\biggr]\biggr\}^{1/5} </math>

: <math>R(q^2) = \tan\biggl\{\frac{1}{2}\arctan\biggl[\frac{\theta_{3}(q)^2}{2\,\theta_{3}(q^5)^2} - \frac{1}{2}\biggr]\biggr\}^{2/5} \tan\biggl\{\frac{1}{2}\arccot\biggl[\frac{\theta_{3}(q)^2}{2\,\theta_{3}(q^5)^2} - \frac{1}{2}\biggr]\biggr\}^{1/5} </math>

The alternating Rogers-Ramanujan continued fraction function S(q) has the following two identities:

: <math>S(q) = \frac{R(q^4)}{R(q^2)R(q)} = \tan\biggl\{\frac{1}{2}\arctan\biggl [\frac{\theta_{3}(q)^2}{2\,\theta_{3}(q^5)^2} - \frac{1}{2}\biggr]\biggr\}^{1/5} \cot\biggl\{\frac{1}{2}\arccot\biggl[\frac{\theta_{3}(q)^2}{2\,\theta_{3}(q^5)^2} - \frac{1}{2}\biggr]\biggr\}^{2/5}</math>

The theta function values from the fifth root of the nome can be represented as a rational combination of the continued fractions R and S and the theta function values from the fifth power of the nome and the nome itself. The following four equations are valid for all values q between 0 and 1:

: <math>\frac{\theta_{3}(q^{1/5})}{\theta_{3}(q^5)} - 1 = \frac{1}{S(q)}\bigl[S(q)^2 + R(q^2)\bigr]\bigl[1 + R(q^2)S(q)\bigr] </math>
: <math>1 - \frac{\theta_{4}(q^{1/5})}{\theta_{4}(q^5)} = \frac{1}{R(q)}\bigl[R(q^2) + R(q)^2\bigr]\bigl[1 - R(q^2)R(q)\bigr] </math>
: <math>\theta_{3}(q^{1/5})^2 - \theta_{3}(q)^2 = \bigl[\theta_{3}(q)^2 - \theta_{3}(q^5)^2\bigr]\biggl[1+\frac{1}{R(q^2)S(q)}+R(q^2)S(q)+\frac{1}{R(q^2)^2}+R(q^2)^2+\frac{1}{S(q)}-S(q)\biggr] </math>
: <math>\theta_{4}(q)^2 - \theta_{4}(q^{1/5})^2 = \bigl[\theta_{4}(q^5)^2 - \theta_{4}(q)^2\bigr]\biggl[1-\frac{1}{R(q^2)R(q)}-R(q^2)R(q)+\frac{1}{R(q^2)^2}+R(q^2)^2-\frac{1}{R(q)}+R(q)\biggr] </math>

=== Modulus dependent theorems===

In combination with the elliptic modulus, the following formulas can be displayed:

These are the formulas for the square of the elliptic nome:

:<math>\theta_{4}[q(k)] = \theta_{4}[q(k)^2]\sqrt[8]{1 - k^2}</math>
:<math>\theta_{4}[q(k)^2] = \theta_{3}[q(k)]\sqrt[8]{1 - k^2}</math>
:<math>\theta_{3}[q(k)^2] = \theta_{3}[q(k)]\cos[\tfrac{1}{2}\arcsin(k)]</math>

And this is an efficient formula for the cube of the nome:

:<math> \theta_{4}\biggl\langle q\bigl\{\tan\bigl[\tfrac{1}{2}\arctan(t^3)\bigr]\bigr\}^3 \biggr\rangle = 
\theta_{4}\biggl\langle q\bigl\{\tan\bigl[\tfrac{1}{2}\arctan(t^3)\bigr]\bigr\} \biggr\rangle \,3^{-1/2} \bigl(\sqrt{2\sqrt{t^4 - t^2 + 1} - t^2 + 2} + \sqrt{t^2 + 1}\,\bigr)^{1/2} </math>

For all real values <math> t \in \R </math> the now mentioned formula is valid.

And for this formula two examples shall be given:

First calculation example with the value <math> t = 1 </math> inserted:

:{| class="wikitable"
|
<math> \theta_{4}\biggl\langle q\bigl\{\tan\bigl[\tfrac{1}{2}\arctan(1)\bigr]\bigr\}^3 \biggr\rangle = 
\theta_{4}\biggl\langle q\bigl\{\tan\bigl[\tfrac{1}{2}\arctan(1)\bigr]\bigr\} \biggr\rangle \,3^{-1/2} \bigl(\sqrt{3} + \sqrt{2}\,\bigr)^{1/2} </math>
|-
|
<math> \theta_{4}\bigl[\exp(-3\sqrt{2}\,\pi)\bigr] = 
\theta_{4}\bigl[\exp(-\sqrt{2}\,\pi)\bigr] \,3^{-1/2} \bigl(\sqrt{3} + \sqrt{2}\,\bigr)^{1/2} </math>
|}

Second calculation example with the value <math> t = \Phi^{-2} </math> inserted:

:{| class="wikitable"
|
<math> \theta_{4}\biggl\langle q\bigl\{\tan\bigl[\tfrac{1}{2}\arctan(\Phi^{-6})\bigr]\bigr\}^3 \biggr\rangle = 
\theta_{4}\biggl\langle q\bigl\{\tan\bigl[\tfrac{1}{2}\arctan(\Phi^{-6})\bigr]\bigr\} \biggr\rangle \,3^{-1/2} \bigl(\sqrt{2\sqrt{\Phi^{-8} - \Phi^{-4} + 1} - \Phi^{-4} + 2} + \sqrt{\Phi^{-4} + 1}\,\bigr)^{1/2} </math>
|-
|
<math> \theta_{4}\bigl[\exp(-3\sqrt{10}\,\pi)\bigr] = 
\theta_{4}\bigl[\exp(-\sqrt{10}\,\pi)\bigr] \,3^{-1/2} \bigl(\sqrt{2\sqrt{\Phi^{-8} - \Phi^{-4} + 1} - \Phi^{-4} + 2} + \sqrt{\Phi^{-4} + 1}\,\bigr)^{1/2} </math>
|}

The constant <math> \Phi </math> represents the [[golden ratio]] number <math> \Phi = \tfrac{1}{2}(\sqrt{5} + 1)</math> exactly.

==Some series identities==

===Sums with theta function in the result===

The infinite sum<ref>Landau (1899)<!-- Vermutl: {{Citation | author = Landau, E. | title = Sur la Série des Invers de Nombres de Fibonacci | journal = Bull. Soc. Math. France | volume = 27 | year = 1899 | pages = 298–300}}
--> zitiert nach [[Fibonacci-Folge#Borwein|Borwein]], Page 94, Exercise 3.</ref><ref>{{cite web|access-date=2021-07-18|title=Number-theoretical, combinatorial and integer functions – mpmath 1.1.0 documentation|url=https://mpmath.org/doc/1.1.0/functions/numtheory.html}}<!-- auto-translated by Module:CS1 translator --></ref> of the reciprocals of [[Fibonacci sequence|Fibonacci numbers]] with odd indices has the identity:

:<math>\sum_{n=1}^\infty \frac{1}{F_{2n-1}} = \frac{\sqrt{5}}{2}\,\sum_{n=1}^\infty \frac{2(\Phi^{-2})^{n - 1/2}}{1 + (\Phi^{-2})^{2n - 1}} = \frac{\sqrt{5}}{4} \sum_{a=-\infty}^\infty \frac{2(\Phi^{-2})^{a - 1/2}}{1 + (\Phi^{-2})^{2a - 1}} =</math>
:<math>= \frac{\sqrt{5}}{4}\,\theta_{2}(\Phi^{-2})^2 = \frac{\sqrt{5}}{8}\bigl[\theta_{3}(\Phi^{-1})^2 - \theta_{4}(\Phi^{-1})^2\bigr]</math>

By not using the theta function expression, following identity between two sums can be formulated:

:<math>\sum_{n=1}^\infty \frac{1}{F_{2n-1}} = \frac{\sqrt{5}}{4}\,\biggl[ \sum_{n=1}^\infty 2 \,\Phi^{- (2n - 1)^2 /2} \biggr]^2 </math>

:<math>\sum_{n=1}^\infty \frac{1}{F_{2n-1}} = 1.82451515740692456814215840626732817332\ldots </math>

Also in this case <math> \Phi = \tfrac{1}{2}(\sqrt{5} + 1)</math> is [[Golden ratio]] number again.

Infinite sum of the reciprocals of the Fibonacci number squares:
:<math>\sum_{n=1}^\infty \frac{1}{F_{n}^2} = \frac{5}{24}\bigl[2\,\theta_{2}(\Phi^{-2})^4 - \theta_{3}(\Phi^{-2})^4 + 1\bigr] = \frac{5}{24}\bigl[\theta_{3}(\Phi^{-2})^4 - 2\,\theta_{4}(\Phi^{-2})^4 + 1\bigr]</math>

Infinite sum of the reciprocals of the [[Pell sequence|Pell numbers]] with odd indices:
:<math>\sum_{n=1}^\infty \frac{1}{P_{2n-1}} = \frac{1}{\sqrt{2}}\,\theta_{2}\bigl[(\sqrt{2}-1)^2\bigr]^2 = \frac{1}{2\sqrt{2}}\bigl[\theta_{3}(\sqrt{2}-1)^2 - \theta_{4}(\sqrt{2}-1)^2\bigr]</math>

===Sums with theta function in the summand===
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}
\theta_4^2(q)&=iq^{\frac14}\sum_{k=-\infty}^\infty q^{2k^2-k}\theta_1\left(\frac{2k-1}{2i}\ln q,q\right),\\[6pt]
\theta_4^2(q)&=\sum_{k=-\infty}^\infty q^{2k^2}\theta_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>\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)} \theta_1 \left(\frac{i\pi}{2}(2k-1),e^{-\pi}\right)</math>

:<math>\sqrt{\frac{\pi}{2}}\cdot\frac{1}{\Gamma^2\left(\frac34\right)} =\sum_{k=-\infty}^\infty\frac{\theta_4\left(ik\pi,e^{-\pi}\right)}{e^{2\pi k^2}}</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_{00}(z;\tau) &= 0 \quad &\Longleftrightarrow&& \quad z &= m + n \tau + \frac{1}{2} + \frac{\tau}{2}
\\[3pt]
\vartheta_{11}(z;\tau) &= 0 \quad &\Longleftrightarrow&& \quad z &= m + n \tau
\\[3pt]
\vartheta_{10}(z;\tau) &= 0 \quad &\Longleftrightarrow&& \quad z &= m + n \tau + \frac{1}{2}
\\[3pt]
\vartheta_{01}(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)=\left(-i\tau\right)^\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\bigl(\vartheta(0;it)-1\bigr)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 | hdl=2437/166217 | hdl-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)} \theta_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>''πiτ''</sup>}}. Then,

:<math>\begin{align}
\theta_2(q) = \vartheta_{10}(0;\tau) &= \frac{2\eta^2(2\tau)}{\eta(\tau)},\\[3pt]
\theta_3(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(q) = \vartheta_{01}(0;\tau) &= \frac{\eta^2\left(\frac{1}{2}\tau\right)}{\eta(\tau)},
\end{align}</math>

and,

:<math>\theta_2(q)\,\theta_3(q)\,\theta_4(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>

== Derivatives of theta functions ==

These are two identical definitions of the complete elliptic integral of the second kind:

:<math>E(k) = \int_{0}^{\pi/2} \sqrt{1 - k^2\sin(\varphi)^2} d\varphi</math>

:<math>E(k) = \frac{\pi}{2}\sum_{a = 0}^{\infty} \frac{[(2a)!]^2}{(1 - 2a)16^{a}(a!)^4} k^{2a}</math>

The derivatives of the Theta Nullwert functions have these MacLaurin series:

:<math>\theta_{2}'(x) = \frac{\mathrm{d}}{\mathrm{d}x}\,\theta_{2}(x) = \frac{1}{2} x^{-3/4}+\sum_{n = 1}^{\infty} \frac{1}{2}(2n + 1)^2 x^{(2n-1)(2n+3)/4}</math>

:<math>\theta_{3}'(x) = \frac{\mathrm{d}}{\mathrm{d}x}\,\theta_{3}(x) = 2+\sum_{n = 1}^{\infty} 2(n + 1)^2 x^{n(n+2)}</math>

:<math>\theta_{4}'(x) = \frac{\mathrm{d}}{\mathrm{d}x}\,\theta_{4}(x) = -2+\sum_{n = 1}^{\infty} 2(n + 1)^2 (-1)^{n+1} x^{n(n+2)}</math>

The derivatives of theta zero-value functions<ref>{{MathWorld|EllipticAlphaFunction|Elliptic Alpha Function}}</ref> are as follows:
:<math>\theta_{2}'(x) = \frac{\mathrm{d}}{\mathrm{d}x} \,\theta_{2}(x) = \frac{1}{2\pi x} \theta_{2}(x)\theta_{3}(x)^2 E\biggl[\frac{\theta_{2}(x)^2}{\theta_{3}(x)^2}\biggr]</math>
:<math>\theta_{3}'(x) = \frac{\mathrm{d}}{\mathrm{d}x} \,\theta_{3}(x) = \theta_{3}(x)\bigl[\theta_{3}(x)^2 + \theta_{4}(x)^2\bigr]\biggl\{\frac{1}{2\pi x}E\biggl[\frac{\theta_{3}(x)^2 - \theta_{4}(x)^2}{\theta_{3}(x)^2 + \theta_{4}(x)^2}\biggr] - \frac{\theta_{4}(x)^2}{4\,x}\biggr\}</math>
:<math>\theta_{4}'(x) = \frac{\mathrm{d}}{\mathrm{d}x} \,\theta_{4}(x) = \theta_{4}(x)\bigl[\theta_{3}(x)^2 + \theta_{4}(x)^2\bigr]\biggl\{\frac{1}{2\pi x}E\biggl[\frac{\theta_{3}(x)^2 - \theta _{4}(x)^2}{\theta_{3}(x)^2+\theta_{4}(x)^2}\biggr] - \frac{\theta _{3}(x)^ 2}{4\,x}\biggr\}</math>

The two last mentioned formulas are valid for all real numbers of the real definition interval: <math> -1 < x < 1 \,\cap \,x \in \R </math>

And these two last named theta derivative functions are related to each other in this way:
:<math>\vartheta _{4}(x)\biggl[\frac{\mathrm{d}}{\mathrm{d}x} \,\vartheta _{3}(x)\biggr] - \vartheta _{3}(x)\biggl[\frac{\mathrm{d}}{\mathrm{d}x} \,\theta _{4}(x)\biggr] = \frac{1}{4\,x}\,\theta_{3}(x)\,\theta_{4}(x)\bigl[\theta_{3}(x)^4 - \theta_{4}(x)^4\bigr] </math>

The derivatives of the quotients from two of the three theta functions mentioned here always have a rational relationship to those three functions:
:<math>\frac{\mathrm{d}}{\mathrm{d}x} \,\frac{\theta _{2}(x)}{\theta _{3}(x)} = \frac{\theta_{2}(x)\,\theta _{4}(x)^4}{4\,x\,\theta _{3}(x)}</math>
:<math>\frac{\mathrm{d}}{\mathrm{d}x} \,\frac{\theta _{2}(x)}{\theta _{4}(x)} = \frac{\theta_{2}(x)\,\theta _{3}(x)^4}{4\,x\,\theta _{4}(x)}</math>
:<math>\frac{\mathrm{d}}{\mathrm{d}x} \,\frac{\theta _{3}(x)}{\theta _{4}(x)} = \frac{\theta_{3}(x)^5 - \theta _{3}(x)\,\theta _{4}(x)^4}{4\,x\,\theta _{4}(x)}</math>

For the derivation of these derivation formulas see the articles [[Nome (mathematics)]] and [[Modular lambda function]]!

== Integrals of theta functions ==

For the theta functions these integrals<ref>{{Cite web |url=https://math.stackexchange.com/q/1865615 |title=integration - Curious integrals for Jacobi Theta Functions $\int_0^1 \vartheta_n(0,q)dq$ |language=en |date=2022-08-13}}</ref> are valid:

:<math>\int_{0}^{1} \theta_{2}(x) \,\mathrm{d}x = \sum _{k = -\infty}^{\infty} \frac{4}{ (2k+1)^2+4} = \pi\tanh(\pi) \approx 3.129881</math>
:<math>\int_{0}^{1} \theta_{3}(x) \,\mathrm{d}x = \sum _{k = -\infty}^{\infty} \frac{1}{ k^2+1} = \pi\coth(\pi) \approx 3.153348</math>
:<math>\int_{0}^{1} \theta_{4}(x) \,\mathrm{d}x = \sum _{k = -\infty}^{\infty} \frac{(-1 )^{k}}{k^2+1} = \pi\,\operatorname{csch}(\pi) \approx 0.272029</math>

The final results now shown are based on the general Cauchy sum formulas.

==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>\mathbb{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 {{mvar|χ}} a primitive [[Dirichlet character]] modulo {{mvar|q}} and {{math|''ν'' {{=}} {{sfrac|1 − ''χ''(−1)|2}}}} 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|{{sfrac|1|2}} + ''ν''}} modular form of level {{math|4''q''<sup>2</sup>}} and character
:<math>\chi(d) \left(\frac{-1}{d}\right)^\nu,</math>
which means<ref>Shimura, On modular forms of half integral weight</ref>

:<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>

===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>

be 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|Sp(2''n'',<math>\mathbb{Z}</math>)}}; for {{math|''n'' {{=}} 1}}, {{math|Sp(2,<math>\mathbb{Z}</math>) {{=}} SL(2,<math>\mathbb{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|''τ'' ∈ <math>\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'' ∈ <math>\mathbb{C}^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|''τ'' ∈ <math>\mathbb{H}</math>}} where {{math|<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 surface]]s, as well as other auxiliary objects that figure prominently in their function theory, by taking {{mvar|τ}} 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>\mathbb{C}^n \times \mathbb{H}_n</math>.

The functional equation is

:<math>\theta (z+a+\tau b, \tau) = \exp\left( 2\pi i \left(-b^\mathsf{T}z-\tfrac12 b^\mathsf{T}\tau b\right)\right) \theta (z,\tau)</math>

which holds for all vectors {{math|''a'', ''b'' ∈ <math>\mathbb{Z}^n</math>}}, and for all {{math|''z'' ∈ <math>\mathbb{C}^n</math>}} and {{math|''τ'' ∈ <math>\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.

== Derivation of the theta values ==

=== Identity of the Euler beta function ===
In the following, three important theta function values are to be derived as examples:

This is how the [[Euler beta function]] is defined in its reduced form:

:<math>\beta(x) = \frac{\Gamma(x)^2}{\Gamma(2x)}</math>

In general, for all natural numbers <math>n \isin \mathbb{N}</math> this formula of the Euler beta function is valid:

:<math>\frac{4^{-1/(n + 2)}}{n + 2}\csc\bigl(\frac{\pi}{n + 2}\bigr)\beta\biggl[\frac{n}{2(n + 2)}\biggr] = \int_{0}^{\infty} \frac{1}{\sqrt{x^{n+2} + 1}} \,\mathrm {d}x</math>

=== Exemplary elliptic integrals ===

In the following some ''Elliptic Integral Singular Values''<ref>{{cite web|access-date=2023-04-07|title=Elliptic Integral Singular Value
|url=https://archive.lib.msu.edu/crcmath/math/math/e/e103.htm|website=msu.edu}}<!-- auto-translated by Module:CS1 translator --></ref> are derived:

{| class="wikitable"
|
The ensuing function has the following lemniscatically elliptic antiderivative:

:<math>\frac{1}{\sqrt{x^4 + 1}} = \frac{\mathrm{d}}{\mathrm{d}x}\,\frac{1}{2} F\biggl[2\arctan(x); \frac{1}{2}\sqrt{2}\,\biggr]</math>

For the value <math>n = 2</math> this identity appears:

:<math>\frac{1}{4\sqrt{2}}\csc\bigl(\frac{\pi}{4}\bigr)\beta\bigl(\frac{1}{4}\bigr) = \int_{0}^{\infty} \frac{1}{\sqrt{x^4 + 1}} \,\mathrm{d}x = \biggl\{{\color{blue}\frac{1}{2} F\biggl[2\arctan(x); \frac{1}{2}\sqrt{2}\,\biggr]}\biggr\}_{x = 0}^{x = \infty} =</math>

:<math>= \frac{1}{2} F\bigl(\pi; \frac{1}{2}\sqrt{2}\bigr) = K\bigl(\frac{1}{2}\sqrt{2}\bigr) </math>

This result follows from that equation chain:

:<math>{\color{ForestGreen}K\bigl(\frac{1}{2}\sqrt{2}\bigr) = \frac{1}{4}\beta\bigl(\frac{1}{ 4}\bigr)}</math>
|}

{| class="wikitable"
|
The following function has the following equianharmonic elliptic antiderivative:

:<math>\frac{1}{\sqrt{x^6 + 1}} = \frac{\mathrm{d}}{\mathrm{d}x}\,\frac{1}{6}\sqrt [4]{27}F\biggl[2\arctan\biggl(\frac{\sqrt[4]{3}\,x}{\sqrt{x^2 + 1}}\biggr);\frac{1 }{4}(\sqrt{6} + \sqrt{2})\biggr]</math>

For the value <math>n = 4</math> that identity appears:

:<math>\frac{1}{6\sqrt[3]{2}}\csc\bigl(\frac{\pi}{6}\bigr)\beta\bigl(\frac{1}{3} \bigr) = \int_{0}^{\infty} \frac{1}{\sqrt{x^6 + 1}} \,\mathrm{d}x = \biggl\{{\color{blue}\frac{1}{6}\sqrt[4]{27}F\biggl[2\arctan\biggl(\frac{\sqrt[4]{3}\,x}{\sqrt{x^2 + 1}}\biggr);\frac{1}{4}(\sqrt{6} + \sqrt{2})\biggr]}\biggr\}_{x = 0}^{x = \infty} =</math>

:<math>= \frac{1}{6}\sqrt[4]{27} F\bigl[2\arctan(\sqrt[4]{3});\frac{1}{4}(\sqrt {6} + \sqrt{2})\bigr] = \frac{2}{9}\sqrt[4]{27} K\bigl[\frac{1}{4}(\sqrt{6} + \sqrt{2})\bigr] = \frac{2}{3}\sqrt[4]{3} K\bigl[\frac{1}{4}(\sqrt{6} - \sqrt{2}) \bigr] </math>

This result follows from that equation chain:

:<math>{\color{ForestGreen}K\bigl[\frac{1}{4}(\sqrt{6} - \sqrt{2})\bigr] = \frac{1}{2\sqrt[3 ]{2}\sqrt[4]{3}}\beta\bigl(\frac{1}{3}\bigr)}</math>
|}

{| class="wikitable"
|
And the following function has the following elliptic antiderivative:
:<math>\frac{1}{\sqrt{x^8 + 1}} =</math>

:<math>= \frac{\mathrm{d}}{\mathrm{d}x}\,\frac{1}{4}\sec \bigl(\frac{\pi}{8}\bigr)F\biggl\{2\arctan\biggl[\frac{2\cos(\pi/8)\,x}{\sqrt{x^4 + \sqrt{2}\,x^2 + 1} - x^2 + 1}\biggr];2\sqrt[4]{2}\sin\bigl(\frac{\pi}{8}\bigr) \biggr\} + \frac{1}{4}\sec\bigl(\frac{\pi}{8}\bigr)F\biggl\{\arcsin\biggl[\frac{2\cos(\pi/ 8)\,x}{x^2 + 1}\biggr];\tan\bigl(\frac{\pi}{8}\bigr)\biggr\}</math>

For the value <math>n = 6</math> the following identity appears:

:<math>\frac{1}{8\sqrt[4]{2}}\csc\bigl(\frac{\pi}{8}\bigr)\beta\bigl(\frac{3}{8} \bigr) = \int_{0}^{\infty} \frac{1}{\sqrt{x^8 + 1}} \,\mathrm{d}x =</math>

:<math>= \biggl\langle{\color{blue}\frac{1}{4}\sec\bigl(\frac{\pi}{8}\bigr)F\biggl\{2\arctan\biggl[\frac{2\cos(\pi/8)\,x}{\sqrt{x^4 + \sqrt{2}\,x^2 + 1} - x^2 + 1}\biggr];2\sqrt[4]{ 2}\sin\bigl(\frac{\pi}{8}\bigr)\biggr\} + \frac{1}{4}\sec\bigl(\frac{\pi}{8}\bigr)F \biggl\{\arcsin\biggl[\frac{2\cos(\pi/8)\,x}{x^2 + 1}\biggr];\tan\bigl(\frac{\pi}{8} \bigr)\biggr\}}\biggr\rangle_{x = 0}^{x = \infty} =</math>

:<math>= \frac{1}{4}\sec\bigl(\frac{\pi}{8}\bigr)F\bigl[\pi;2\sqrt[4]{2}\sin\bigl (\frac{\pi}{8}\bigr)\bigr] = \frac{1}{2}\sec\bigl(\frac{\pi}{8}\bigr)K(\sqrt{2\sqrt {2} - 2}\bigr) = 2\sin\bigl(\frac{\pi}{8}\bigr)K(\sqrt{2} - 1)</math>

This result follows from that equation chain:
:<math>{\color{ForestGreen}K(\sqrt{2} - 1) = \frac{1}{8}\sqrt[4]{2}\,(\sqrt{2} + 1)\, \beta\bigl(\frac{3}{8}\bigr)}</math>
|}

=== Combination of the integral identities with the nome ===

The elliptic nome function has these important values:
:<math>q(\tfrac{1}{2}\sqrt{2}) = \exp(-\pi)</math>

:<math>q[\tfrac{1}{4}(\sqrt{6} - \sqrt{2})] = \exp(-\sqrt{3}\,\pi)</math>

:<math>q(\sqrt{2} - 1) = \exp(-\sqrt{2}\,\pi)</math>

For the proof of the correctness of these nome values, see the article [[Nome (mathematics)]]!

On the basis of these integral identities and the above-mentioned '''Definition and identities to the theta functions''' in the same section of this article, exemplary theta zero values shall be determined now:
:{| class="wikitable"
|<math>\theta_{3}[q(k)] = \sqrt{2\pi^{-1} K(k)}</math>

|}
:<math>\theta_{3}[\exp(-\pi)] = \theta_{3}[q(\tfrac{1}{2}\sqrt{2})] = \sqrt{2\pi^ {-1}K(\tfrac{1}{2}\sqrt{2})} = 2^{-1/2}\pi^{-1/2}\beta(\tfrac{1}{4} )^{1/2} = 2^{-1/4}\sqrt[4]{\pi}\,{\Gamma\bigl(\tfrac{3}{4}\bigr)}^{-1} </math>

:<math>\theta _{3}[\exp(-\sqrt{3}\,\pi )] = \theta _{3}\bigl\{q\bigl[\tfrac{1}{4}(\sqrt {6} - \sqrt{2})\bigr]\bigr\} = \sqrt{2\pi^{-1}K\bigl[\tfrac{1}{4}(\sqrt{6} - \sqrt {2})\bigr]} = 2^{-1/6}3^{-1/8}\pi^{-1/2}\beta(\tfrac{1}{3})^{1/ 2}</math>

:<math>\theta _{3}[\exp(-\sqrt{2}\,\pi )] = \theta _{3}[q(\sqrt{2} - 1)] = \sqrt{2\pi ^{-1}K(\sqrt{2} - 1)} = 2^{-1/8}\cos(\tfrac{1}{8}\pi)\,\pi^{-1/2} \beta(\tfrac{3}{8})^{1/2}</math>

:{| class="wikitable"
|<math>\theta_{4}[q(k)] = \sqrt[4]{1 - k^2}\,\sqrt{2\pi^{-1} K(k)}</math>
|}

:<math>\theta_{4}[\exp(-\sqrt{2}\,\pi)] = \theta_{4}[q(\sqrt{2} - 1)] = \sqrt[4]{ 2\sqrt{2} - 2}\,\sqrt{2\pi^{-1}K(\sqrt{2} - 1)} = 2^{-1/4}\cos(\tfrac{1} {8}\pi)^{1/2}\,\pi^{-1/2}\beta(\tfrac{3}{8})^{1/2}</math>

== Partition sequences and Pochhammer products ==

=== Regular partition number sequence ===

The regular partition sequence <math>P(n)</math> itself indicates the number of ways in which a positive [[Integer|integer number]] <math>n</math> can be split into positive integer summands. For the numbers <math>n = 1</math> to <math>n = 5</math>, the associated partition numbers <math>P</math> with all associated number partitions are listed in the following table:

{| class="wikitable"
|+Example values of P(n) and associated number partitions
!n
!P(n)
!paying partitions
|-
|0
|1
|() empty partition/[[empty sum]]
|-
|1
|1
|(1)
|-
|2
|2
|(1+1), (2)
|-
|3
|3
|(1+1+1), (1+2), (3)
|-
|4
|5
|(1+1+1+1), (1+1+2), (2+2), (1+3), (4)
|-
|5
|7
|(1+1+1+1+1), (1+1+1+2), (1+2+2), (1+1+3), (2+3), (1+4), (5)
|}

The generating function of the regular partition number sequence can be represented via Pochhammer product in the following way:
: <math>\sum _{k = 0}^\infty P(k)x^k = \frac{1}{(x;x)_{\infty}} = \theta_{3}(x)^{-1/6}\theta_{4}(x)^{-2/3} \biggl[\frac{\theta_{3}(x)^4 - \theta_{4}(x)^4}{16\,x}\biggr]^{-1/24}</math>

The summandization of the now mentioned [[Pochhammer symbol|Pochhammer product]] is described by the [[Pentagonal number theorem]] in this way:

:<math>(x;x)_{\infty} = 1 + \sum_{n = 1}^{\infty} \bigl[- x^{\text{Fn}(2n-1)} - x^ {\text{Kr}(2n-1)} + x^{\text{Fn}(2n)} + x^{\text{Kr}(2n)}\bigr]</math>

The following basic definitions apply to the [[pentagonal number]]s and the card house numbers:

: <math>\text{Fn}(z) = \tfrac{1}{2}z(3z-1)</math>
: <math>\text{Kr}(z) = \tfrac{1}{2}z(3z+1)</math>

As a further application<ref>[https://www.researchgate.net/publication/235432739_Ramanujan%27s_theta-function_identities_involving_Lambert_series Ramanujan's theta-function identities involving Lambert series]</ref> one obtains a formula for the third power of the Euler product:
:<math>(x;x)^3 = \prod_{n=1}^\infty (1-x^n)^3 = \sum _{m=0}^\infty (-1)^m(2m +1)x^{m(m+1)/2}</math>

=== Strict partition number sequence ===

And the strict partition sequence <math>Q(n)</math> indicates the number of ways in which such a positive integer number <math>n</math> can be splitted into positive integer summands such that each summand appears at most once<ref>{{cite web|accessdate=2022-03-09|title=code golf - Strict partitions of a positive integer|url=https://codegolf.stackexchange.com/questions/71941/strict-partitions-of-a-positive-integer}}</ref> and no summand value occurs repeatedly. Exactly the same sequence<ref>{{cite web|date=2022-03-09|title=A000009 - OEIS|url=https://oeis.org/A000009}}</ref> is also generated if in the partition only odd summands are included, but these odd summands may occur more than once. Both representations for the strict partition number sequence are compared in the following table:

{| class="wikitable"
|+Example values of Q(n) and associated number partitions
!n
!Q(n)
!Number partitions without repeated summands
!Number partitions with only odd addends
|-
|0
|1
|() empty partition/[[empty sum]]
|() empty partition/[[empty sum]]
|-
|1
|1
|(1)
|(1)
|-
|2
|1
|(2)
|(1+1)
|-
|3
|2
|(1+2), (3)
|(1+1+1), (3)
|-
|4
|2
|(1+3), (4)
|(1+1+1+1), (1+3)
|-
|5
|3
|(2+3), (1+4), (5)
|(1+1+1+1+1), (1+1+3), (5)
|-
|6
|4
|(1+2+3), (2+4), (1+5), (6)
|(1+1+1+1+1+1), (1+1+1+3), (3+3), (1+5)
|-
|7
|5
|(1+2+4), (3+4), (2+5), (1+6), (7)
|(1+1+1+1+1+1+1), (1+1+1+1+3), (1+3+3), (1+1+5), (7)
|-
|8
|6
|(1+3+4), (1+2+5), (3+5), (2+6), (1+7), (8)
|(1+1+1+1+1+1+1+1), (1+1+1+1+1+3), (1+1+3+3), (1+1+1+ 5), (3+5), (1+7)
|}

The generating function of the strict partition number sequence can be represented using Pochhammer's product:
: <math>\sum _{k = 0}^\infty Q(k)x^k = \frac{1}{(x;x^2)_{\infty }} = \theta_{3}(x)^{1/6}\theta_{4}(x)^{-1/3} \biggl[\frac{\theta_{3}(x)^4 - \theta_{4}(x)^4}{16\,x}\biggr]^{1/24}</math>

=== Overpartition number sequence ===

The [[Maclaurin series]] for the reciprocal of the function {{math|''{{not a typo|ϑ}}<sub>01</sub>''}} has the numbers of [[Partition function (number theory)|over partition sequence]] as coefficients with a positive sign:<ref>{{cite journal | last=Mahlburg | first=Karl | title=The overpartition function modulo small powers of 2 | journal=Discrete Mathematics | volume=286 | issue=3 | date=2004 | doi=10.1016/j.disc.2004.03.014 | pages=263–267}}</ref>

: <math>\frac{1}{\theta_{4}(x)} = \prod_{n=1}^{\infty} \frac{1 + x^{n}}{1 - x^{n }} = \sum_{k=0}^{\infty} \overline{P}(k)x^{k}</math>
: <math>\frac{1}{\theta_{4}(x)} = 1+2x+4x^2+8x^3+14x^4+24x^5+40x^6+64x^7+100x^ 8+154x^9+232x^{10} + \dots</math>

If, for a given number <math>k</math>, all partitions are set up in such a way that the summand size never increases, and all those summands that do not have a summand of the same size to the left of themselves can be marked for each partition of this type, then it will be the resulting number<ref>{{cite journal|language=en|title=Elsevier Enhanced Reader|journal=Discrete Mathematics |date=28 April 2009 |volume=309 |issue=8 |pages=2528–2532 |doi=10.1016/j.disc.2008.05.007 |last1=Kim |first1=Byungchan |doi-access=free }}<!-- auto-translated by Module:CS1 translator --></ref> of the marked partitions depending on <math>k</math> by the overpartition function <math>\overline{P}(k)</math> .

First example:

: <math>\overline{P}(4) = 14</math>

These 14 possibilities of partition markings exist for the sum 4:

{| class="wikitable"
|(4), ('''4'''), (3+1), ('''3'''+1), (3+'''1'''), ('''3'''+'''1'''), (2+2), ('''2'''+2), (2+1+1), ('''2'''+1+1), (2+'''1'''+1), ('''2'''+'''1'''+1), (1+1+1+1), ('''1'''+1+1+1)
|}

Second example:

: <math>\overline{P}(5) = 24</math>

These 24 possibilities of partition markings exist for the sum 5:

{| class="wikitable"
|(5), ('''5'''), (4+1), ('''4'''+1), (4+'''1'''), ('''4'''+'''1'''), (3+2), ('''3'''+2), (3+'''2'''), ('''3'''+'''2'''), (3+1+1), ('''3'''+1+1), (3+'''1'''+1), ('''3'''+'''1'''+1), (2+2+1), ('''2'''+2+1), (2+2+'''1'''), ('''2'''+2+'''1'''),
(2+1+1+1), ('''2'''+1+1+1), (2+'''1'''+1+1), ('''2'''+'''1'''+1+1), (1+1+1+1+1), ('''1'''+1+1+1+1)
|}

=== Relations of the partition number sequences to each other ===
In the Online Encyclopedia of Integer Sequences (OEIS), the sequence of regular partition numbers <math>P(n)</math> is under the code A000041, the sequence of strict partitions is <math> Q(n)</math> under the code A000009 and the sequence of superpartitions <math>\overline{P}(n)</math> under the code A015128. All parent partitions from index <math>n = 1</math> are even.

The sequence of superpartitions <math>\overline{P}(n)</math> can be written with the regular partition sequence P<ref>{{cite web|date=2022-03-11|author=Eric W. Weisstein|language=en|title=Partition Function P|url=https://mathworld.wolfram.com/PartitionFunctionP.html}}</ref> and the strict partition sequence Q<ref>{{cite web|date=2022-03-11|author=Eric W. Weisstein|language=en|title=Partition Function Q|url=https://mathworld.wolfram.com/PartitionFunctionQ.html}}</ref> can be generated like this:
: <math>\overline{P}(n) = \sum_{k=0}^{n} P(n - k)Q(k)</math>
In the following table of sequences of numbers, this formula should be used as an example:
{| class="wikitable"
!n
!P(n)
!Q(n)
!<math>\overline{P}(n)</math>
|-
|0
|1
|1
|1 = 1*1
|-
|1
|1
|1
|2 = 1 * 1 + 1 * 1
|-
|2
|2
|1
|4 = 2 * 1 + 1 * 1 + 1 * 1
|-
|3
|3
|2
|8 = 3 * 1 + 2 * 1 + 1 * 1 + 1 * 2
|-
|4
|5
|2
|14 = 5 * 1 + 3 * 1 + 2 * 1 + 1 * 2 + 1 * 2
|-
|5
|7
|3
|24 = 7 * 1 + 5 * 1 + 3 * 1 + 2 * 2 + 1 * 2 + 1 * 3
|}
Related to this property, the following combination of two series of sums can also be set up via the function {{math|{{not a typo|ϑ}}<sub>01</sub>}}:

:<math>\theta_{4}(x) = \biggl[\sum_{k = 0}^{\infty} P(k) x^k \biggr]^{-1} \biggl[\sum_{k = 0}^{\infty} Q(k) x^k \biggr]^{-1}</math>

==Notes==
{{reflist|group=note}}

==References==
{{reflist}}

*{{Cite book |first1=Milton |last1=Abramowitz |author-link=Milton Abramowitz |first2=Irene A. |last2=Stegun |author-link2=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 |author-link=Naum Akhiezer |title=Elements of the Theory of Elliptic Functions |orig-year=1970 |series=[[American Mathematical Society|AMS]] Translations of Mathematical Monographs |volume=79 |year=1990 |publisher=[[American Mathematical Society|AMS]] |location=Providence, RI |isbn=978-0-8218-4532-5 }}
*{{Cite book|first1=Hershel M. |last1=Farkas |author-link=Hershel M. Farkas |first2=Irwin |last2=Kra |author-link2=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|first1=G. H. |last1=Hardy |author-link=G. H. Hardy |first2=E. M. |last2=Wright |author-link2=E. M. Wright |title=An Introduction to the Theory of Numbers |edition=4th |year=1959 |publisher=[[Clarendon Press]] |location=Oxford }}
*{{Cite book|first=David |last=Mumford |author-link=David Mumford |title=Tata Lectures on Theta I |year=1983 |publisher=[[Birkhauser]] |location=Boston |isbn=978-3-7643-3109-2 }}
*{{Cite book|author-link=James Pierpont (mathematician) |first=James |last=Pierpont |title=Functions of a Complex Variable |location=New York |publisher=[[Dover Publications]] |year=1959 }}
*{{Cite book|first1=Harry E. |last1=Rauch |author-link=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|author-link=E. T. Whittaker |first1=E. T. |last1=Whittaker |first2=G. N. |last2=Watson |author-link2=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 | author-link= 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 }}
* {{cite journal |last1=Ackerman |first1=Michael |title=On the generating functions of certain Eisenstein series |journal=Mathematische Annalen |date=1 February 1979 |volume=244 |issue=1 |pages=75–81 |doi=10.1007/BF01420339 |s2cid=120045753 }}
Harry Rauch with Hershel M. Farkas: Theta functions with applications to Riemann Surfaces, Williams and Wilkins, Baltimore MD 1974, {{ISBN|0-683-07196-3}}.
* Charles Hermite: Sur la résolution de l'Équation du cinquiéme degré Comptes rendus, C. R. Acad. Sci. Paris, Nr. 11, March 1858.

==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]]