Theta function

 ( Elliptic Functions ) 

[[File:Cplot of Jacobi theta 1.svg|400px|thumb|upright=1.2|Jacobi's theta function with nome : ]]

In mathematics, theta functions are of several complex variables. Fundamentally, they are a family of continuous functions which encode the behavior of discrete multi-dimensional periodic systems, such as Bravais lattice or points on a torus. Because they are smooth, they allow the study and manipulation of discrete combinatorial systems using the tools of Complex analysis.

For this reason, theta functions have useful applications in topics such as:

• Number theory ("In how many ways can a number be written as a sum of squares?")
• Physics ("How does heat flow on a toroidal ring?", "How do quantum particles behave when arranged in a lattice?")
• Geometry ("What are the shape properties of Elliptic curve?")

and others, including Abelian variety, , , and .

Theta functions in two dimensions are functions of two complex arguments. In one choice of parameter, for example, z encodes position on a two-dimensional lattice, and τ or q encodes the shape of the lattice. In higher dimensions, the shape of the lattice is dictated by a matrix; in general, theta functions are parametrized by points in a tube domain inside a complex Lagrangian Grassmannian, namely the Siegel upper half space.

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Basic example One example of a theta function is:

$\backslash theta(z,q)\; \backslash equiv\; \backslash sum\_\{n=-\backslash infty\}^\backslash infty\; q^\{(n^2)\}\; \backslash exp\{(2\backslash pi\; i\backslash ,nz)\}$

where z and q are complex numbers and |q| < 1 so that the sum converges.

This analytic function can be used to solve a combinatorics problem: in how many different ways can each integer n be written as the sum of two squares?

When z=0, this function becomes $\backslash theta(0,q)\; =\; \backslash sum\_\{n=-\backslash infty\}^\backslash infty\; q^\{(n^2)\}\; =\; 1\; +\; 2q\; +\; 2q^4\; +\; 2q^9\; +\; \backslash ldots\; +\; 2q^\{(n^2)\}\; +\; \backslash ldots$

This is a generating function where the coefficient on $q^k$ represents how many ways there are to write k as a perfect square—when k=0, there is just one way. When k is any other perfect square, there are two ways: $n^2\; =\; (-n)^2$. When k is not a perfect square, there are zero ways.

If you square this generating function, you obtain $\backslash left\backslash theta(0,q)\backslash right^2\; =\; (\backslash sum\_m\; q^\{(m^2)\})(\backslash sum\_n\; q^\{(n^2)\})\; =\; \backslash sum\_\{m,n\}\; q^\{m^2+n^2\}$. If you collect terms by exponent, you find that $\backslash theta(0,q)^2$ is a generating function where the coefficient on $q^k$ counts how many ways there are to write k as the sum of any two squares. This count includes negative integers and order, such that (3,4), (4,3), and (-3,4) each count as separate ways of making 3² + 4² = 25.

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Application to elliptic functions The most common form of theta function is that occurring in the theory of elliptic functions. With respect to one of the complex variables (conventionally called ), 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 of a Complex torus, a condition of 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".

Throughout this article, $(e^\{\backslash pi\; i\backslash tau\})^\{\backslash alpha\}$ should be interpreted as $e^\{\backslash alpha\; \backslash pi\; i\backslash tau\}$ (in order to resolve issues of choice of branch). Note that this is, in general, not equivalent to the usual interpretation $(e^z)^\backslash alpha=e^\{\backslash alpha\; \backslash operatorname\{Log\}\; e^z\}$ when $z$ is outside the strip . Here, $\backslash operatorname\{Log\}$ denotes the principal branch of the complex logarithm.

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Jacobi theta function 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 and , where can be any complex number and 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

$\backslash begin\{align\}$

\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}

where is the nome and . It is a Jacobi form. The restriction ensures that it is an absolutely convergent series. At fixed , this is a Fourier series for a 1-periodic entire function of . Accordingly, the theta function is 1-periodic in :

$\backslash vartheta(z+1;\; \backslash tau)\; =\; \backslash vartheta(z;\; \backslash tau).$

By completing the square, it is also -quasiperiodic in , with

$\backslash vartheta(z+\backslash tau;\backslash tau)\; =\; \backslash exp\backslash bigl(-\backslash pi\; i\; (\backslash tau\; +\; 2\; z)\backslash bigr)\; \backslash vartheta(z;\backslash tau).$

Thus, in general,

$\backslash vartheta(z+a+b\backslash tau;\backslash tau)\; =\; \backslash exp\backslash left(-\backslash pi\; i\; b^2\; \backslash tau\; -2\; \backslash pi\; i\; b\; z\backslash right)\; \backslash vartheta(z;\backslash tau)$

for any integers and .

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

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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:

$\backslash vartheta\_\{00\}(z;\backslash tau)\; =\; \backslash vartheta(z;\backslash tau)$

The auxiliary (or half-period) functions are defined by

$\backslash begin\{align\}$

\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}

This notation follows Bernhard Riemann and David Mumford; Jacobi's original formulation was in terms of the nome rather than . In Jacobi's notation the -functions are written:

$\backslash begin\{align\}$

\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}

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 in the above theta functions, we obtain four functions of 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 only, defined on the unit disk . They are sometimes called :$\backslash theta\_1(q)=0$ for all $q\backslash in\backslash mathbb\{C\}$ with .

$\backslash begin\{align\}$

\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}

with the nome . Observe that $\backslash theta\_1(q)=0$. These can be used to define a variety of modular forms, and to parametrize certain curves; in particular, the Jacobi identity is

$\backslash theta\_2(q)^4\; +\; \backslash theta\_4(q)^4\; =\; \backslash theta\_3(q)^4$

or equivalently,

$\backslash vartheta\_\{01\}(0;\backslash tau)^4\; +\; \backslash vartheta\_\{10\}(0;\backslash tau)^4\; =\; \backslash vartheta\_\{00\}(0;\backslash tau)^4$

which is the Fermat curve of degree four.

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Jacobi identities Jacobi's identities describe how theta functions transform under the modular group, which is generated by and . Equations for the first transform are easily found since adding one to in the exponent has the same effect as adding to (). For the second, let

$\backslash alpha\; =\; (-i\; \backslash tau)^\backslash frac12\; \backslash exp\backslash left(\backslash frac\{\backslash pi\}\{\backslash tau\}\; i\; z^2\; \backslash right).$

Then

$\backslash begin\{align\}$

\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}

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Theta functions in terms of the nome Instead of expressing the Theta functions in terms of and , we may express them in terms of arguments and the nome , where and . In this form, the functions become

$\backslash begin\{align\}$

\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}

We see that the theta functions can also be defined in terms of and , without a direct reference to the exponential function. These formulas can, therefore, be used to define the Theta functions over other fields where the exponential function might not be everywhere defined, such as fields of p-adic number.

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Product representations The Jacobi triple product (a special case of the Macdonald identities) tells us that for complex numbers and with and we have

$\backslash prod\_\{m=1\}^\backslash infty$

\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}.

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 (noting some authors instead set ) and take then

$\backslash vartheta(z;\; \backslash tau)\; =\; \backslash sum\_\{n=-\backslash infty\}^\backslash infty\; \backslash exp(\backslash pi\; i\; \backslash tau\; n^2)\; \backslash exp(2\backslash pi\; i\; z\; n)\; =\; \backslash sum\_\{n=-\backslash infty\}^\backslash infty\; w^\{2n\}q^\{n^2\}.$

We therefore obtain a product formula for the theta function in the form

$\backslash vartheta(z;\; \backslash tau)\; =\; \backslash prod\_\{m=1\}^\backslash infty$

\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).

In terms of and :

$\backslash begin\{align\}$

\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}

where is the -Pochhammer symbol and is the q-theta function. Expanding terms out, the Jacobi triple product can also be written

$\backslash prod\_\{m=1\}^\backslash infty$

\left( 1 - q^{2m}\right) \Big( 1 + \left(w^2+w^{-2}\right)q^{2m-1}+q^{4m-2}\Big),

which we may also write as

$\backslash vartheta(z\backslash mid\; q)\; =\; \backslash prod\_\{m=1\}^\backslash infty$

\left( 1 - q^{2m}\right) \left( 1 + 2 \cos(2 \pi z)q^{2m-1}+q^{4m-2}\right).

This form is valid in general but clearly is of particular interest when is real. Similar product formulas for the auxiliary theta functions are

$\backslash begin\{align\}$

\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} In particular, $$\backslash lim\_\{q\backslash to\; 0\}\backslash frac\{\backslash vartheta\_\{10\}(z\backslash mid\; q)\}\{2\; q^\{\backslash frac14\}\}\; =\; \backslash cos(\backslash pi\; z),\backslash quad\; \backslash lim\_\{q\backslash to\; 0\}\backslash frac\{-\backslash vartheta\_\{11\}(z\backslash mid\; q)\}\{2\; q^\{\backslash frac14\}\}\; =\; \backslash sin(\backslash pi\; z)$$so we may interpret them as one-parameter deformations of the periodic functions $\backslash sin,\; \backslash cos$, again validating the interpretation of the theta function as the most general 2 quasi-period function.

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Integral representations The Jacobi theta functions have the following integral representations:

$\backslash begin\{align\}$

\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}

The Theta Nullwert function $\backslash theta\_\{3\}(q)$ as this integral identity:

$\backslash theta\_\{3\}(q)\; =\; 1\; +\; \backslash frac\{4q\backslash sqrt\{\backslash ln(1/q)\}\}\{\backslash sqrt\{\backslash pi\}\}\; \backslash int\_\{0\}^\{\backslash infty\}\; \backslash frac\{\backslash exp-\backslash ln(1/q)\backslash ,x^2\backslash \{1\; -\; q^2\backslash cos2\backslash ln(1/q)\backslash ,x\backslash \}\}\{1\; -\; 2q^2\backslash cos2\backslash ln(1/q)\backslash ,x\; +\; q^4\}\; \backslash ,\backslash mathrm\{d\}x$

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:

$\backslash biggl\backslash frac\{2\}\{\backslash pi\}K\backslash bigl(\backslash frac\{1\}\{2\}\backslash sqrt\{2\}\backslash bigr)\backslash biggr^\{1/2\}\; =\; \backslash theta\_\{3\}\backslash bigl\backslash exp(-\backslash pi)\backslash bigr\; =\; 1\; +\; 4\backslash exp(-\backslash pi)\; \backslash int\_\{0\}^\{\backslash infty\}\; \backslash frac\{\backslash exp(-\backslash pi\; x^2)1\}\{1\; -\; 2\backslash exp(-2\backslash pi)\backslash cos(2\backslash pi\; x)\; +\; \backslash exp(-4\backslash pi)\}\; \backslash ,\backslash mathrm\{d\}x$

$\backslash biggl\backslash frac\{2\}\{\backslash pi\}K(\backslash sqrt\{2\}^\{1/2\}\; =\; \backslash theta\_\{3\}\backslash bigl\backslash exp(-\backslash sqrt\{2\}\backslash ,\backslash pi)\backslash bigr\; =\; 1\; +\; 4\backslash ,\backslash sqrt4\{2\}\backslash exp(-\backslash sqrt\{2\}\backslash ,\backslash pi)\; \backslash int\_\{0\}^\{\backslash infty\}\; \backslash frac\{\backslash exp(-\backslash sqrt\{2\}\backslash ,\backslash pi\; x^2)1\}\{1\; -\; 2\backslash exp(-2\backslash sqrt\{2\}\backslash ,\backslash pi)\backslash cos(2\backslash sqrt\{2\}\backslash ,\backslash pi\; x)\; +\; \backslash exp(-4\backslash sqrt\{2\}\backslash ,\backslash pi)\}\; \backslash ,\backslash mathrm\{d\}x$

$\backslash biggl\backslash \{\backslash frac\{2\}\{\backslash pi\}K\backslash bigl\backslash sin\backslash bigl(\backslash frac\{\backslash pi\}\{12\}\backslash bigr)\backslash bigr\backslash biggr\backslash \}^\{1/2\}\; =\; \backslash theta\_\{3\}\backslash bigl\backslash exp(-\backslash sqrt\{3\}\backslash ,\backslash pi)\backslash bigr\; =\; 1\; +\; 4\backslash ,\backslash sqrt4\{3\}\backslash exp(-\backslash sqrt\{3\}\backslash ,\backslash pi)\; \backslash int\_\{0\}^\{\backslash infty\}\; \backslash frac\{\backslash exp(-\backslash sqrt\{3\}\backslash ,\backslash pi\; x^2)1\}\{1\; -\; 2\backslash exp(-2\backslash sqrt\{3\}\backslash ,\backslash pi)\backslash cos(2\backslash sqrt\{3\}\backslash ,\backslash pi\; x)\; +\; \backslash exp(-4\backslash sqrt\{3\}\backslash ,\backslash pi)\}\; \backslash ,\backslash mathrm\{d\}x$

Furthermore, the theta examples $\backslash theta\_\{3\}(\backslash tfrac\{1\}\{2\})$ and $\backslash theta\_\{3\}(\backslash tfrac\{1\}\{3\})$ shall be displayed:

$\backslash theta\_\{3\}\backslash left(\backslash frac\{1\}\{2\}\backslash right)\; =\; 1+2\backslash sum\_\{n\; =\; 1\}^\{\backslash infty\}\; \backslash frac\{1\}\{2^\{n^2\}\}\; =\; 1\; +\; 2\backslash pi^\{-1/2\}\backslash sqrt\{\backslash ln(2)\}\; \backslash int\_\{0\}^\{\backslash infty\}\; \backslash frac\{\backslash exp-\backslash ln(2)\backslash ,x^2\backslash \{16\; -\; 4\backslash cos2\backslash ln(2)\backslash ,x\backslash \}\}\{17\; -\; 8\backslash cos2\backslash ln(2)\backslash ,x\}\; \backslash ,\backslash mathrm\{d\}x$

$\backslash theta\_\{3\}\backslash left(\backslash frac\{1\}\{2\}\backslash right)\; =\; 2.128936827211877158669\backslash ldots$

$\backslash theta\_\{3\}\backslash left(\backslash frac\{1\}\{3\}\backslash right)\; =\; 1+2\backslash sum\_\{n\; =\; 1\}^\{\backslash infty\}\; \backslash frac\{1\}\{3^\{n^2\}\}\; =\; 1\; +\; \backslash frac\{4\}\{3\}\backslash pi^\{-1/2\}\backslash sqrt\{\backslash ln(3)\}\; \backslash int\_\{0\}^\{\backslash infty\}\; \backslash frac\{\backslash exp-\backslash ln(3)\backslash ,x^2\backslash \{81\; -\; 9\backslash cos2\backslash ln(3)\backslash ,x\backslash \}\}\{82\; -\; 18\backslash cos2\backslash ln(3)\backslash ,x\}\; \backslash ,\backslash mathrm\{d\}x$

$\backslash theta\_\{3\}\backslash left(\backslash frac\{1\}\{3\}\backslash right)\; =\; 1.691459681681715341348\backslash ldots$

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Explicit values

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[[:Wiktionary:lemniscatic/" itemprop="url" title="Wiki: 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). Define,

$\backslash quad\; \backslash varphi(q)\; =\backslash vartheta\_\{00\}(0;\backslash tau)\; =\backslash theta\_3(0;q)=\backslash sum\_\{n=-\backslash infty\}^\backslash infty\; q^\{n^2\}$

with the nome $q\; =e^\{\backslash pi\; i\; \backslash tau\},$ $\backslash tau\; =\; n\backslash sqrt\{-1\},$ and Dedekind eta function $\backslash eta(\backslash tau).$ Then for $n\; =\; 1,2,3,\backslash dots$

$\backslash begin\{align\}$

\varphi\left(e^{-\pi} \right) &= \frac{\sqrt4{\pi}}{\Gamma\left(\frac34\right)} = \sqrt2\,\eta\left(\sqrt{-1}\right)\\ \varphi\left(e^{-2\pi}\right) &= \frac{\sqrt4{\pi}}{\Gamma\left(\frac34\right)} \frac{\sqrt{2+\sqrt2}}{2}\\ \varphi\left(e^{-3\pi}\right) &= \frac{\sqrt4{\pi}}{\Gamma\left(\frac34\right)} \frac{\sqrt{1+\sqrt3}}{\sqrt8{108}}\\ \varphi\left(e^{-4\pi}\right) &= \frac{\sqrt4{\pi}}{\Gamma\left(\frac34\right)} \frac{2+\sqrt4{8}}{4}\\ \varphi\left(e^{-5\pi}\right) &= \frac{\sqrt4{\pi}}{\Gamma\left(\frac34\right)} \sqrt{\frac{2+\sqrt5}{5}}\\ \varphi\left(e^{-6\pi}\right) &= \frac{\sqrt4{\pi}}{\Gamma\left(\frac34\right)} \frac{\sqrt{\sqrt4{1}+\sqrt4{3}+\sqrt4{4}+\sqrt4{9}}}{\sqrt8{12^3}}\\ \varphi\left(e^{-7\pi}\right) &= \frac{\sqrt4{\pi}}{\Gamma\left(\frac34\right)} \frac{\sqrt{\sqrt{13+\sqrt{7}}+\sqrt{7+3\sqrt{7}}}}{\sqrt8{14^3}\cdot\sqrt16{7}}\\ \varphi\left(e^{-8\pi}\right) &= \frac{\sqrt4{\pi}}{\Gamma\left(\frac34\right)} \frac{\sqrt{2+\sqrt{2}}+\sqrt8{128}}{4}\\ \varphi\left(e^{-9\pi}\right) &= \frac{\sqrt4{\pi}}{\Gamma\left(\frac34\right)} \frac{1+\sqrt3{2+2\sqrt{3}}}{3}\\ \varphi\left(e^{-10\pi}\right) &= \frac{\sqrt4{\pi}}{\Gamma\left(\frac34\right)} \frac{\sqrt{\sqrt4{64}+\sqrt4{80}+\sqrt4{81}+\sqrt4{100}}}{\sqrt4{200}}\\ \varphi\left(e^{-11\pi}\right) &= \frac{\sqrt4{\pi}}{\Gamma\left(\frac34\right)} \frac{\sqrt{11+\sqrt{11}+(5+3\sqrt{3}+\sqrt{11}+\sqrt{33})\sqrt3{-44+33\sqrt{3}}+(-5+3\sqrt{3}-\sqrt{11}+\sqrt{33})\sqrt3{44+33\sqrt{3}}}}{\sqrt8{52180524}}\\ \varphi\left(e^{-12\pi}\right) &= \frac{\sqrt4{\pi}}{\Gamma\left(\frac34\right)} \frac{\sqrt{\sqrt4{1}+\sqrt4{2}+\sqrt4{3}+\sqrt4{4}+\sqrt4{9}+\sqrt4{18}+\sqrt4{24}}}{2\sqrt8{108}}\\ \varphi\left(e^{-13\pi}\right) &= \frac{\sqrt4{\pi}}{\Gamma\left(\frac34\right)} \frac{\sqrt{13+8\sqrt{13}+(11-6\sqrt{3}+\sqrt{13})\sqrt3{143+78\sqrt{3}}+(11+6\sqrt{3}+\sqrt{13})\sqrt3{143-78\sqrt{3}}}}{\sqrt4{19773}}\\ \varphi\left(e^{-14\pi}\right) &= \frac{\sqrt4{\pi}}{\Gamma\left(\frac34\right)} \frac{\sqrt{\sqrt{13+\sqrt{7}}+\sqrt{7+3\sqrt{7}}+\sqrt{10+2\sqrt{7}}+\sqrt8{28}\sqrt{4+\sqrt{7}}}}{\sqrt16{28^7}}\\ \varphi\left(e^{-15\pi}\right) &= \frac{\sqrt4{\pi}}{\Gamma\left(\frac34\right)} \frac{\sqrt{7+3\sqrt{3}+\sqrt{5}+\sqrt{15}+\sqrt4{60}+\sqrt4{1500}}}{\sqrt8{12^3}\cdot\sqrt{5}}\\ 2\varphi\left(e^{-16\pi}\right) &= \varphi\left(e^{-4\pi}\right) + \frac{\sqrt4{\pi}}{\Gamma\left(\frac34\right)} \frac{\sqrt4{1+\sqrt{2}}}{\sqrt16{128}}\\ \varphi\left(e^{-17\pi}\right) &= \frac{\sqrt4{\pi}}{\Gamma\left(\frac34\right)} \frac{\sqrt{2}(1+\sqrt4{17})+\sqrt8{17}\sqrt{5+\sqrt{17}}}{\sqrt{17+17\sqrt{17}}}\\ 2\varphi\left(e^{-20\pi}\right) &= \varphi\left(e^{-5\pi}\right) + \frac{\sqrt4{\pi}}{\Gamma\left(\frac34\right)} \sqrt{\frac{3+2\sqrt4{5}}{5\sqrt2}}\\ \end{align}

If the reciprocal of the Gelfond constant is raised to the power of the reciprocal of an odd number, then the corresponding $\backslash vartheta\_\{00\}$ values or $\backslash phi$ values can be represented in a simplified way by using the hyperbolic lemniscatic sine:

$\backslash varphi\backslash bigl\backslash exp(-\backslash tfrac\{1\}\{5\}\backslash pi)\backslash bigr\; =\; \backslash sqrt4\{\backslash pi\}\backslash ,\{\backslash Gamma\backslash left(\backslash tfrac\{3\}\{4\}\backslash right)\}^\{-1\}\; \backslash operatorname\{slh\}\backslash bigl(\backslash tfrac\{1\}\{5\}\backslash sqrt\{2\}\backslash ,\backslash varpi\backslash bigr)\backslash operatorname\{slh\}\backslash bigl(\backslash tfrac\{2\}\{5\}\backslash sqrt\{2\}\backslash ,\backslash varpi\backslash bigr)$

$\backslash varphi\backslash bigl\backslash exp(-\backslash tfrac\{1\}\{7\}\backslash pi)\backslash bigr\; =\; \backslash sqrt4\{\backslash pi\}\backslash ,\{\backslash Gamma\backslash left(\backslash tfrac\{3\}\{4\}\backslash right)\}^\{-1\}\; \backslash operatorname\{slh\}\backslash bigl(\backslash tfrac\{1\}\{7\}\backslash sqrt\{2\}\backslash ,\backslash varpi\backslash bigr)\backslash operatorname\{slh\}\backslash bigl(\backslash tfrac\{2\}\{7\}\backslash sqrt\{2\}\backslash ,\backslash varpi\backslash bigr)\; \backslash operatorname\{slh\}\backslash bigl(\backslash tfrac\{3\}\{7\}\backslash sqrt\{2\}\backslash ,\backslash varpi\backslash bigr)$

$\backslash varphi\backslash bigl\backslash exp(-\backslash tfrac\{1\}\{9\}\backslash pi)\backslash bigr\; =\; \backslash sqrt4\{\backslash pi\}\backslash ,\{\backslash Gamma\backslash left(\backslash tfrac\{3\}\{4\}\backslash right)\}^\{-1\}\; \backslash operatorname\{slh\}\backslash bigl(\backslash tfrac\{1\}\{9\}\backslash sqrt\{2\}\backslash ,\backslash varpi\backslash bigr)\backslash operatorname\{slh\}\backslash bigl(\backslash tfrac\{2\}\{9\}\backslash sqrt\{2\}\backslash ,\backslash varpi\backslash bigr)\; \backslash operatorname\{slh\}\backslash bigl(\backslash tfrac\{3\}\{9\}\backslash sqrt\{2\}\backslash ,\backslash varpi\backslash bigr)\; \backslash operatorname\{slh\}\; \backslash bigl(\backslash tfrac\{4\}\{9\}\backslash sqrt\{2\}\backslash ,\backslash varpi\backslash bigr)$

$\backslash varphi\backslash bigl\backslash exp(-\backslash tfrac\{1\}\{11\}\backslash pi)\backslash bigr\; =\; \backslash sqrt4\{\backslash pi\}\backslash ,\{\backslash Gamma\backslash left(\backslash tfrac\{3\}\{4\}\backslash right)\}^\{-1\}\; \backslash operatorname\{slh\}\backslash bigl(\backslash tfrac\{1\}\{11\}\backslash sqrt\{2\}\backslash ,\backslash varpi\backslash bigr)\backslash operatorname\{slh\}\backslash bigl(\backslash tfrac\{2\}\{11\}\backslash sqrt\{2\}\backslash ,\backslash varpi\backslash bigr)\; \backslash operatorname\{slh\}\backslash bigl(\backslash tfrac\{3\}\{11\}\backslash sqrt\{2\}\backslash ,\backslash varpi\backslash bigr)\; \backslash operatorname\{slh\}\; \backslash bigl(\backslash tfrac\{4\}\{11\}\backslash sqrt\{2\}\backslash ,\backslash varpi\backslash bigr)\; \backslash operatorname\{slh\}\backslash bigl(\backslash tfrac\{5\}\{11\}\backslash sqrt\{2\}\backslash ,\backslash varpi\backslash bigr)$

With the letter $\backslash varpi$ the Lemniscate constant is represented.

Note that the following modular identities hold:

$\backslash begin\{align\}$

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)+\sqrt3{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}

where $s(q)=s\backslash left(e^\{\backslash pi\; i\backslash tau\}\backslash right)=-R\backslash left(-e^\{-\backslash pi\; i/(5\backslash tau)\}\backslash right)$ is the Rogers–Ramanujan continued fraction:

$\backslash begin\{align\}$

s(q) &= \sqrt5{\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)}\\ \end{align}

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[[Equianharmonic/" itemprop="url" title="Wiki: equianharmonic"> <hr class="us2411627114"> <span class="us3003804241 us1353177739">[[Equianharmonic">equianharmonic">

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[[Equianharmonic values The mathematician Bruce Berndt found out further values of the theta function:

$\backslash begin\{array\}\{lll\}$

\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}(\sqrt3{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}\sqrt3{100}+\tfrac{2}{5}\sqrt3{10}+\tfrac{3}{5}\sqrt{5}+1) \end{array}

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Further values Many values of the theta function and especially of the shown phi function can be represented in terms of the gamma function:

$\backslash begin\{array\}\{lll\}$

\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+\sqrt4{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}

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Nome power theorems

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Direct power theorems For the transformation of the nomeAndreas Dieckmann: Table of Infinite Products Infinite Sums Infinite Series, Elliptic Theta. Physikalisches Institut Universität Bonn, Abruf am 1. Oktober 2021. in the theta functions these formulas can be used:

$\backslash theta\_\{2\}(q^2)\; =\; \backslash tfrac\{1\}\{2\}\backslash sqrt\{2\backslash theta\_\{3\}(q)^2\}$

$\backslash theta\_\{3\}(q^2)\; =\; \backslash tfrac\{1\}\{2\}\backslash sqrt\{2\backslash theta\_\{3\}(q)^2\}$

$\backslash theta\_\{4\}(q^2)\; =\; \backslash sqrt\{\backslash theta\_\{4\}(q)\backslash theta\_\{3\}(q)\}$

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 triples according to the Jacobi identity. Furthermore, those transformations are valid:

$\backslash theta\_\{3\}(q^4)\; =\; \backslash tfrac\{1\}\{2\}\backslash theta\_\{3\}(q)\; +\; \backslash tfrac\{1\}\{2\}\backslash theta\_\{4\}(q)$

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

$27\backslash ,\backslash theta\_\{3\}(q^3)^8\; -\; 18\backslash ,\backslash theta\_\{3\}(q^3)^4\backslash theta\_\{3\}(q)^4\; -\; \backslash ,\backslash theta\_\{3\}(q)^8\; =\; 8\backslash ,\backslash theta\_\{3\}(q^3)^2\backslash theta\_\{3\}(q)^22\backslash ,\backslash theta\_\{4\}(q)^4$

$27\backslash ,\backslash theta\_\{4\}(q^3)^8\; -\; 18\backslash ,\backslash theta\_\{4\}(q^3)^4\backslash theta\_\{4\}(q)^4\; -\; \backslash ,\backslash theta\_\{4\}(q)^8\; =\; 8\backslash ,\backslash theta\_\{4\}(q^3)^2\backslash theta\_\{4\}(q)^22\backslash ,\backslash theta\_\{3\}(q)^4$

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

$\backslash theta\_\{3\}(q)^25\backslash ,\backslash theta\_\{3\}(q^5)^2^5\; =\; 256\backslash ,\backslash theta\_\{3\}(q^5)^2\backslash theta\_\{3\}(q)^2\backslash theta\_\{4\}(q)^4\; \backslash theta\_\{3\}(q)^4$

$\backslash theta\_\{4\}(q^5)^25\backslash ,\backslash theta\_\{4\}(q^5)^2^5\; =\; 256\backslash ,\backslash theta\_\{4\}(q^5)^2\backslash theta\_\{4\}(q)^2\backslash theta\_\{3\}(q)^4\; \backslash theta\_\{3\}(q)^4$

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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:

$\backslash biggl\backslash frac\{\backslash theta\_\{3\}(q^\{1/3\})^2\}\{\backslash theta\_\{3\}(q)^2\}^2\; =\; 4\; -\; 4\backslash biggl\backslash frac\{2\backslash ,\backslash theta\_\{2\}(q)^2^\{2/3\}$

$\backslash biggl\backslash frac\{3\backslash ,\backslash theta\_\{4\}(q^\{3\})^2\}\{\backslash theta\_\{4\}(q)^2\}^2\; =\; 4\; +\; 4\backslash biggl\backslash frac\{2\backslash ,\backslash theta\_\{2\}(q)^2^\{2/3\}$

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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:

$R(q)\; =\; \backslash tan\backslash biggl\backslash \{\backslash frac\{1\}\{2\}\backslash arctan\backslash biggl\backslash frac\{1\}\{2\}\backslash biggr\backslash \}^\{1/5\}\; \backslash tan\backslash biggl\backslash \{\backslash frac\{1\}\{2\}\backslash arccot\backslash biggl\backslash frac\{1\}\{2\}\backslash biggr\backslash \}^\{2/5\}$

$R(q^2)\; =\; \backslash tan\backslash biggl\backslash \{\backslash frac\{1\}\{2\}\backslash arctan\backslash biggl\backslash frac\{1\}\{2\}\backslash biggr\backslash \}^\{2/5\}\; \backslash cot\backslash biggl\backslash \{\backslash frac\{1\}\{2\}\backslash arccot\backslash biggl\backslash frac\{1\}\{2\}\backslash biggr\backslash \}^\{1/5\}$

$R(q^2)\; =\; \backslash tan\backslash biggl\backslash \{\backslash frac\{1\}\{2\}\backslash arctan\backslash biggl\backslash frac\{\backslash theta\_\{3\}(q)^2\}\{2\backslash ,\backslash theta\_\{3\}(q^5)^2\}\backslash biggr\backslash \}^\{2/5\}\; \backslash tan\backslash biggl\backslash \{\backslash frac\{1\}\{2\}\backslash arccot\backslash biggl\backslash frac\{\backslash theta\_\{3\}(q)^2\}\{2\backslash ,\backslash theta\_\{3\}(q^5)^2\}\backslash biggr\backslash \}^\{1/5\}$

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

$S(q)\; =\; \backslash frac\{R(q^4)\}\{R(q^2)R(q)\}\; =\; \backslash tan\backslash biggl\backslash \{\backslash frac\{1\}\{2\}\backslash arctan\backslash biggl\; \backslash frac\{\backslash theta\_\{3\}(q)^2\}\{2\backslash ,\backslash theta\_\{3\}(q^5)^2\}\backslash biggr\backslash \}^\{1/5\}\; \backslash cot\backslash biggl\backslash \{\backslash frac\{1\}\{2\}\backslash arccot\backslash biggl\backslash frac\{\backslash theta\_\{3\}(q)^2\}\{2\backslash ,\backslash theta\_\{3\}(q^5)^2\}\backslash biggr\backslash \}^\{2/5\}$

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:

$\backslash frac\{\backslash theta\_\{3\}(q^\{1/5\})\}\{\backslash theta\_\{3\}(q^5)\}\; -\; 1\; =\; \backslash frac\{1\}\{S(q)\}\backslash biglS(q)^2\backslash bigl1$

$1\; -\; \backslash frac\{\backslash theta\_\{4\}(q^\{1/5\})\}\{\backslash theta\_\{4\}(q^5)\}\; =\; \backslash frac\{1\}\{R(q)\}\backslash biglR(q^2)\backslash bigl1$

$\backslash theta\_\{3\}(q^\{1/5\})^2\; -\; \backslash theta\_\{3\}(q)^2\; =\; \backslash bigl\backslash theta\_\{3\}(q)^2\backslash biggl1+\backslash frac\{1\}\{R(q^2)S(q)\}+R(q^2)S(q)+\backslash frac\{1\}\{R(q^2)^2\}+R(q^2)^2+\backslash frac\{1\}\{S(q)\}-S(q)\backslash biggr$

$\backslash theta\_\{4\}(q)^2\; -\; \backslash theta\_\{4\}(q^\{1/5\})^2\; =\; \backslash bigl\backslash theta\_\{4\}(q^5)^2\backslash biggl1-\backslash frac\{1\}\{R(q^2)R(q)\}-R(q^2)R(q)+\backslash frac\{1\}\{R(q^2)^2\}+R(q^2)^2-\backslash frac\{1\}\{R(q)\}+R(q)\backslash biggr$

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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:

$\backslash theta\_\{4\}q(k)\; =\; \backslash theta\_\{4\}q(k)^2\backslash sqrt8\{1\; -\; k^2\}$

$\backslash theta\_\{4\}q(k)^2\; =\; \backslash theta\_\{3\}q(k)\backslash sqrt8\{1\; -\; k^2\}$

$\backslash theta\_\{3\}q(k)^2\; =\; \backslash theta\_\{3\}q(k)\backslash cos\backslash tfrac\{1\}\{2\}\backslash arcsin(k)$

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

$\backslash theta\_\{4\}\backslash biggl\backslash langle\; q\backslash bigl\backslash \{\backslash tan\backslash bigl\backslash tfrac\{1\}\{2\}\backslash arctan(t^3)\backslash bigr\backslash bigr\backslash \}^3\; \backslash biggr\backslash rangle\; =$

\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}

For all real values $t\; \backslash in\; \backslash R$ the now mentioned formula is valid.

And for this formula two examples shall be given:

First calculation example with the value $t\; =\; 1$ inserted:

>

$\backslash theta\_\{4\}\backslash biggl\backslash langle\; q\backslash bigl\backslash \{\backslash tan\backslash bigl\backslash tfrac\{1\}\{2\}\backslash arctan(1)\backslash bigr\backslash bigr\backslash \}^3\; \backslash biggr\backslash rangle\; =\; \backslash theta\_\{4\}\backslash biggl\backslash langle\; q\backslash bigl\backslash \{\backslash tan\backslash bigl\backslash tfrac\{1\}\{2\}\backslash arctan(1)\backslash bigr\backslash bigr\backslash \}\; \backslash biggr\backslash rangle\; \backslash ,3^\{-1/2\}\; \backslash bigl(\backslash sqrt\{3\}\; +\; \backslash sqrt\{2\}\backslash ,\backslash bigr)^\{1/2\}$

$\backslash theta\_\{4\}\backslash bigl\backslash exp(-3\backslash sqrt\{2\}\backslash ,\backslash pi)\backslash bigr\; =\; \backslash theta\_\{4\}\backslash bigl\backslash exp(-\backslash sqrt\{2\}\backslash ,\backslash pi)\backslash bigr\; \backslash ,3^\{-1/2\}\; \backslash bigl(\backslash sqrt\{3\}\; +\; \backslash sqrt\{2\}\backslash ,\backslash bigr)^\{1/2\}$

Second calculation example with the value $t\; =\; \backslash Phi^\{-2\}$ inserted:

>

$\backslash theta\_\{4\}\backslash biggl\backslash langle\; q\backslash bigl\backslash \{\backslash tan\backslash bigl\backslash tfrac\{1\}\{2\}\backslash arctan(\backslash Phi^\{-6\})\backslash bigr\backslash bigr\backslash \}^3\; \backslash biggr\backslash rangle\; =\; \backslash theta\_\{4\}\backslash biggl\backslash langle\; q\backslash bigl\backslash \{\backslash tan\backslash bigl\backslash tfrac\{1\}\{2\}\backslash arctan(\backslash Phi^\{-6\})\backslash bigr\backslash bigr\backslash \}\; \backslash biggr\backslash rangle\; \backslash ,3^\{-1/2\}\; \backslash bigl(\backslash sqrt\{2\backslash sqrt\{\backslash Phi^\{-8\}\; -\; \backslash Phi^\{-4\}\; +\; 1\}\; -\; \backslash Phi^\{-4\}\; +\; 2\}\; +\; \backslash sqrt\{\backslash Phi^\{-4\}\; +\; 1\}\backslash ,\backslash bigr)^\{1/2\}$

$\backslash theta\_\{4\}\backslash bigl\backslash exp(-3\backslash sqrt\{10\}\backslash ,\backslash pi)\backslash bigr\; =\; \backslash theta\_\{4\}\backslash bigl\backslash exp(-\backslash sqrt\{10\}\backslash ,\backslash pi)\backslash bigr\; \backslash ,3^\{-1/2\}\; \backslash bigl(\backslash sqrt\{2\backslash sqrt\{\backslash Phi^\{-8\}\; -\; \backslash Phi^\{-4\}\; +\; 1\}\; -\; \backslash Phi^\{-4\}\; +\; 2\}\; +\; \backslash sqrt\{\backslash Phi^\{-4\}\; +\; 1\}\backslash ,\backslash bigr)^\{1/2\}$

The constant $\backslash Phi$ represents the golden ratio number $\backslash Phi\; =\; \backslash tfrac\{1\}\{2\}(\backslash sqrt\{5\}\; +\; 1)$ exactly.

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Some series identities

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Sums with theta function in the result

The infinite sumLandau (1899) zitiert nach Borwein, Page 94, Exercise 3. of the reciprocals of Fibonacci numbers with odd indices has the identity:

$\backslash sum\_\{n=1\}^\backslash infty\; \backslash frac\{1\}\{F\_\{2n-1\}\}\; =\; \backslash frac\{\backslash sqrt\{5\}\}\{2\}\backslash ,\backslash sum\_\{n=1\}^\backslash infty\; \backslash frac\{2(\backslash Phi^\{-2\})^\{n\; -\; 1/2\}\}\{1\; +\; (\backslash Phi^\{-2\})^\{2n\; -\; 1\}\}\; =\; \backslash frac\{\backslash sqrt\{5\}\}\{4\}\; \backslash sum\_\{a=-\backslash infty\}^\backslash infty\; \backslash frac\{2(\backslash Phi^\{-2\})^\{a\; -\; 1/2\}\}\{1\; +\; (\backslash Phi^\{-2\})^\{2a\; -\; 1\}\}\; =$

$=\; \backslash frac\{\backslash sqrt\{5\}\}\{4\}\backslash ,\backslash theta\_\{2\}(\backslash Phi^\{-2\})^2\; =\; \backslash frac\{\backslash sqrt\{5\}\}\{8\}\backslash bigl\backslash theta\_\{3\}(\backslash Phi^\{-1\})^2$

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

$\backslash sum\_\{n=1\}^\backslash infty\; \backslash frac\{1\}\{F\_\{2n-1\}\}\; =\; \backslash frac\{\backslash sqrt\{5\}\}\{4\}\backslash ,\backslash biggl^2$

$\backslash sum\_\{n=1\}^\backslash infty\; \backslash frac\{1\}\{F\_\{2n-1\}\}\; =\; 1.82451515740692456814215840626732817332\backslash ldots$

Also in this case $\backslash Phi\; =\; \backslash tfrac\{1\}\{2\}(\backslash sqrt\{5\}\; +\; 1)$ is Golden ratio number again.

Infinite sum of the reciprocals of the Fibonacci number squares:

$\backslash sum\_\{n=1\}^\backslash infty\; \backslash frac\{1\}\{F\_\{n\}^2\}\; =\; \backslash frac\{5\}\{24\}\backslash bigl2\backslash ,\backslash theta\_\{2\}(\backslash Phi^\{-2\})^4\; =\; \backslash frac\{5\}\{24\}\backslash bigl\backslash theta\_\{3\}(\backslash Phi^\{-2\})^4$

Infinite sum of the reciprocals of the Pell sequence with odd indices:

$\backslash sum\_\{n=1\}^\backslash infty\; \backslash frac\{1\}\{P\_\{2n-1\}\}\; =\; \backslash frac\{1\}\{\backslash sqrt\{2\}\}\backslash ,\backslash theta\_\{2\}\backslash bigl(\backslash sqrt\{2\}-1)^2\backslash bigr^2\; =\; \backslash frac\{1\}\{2\backslash sqrt\{2\}\}\backslash bigl\backslash theta\_\{3\}(\backslash sqrt\{2\}-1)^2$

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Sums with theta function in the summand

The next two series identities were proved by István Mező:

$\backslash begin\{align\}$

\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}

These relations hold for all . Specializing the values of , we have the next parameter free sums

$\backslash sqrt\{\backslash frac\{\backslash pi\backslash sqrt\{e^\backslash pi\}\}\{2\}\}\backslash cdot\backslash frac\{1\}\{\backslash Gamma^2\backslash left(\backslash frac34\backslash right)\}\; =i\backslash sum\_\{k=-\backslash infty\}^\backslash infty\; e^\{\backslash pi\backslash left(k-2k^2\backslash right)\}\; \backslash theta\_1\; \backslash left(\backslash frac\{i\backslash pi\}\{2\}(2k-1),e^\{-\backslash pi\}\backslash right)$

$\backslash sqrt\{\backslash frac\{\backslash pi\}\{2\}\}\backslash cdot\backslash frac\{1\}\{\backslash Gamma^2\backslash left(\backslash frac34\backslash right)\}\; =\backslash sum\_\{k=-\backslash infty\}^\backslash infty\backslash frac\{\backslash theta\_4\backslash left(ik\backslash pi,e^\{-\backslash pi\}\backslash right)\}\{e^\{2\backslash pi\; k^2\}\}$

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Zeros of the Jacobi theta functions

All zeros of the Jacobi theta functions are simple zeros and are given by the following:

$\backslash begin\{align\}$

\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} where , are arbitrary integers.

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Relation to the Riemann zeta function

The relation

$\backslash vartheta\backslash left(0;-\backslash frac\{1\}\{\backslash tau\}\backslash right)=\backslash left(-i\backslash tau\backslash right)^\backslash frac12\; \backslash vartheta(0;\backslash tau)$

was used by Riemann to prove the functional equation for the Riemann zeta function, by means of the Mellin transform

$\backslash Gamma\backslash left(\backslash frac\{s\}\{2\}\backslash right)\; \backslash pi^\{-\backslash frac\{s\}\{2\}\}\; \backslash zeta(s)\; =\; \backslash frac\{1\}\{2\}\backslash int\_0^\backslash infty\backslash bigl(\backslash vartheta(0;it)-1\backslash bigr)t^\backslash frac\{s\}\{2\}\backslash frac\{\backslash mathrm\{d\}t\}\{t\}$

which can be shown to be invariant under substitution of by . The corresponding integral for is given in the article on the Hurwitz zeta function.

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Relation to the Weierstrass elliptic function

The theta function was used by Jacobi to construct (in a form adapted to easy calculation) 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

$\backslash wp(z;\backslash tau)\; =\; -\backslash big(\backslash log\; \backslash vartheta\_\{11\}(z;\backslash tau)\backslash big)\text{'}\text{'}\; +\; c$

where the second derivative is with respect to and the constant is defined so that the Laurent expansion of at has zero constant term.

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Relation to the q-gamma function

The fourth theta function – and thus the others too – is intimately connected to the q-gamma function via the relation

$\backslash left(\backslash Gamma\_\{q^2\}(x)\backslash Gamma\_\{q^2\}(1-x)\backslash right)^\{-1\}=\backslash frac\{q^\{2x(1-x)\}\}\{\backslash left(q^\{-2\};q^\{-2\}\backslash right)^3\_\backslash infty\backslash left(q^2-1\backslash right)\}\; \backslash theta\_4\backslash left(\backslash frac\{1\}\{2i\}(1-2x)\backslash log\; q,\backslash frac\{1\}\{q\}\backslash right).$

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Relations to Dedekind eta function

Let be the Dedekind eta function, and the argument of the theta function as the nome . Then,

$\backslash begin\{align\}$

\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}

and,

$\backslash theta\_2(q)\backslash ,\backslash theta\_3(q)\backslash ,\backslash theta\_4(q)\; =\; 2\backslash eta^3(\backslash tau).$

See also the Weber modular functions.

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Elliptic modulus

The elliptic modulus is

$k(\backslash tau)\; =\; \backslash frac\{\backslash vartheta\_\{10\}(0;\backslash tau)^2\; \}\{\backslash vartheta\_\{00\}(0;\backslash tau)^2\}$

and the complementary elliptic modulus is

$k\text{'}(\backslash tau)\; =\; \backslash frac\{\backslash vartheta\_\{01\}(0;\backslash tau)^2\; \}\{\backslash vartheta\_\{00\}(0;\backslash tau)^2\}$

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Derivatives of theta functions

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

$E(k)\; =\; \backslash int\_\{0\}^\{\backslash pi/2\}\; \backslash sqrt\{1\; -\; k^2\backslash sin(\backslash varphi)^2\}\; d\backslash varphi$

$E(k)\; =\; \backslash frac\{\backslash pi\}\{2\}\backslash sum\_\{a\; =\; 0\}^\{\backslash infty\}\; \backslash frac\{(2a)!^2\}\{(1\; -\; 2a)16^\{a\}(a!)^4\}\; k^\{2a\}$

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

$\backslash theta\_\{2\}\text{'}(x)\; =\; \backslash frac\{\backslash mathrm\{d\}\}\{\backslash mathrm\{d\}x\}\backslash ,\backslash theta\_\{2\}(x)\; =\; \backslash frac\{1\}\{2\}\; x^\{-3/4\}+\backslash sum\_\{n\; =\; 1\}^\{\backslash infty\}\; \backslash frac\{1\}\{2\}(2n\; +\; 1)^2\; x^\{(2n-1)(2n+3)/4\}$

$\backslash theta\_\{3\}\text{'}(x)\; =\; \backslash frac\{\backslash mathrm\{d\}\}\{\backslash mathrm\{d\}x\}\backslash ,\backslash theta\_\{3\}(x)\; =\; 2+\backslash sum\_\{n\; =\; 1\}^\{\backslash infty\}\; 2(n\; +\; 1)^2\; x^\{n(n+2)\}$

$\backslash theta\_\{4\}\text{'}(x)\; =\; \backslash frac\{\backslash mathrm\{d\}\}\{\backslash mathrm\{d\}x\}\backslash ,\backslash theta\_\{4\}(x)\; =\; -2+\backslash sum\_\{n\; =\; 1\}^\{\backslash infty\}\; 2(n\; +\; 1)^2\; (-1)^\{n+1\}\; x^\{n(n+2)\}$

The derivatives of theta zero-value functions are as follows:

$\backslash theta\_\{2\}\text{'}(x)\; =\; \backslash frac\{\backslash mathrm\{d\}\}\{\backslash mathrm\{d\}x\}\; \backslash ,\backslash theta\_\{2\}(x)\; =\; \backslash frac\{1\}\{2\backslash pi\; x\}\; \backslash theta\_\{2\}(x)\backslash theta\_\{3\}(x)^2\; E\backslash biggl\backslash frac\{\backslash theta\_\{2\}(x)^2\}\{\backslash theta\_\{3\}(x)^2\}\backslash biggr$

$\backslash theta\_\{3\}\text{'}(x)\; =\; \backslash frac\{\backslash mathrm\{d\}\}\{\backslash mathrm\{d\}x\}\; \backslash ,\backslash theta\_\{3\}(x)\; =\; \backslash theta\_\{3\}(x)\backslash bigl\backslash theta\_\{3\}(x)^2\backslash biggl\backslash \{\backslash frac\{1\}\{2\backslash pi\; x\}E\backslash biggl\backslash frac\{\backslash theta\_\{3\}(x)^2\; -\; \backslash frac\{\backslash theta\_\{4\}(x)^2\}\{4\backslash ,x\}\backslash biggr\backslash \}$

$\backslash theta\_\{4\}\text{'}(x)\; =\; \backslash frac\{\backslash mathrm\{d\}\}\{\backslash mathrm\{d\}x\}\; \backslash ,\backslash theta\_\{4\}(x)\; =\; \backslash theta\_\{4\}(x)\backslash bigl\backslash theta\_\{3\}(x)^2\backslash biggl\backslash \{\backslash frac\{1\}\{2\backslash pi\; x\}E\backslash biggl\backslash frac\{\backslash theta\_\{3\}(x)^2\; -\; \backslash frac\{\backslash theta\; \_\{3\}(x)^\; 2\}\{4\backslash ,x\}\backslash biggr\backslash \}$

The two last mentioned formulas are valid for all real numbers of the real definition interval:

And these two last named theta derivative functions are related to each other in this way:

$\backslash vartheta\; \_\{4\}(x)\backslash biggl\backslash frac\{\backslash mathrm\{d\}\}\{\backslash mathrm\{d\}x\}\; -\; \backslash vartheta\; \_\{3\}(x)\backslash biggl\backslash frac\{\backslash mathrm\{d\}\}\{\backslash mathrm\{d\}x\}\; =\; \backslash frac\{1\}\{4\backslash ,x\}\backslash ,\backslash theta\_\{3\}(x)\backslash ,\backslash theta\_\{4\}(x)\backslash bigl\backslash theta\_\{3\}(x)^4$

The derivatives of the quotients from two of the three theta functions mentioned here always have a rational relationship to those three functions:

$\backslash frac\{\backslash mathrm\{d\}\}\{\backslash mathrm\{d\}x\}\; \backslash ,\backslash frac\{\backslash theta\; \_\{2\}(x)\}\{\backslash theta\; \_\{3\}(x)\}\; =\; \backslash frac\{\backslash theta\_\{2\}(x)\backslash ,\backslash theta\; \_\{4\}(x)^4\}\{4\backslash ,x\backslash ,\backslash theta\; \_\{3\}(x)\}$

$\backslash frac\{\backslash mathrm\{d\}\}\{\backslash mathrm\{d\}x\}\; \backslash ,\backslash frac\{\backslash theta\; \_\{2\}(x)\}\{\backslash theta\; \_\{4\}(x)\}\; =\; \backslash frac\{\backslash theta\_\{2\}(x)\backslash ,\backslash theta\; \_\{3\}(x)^4\}\{4\backslash ,x\backslash ,\backslash theta\; \_\{4\}(x)\}$

$\backslash frac\{\backslash mathrm\{d\}\}\{\backslash mathrm\{d\}x\}\; \backslash ,\backslash frac\{\backslash theta\; \_\{3\}(x)\}\{\backslash theta\; \_\{4\}(x)\}\; =\; \backslash frac\{\backslash theta\_\{3\}(x)^5\; -\; \backslash theta\; \_\{3\}(x)\backslash ,\backslash theta\; \_\{4\}(x)^4\}\{4\backslash ,x\backslash ,\backslash theta\; \_\{4\}(x)\}$

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

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Integrals of theta functions

For the theta functions these integrals are valid:

$\backslash int\_\{0\}^\{1\}\; \backslash theta\_\{2\}(x)\; \backslash ,\backslash mathrm\{d\}x\; =\; \backslash sum\; \_\{k\; =\; -\backslash infty\}^\{\backslash infty\}\; \backslash frac\{4\}\{\; (2k+1)^2+4\}\; =\; \backslash pi\backslash tanh(\backslash pi)\; \backslash approx\; 3.129881$

$\backslash int\_\{0\}^\{1\}\; \backslash theta\_\{3\}(x)\; \backslash ,\backslash mathrm\{d\}x\; =\; \backslash sum\; \_\{k\; =\; -\backslash infty\}^\{\backslash infty\}\; \backslash frac\{1\}\{\; k^2+1\}\; =\; \backslash pi\backslash coth(\backslash pi)\; \backslash approx\; 3.153348$

$\backslash int\_\{0\}^\{1\}\; \backslash theta\_\{4\}(x)\; \backslash ,\backslash mathrm\{d\}x\; =\; \backslash sum\; \_\{k\; =\; -\backslash infty\}^\{\backslash infty\}\; \backslash frac\{(-1\; )^\{k\}\}\{k^2+1\}\; =\; \backslash pi\backslash ,\backslash operatorname\{csch\}(\backslash pi)\; \backslash approx\; 0.272029$

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

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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. Taking to be real and with real and positive, we can write

$\backslash vartheta\; (x;it)=1+2\backslash sum\_\{n=1\}^\backslash infty\; \backslash exp\backslash left(-\backslash pi\; n^2\; t\backslash right)\; \backslash cos(2\backslash pi\; nx)$

which solves the heat equation

$\backslash frac\{\backslash partial\}\{\backslash partial\; t\}\; \backslash vartheta(x;it)=\backslash frac\{1\}\{4\backslash pi\}\; \backslash frac\{\backslash partial^2\}\{\backslash partial\; x^2\}\; \backslash vartheta(x;it).$

This theta-function solution is 1-periodic in , and as it approaches the periodic delta function, or Dirac comb, in the sense of distributions

$\backslash lim\_\{t\backslash to\; 0\}\; \backslash vartheta(x;it)=\backslash sum\_\{n=-\backslash infty\}^\backslash infty\; \backslash delta(x-n)$.

General solutions of the spatially periodic initial value problem for the heat equation may be obtained by convolving the initial data at with the theta function.

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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.

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Generalizations

If is a positive-definite quadratic form in variables, then the theta function associated with is

$\backslash theta\_F\; (z)=\; \backslash sum\_\{m\backslash in\; \backslash Z^n\}\; e^\{-\backslash pi\; z\; F(m)\}$

with the sum extending over the lattice of integers $\backslash mathbb\{Z\}^n$. This theta function is a modular form of weight (on an appropriately defined subgroup) of the modular group. In the Fourier expansion,

$\backslash hat\{\backslash theta\}\_F\; (z)\; =\; \backslash sum\_\{k=0\}^\backslash infty\; R\_F(k)\; e^\{2\backslash pi\; ikz\},$

the numbers are called the representation numbers of the form.

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Theta series of a Dirichlet character

For a primitive Dirichlet character modulo and then

$\backslash theta\_\backslash chi(z)\; =\; \backslash frac12\backslash sum\_\{n=-\backslash infty\}^\backslash infty\; \backslash chi(n)\; n^\backslash nu\; e^\{2i\; \backslash pi\; n^2\; z\}$

is a weight modular form of level and character

$\backslash chi(d)\; \backslash left(\backslash frac\{-1\}\{d\}\backslash right)^\backslash nu,$

which meansShimura, On modular forms of half integral weight

$\backslash theta\_\backslash chi\backslash left(\backslash frac\{az+b\}\{cz+d\}\backslash right)\; =\; \backslash chi(d)\; \backslash left(\backslash frac\{-1\}\{d\}\backslash right)^\backslash nu\; \backslash left(\backslash frac\{\backslash theta\_1\backslash left(\backslash frac\{az+b\}\{cz+d\}\backslash right)\}\{\backslash theta\_1(z)\}\backslash right)^\{1+2\backslash nu\}\backslash theta\_\backslash chi(z)$

whenever

$a,b,c,d\backslash in\; \backslash Z^4,\; ad-bc=1,c\; \backslash equiv\; 0\; \backslash bmod\; 4\; q^2.$

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Ramanujan theta function

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Riemann theta function

Let

$\backslash mathbb\{H\}\_n=\backslash left\backslash \{F\backslash in\; M(n,\backslash Complex)\; \backslash ,\backslash big|\backslash ,\; F=F^\backslash mathsf\{T\}\; \backslash ,,\backslash ,\; \backslash operatorname\{Im\}\; F\; >0\; \backslash right\backslash \}$

be the set of symmetric square matrices whose imaginary part is positive definite. $\backslash mathbb\{H\}\_n$ is called the Siegel upper half-space and is the multi-dimensional analog of the upper half-plane. The -dimensional analogue of the modular group is the symplectic group $\backslash operatorname\{Sp\}(2n,\backslash mathbb\{Z\})$; for , $\backslash operatorname\{Sp\}(2,\backslash mathbb\{Z\})\; =\; \backslash operatorname\{SL\}(2,\backslash mathbb\{Z\})$. The -dimensional analogue of the congruence subgroups is played by

$\backslash ker\; \backslash big\backslash \{\backslash operatorname\{Sp\}(2n,\backslash Z)\backslash to\; \backslash operatorname\{Sp\}(2n,\backslash Z/k\backslash Z)\; \backslash big\backslash \}.$

Then, given $\backslash tau\backslash in\backslash mathbb\{H\}\_n$, the Riemann theta function is defined as

$\backslash theta\; (z,\backslash tau)=\backslash sum\_\{m\backslash in\; \backslash Z^n\}\; \backslash exp\backslash left(2\backslash pi\; i\; \backslash left(\backslash tfrac12\; m^\backslash mathsf\{T\}\; \backslash tau\; m\; +m^\backslash mathsf\{T\}\; z\; \backslash right)\backslash right).$

Here, $z\backslash in\backslash mathbb\{C\}^n$ is an -dimensional complex vector, and the superscript T denotes the transpose. The Jacobi theta function is then a special case, with and $\backslash tau\; \backslash in\; \backslash mathbb\{H\}$ where $\backslash mathbb\{H\}$ 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 , as well as other auxiliary objects that figure prominently in their function theory, by taking to be the period matrix with respect to a canonical basis for its first homology group.

The Riemann theta converges absolutely and uniformly on compact subsets of $\backslash mathbb\{C\}^n\; \backslash times\; \backslash mathbb\{H\}\_n$.

The functional equation is

$\backslash theta\; (z+a+\backslash tau\; b,\; \backslash tau)\; =\; \backslash exp\backslash left(\; 2\backslash pi\; i\; \backslash left(-b^\backslash mathsf\{T\}z-\backslash tfrac12\; b^\backslash mathsf\{T\}\backslash tau\; b\backslash right)\backslash right)\; \backslash theta\; (z,\backslash tau)$

which holds for all vectors $a,b\backslash in\backslash mathbb\{Z\}^n$, and for all $z\backslash in\backslash mathbb\{C\}^n$ and $\backslash tau\backslash in\backslash mathbb\{H\}\_n$.

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Poincaré series

The Poincaré series generalizes the theta series to automorphic forms with respect to arbitrary .

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Derivation of the theta values

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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:

$\backslash beta(x)\; =\; \backslash frac\{\backslash Gamma(x)^2\}\{\backslash Gamma(2x)\}$

In general, for all natural numbers $n\; \backslash isin\; \backslash mathbb\{N\}$ this formula of the Euler beta function is valid:

$\backslash frac\{4^\{-1/(n\; +\; 2)\}\}\{n\; +\; 2\}\backslash csc\backslash bigl(\backslash frac\{\backslash pi\}\{n\; +\; 2\}\backslash bigr)\backslash beta\backslash biggl\backslash frac\{n\}\{2(n\; =\; \backslash int\_\{0\}^\{\backslash infty\}\; \backslash frac\{1\}\{\backslash sqrt\{x^\{n+2\}\; +\; 1\}\}\; \backslash ,\backslash mathrm\; \{d\}x$

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Exemplary elliptic integrals

In the following some Elliptic Integral Singular Values are derived:

The ensuing function has the following lemniscatically elliptic antiderivative: $\backslash frac\{1\}\{\backslash sqrt\{x^4\; +\; 1\}\}\; =\; \backslash frac\{\backslash mathrm\{d\}\}\{\backslash mathrm\{d\}x\}\backslash ,\backslash frac\{1\}\{2\}\; F\backslash biggl2\backslash arctan(x);$ For the value $n\; =\; 2$ this identity appears: $\backslash frac\{1\}\{4\backslash sqrt\{2\}\}\backslash csc\backslash bigl(\backslash frac\{\backslash pi\}\{4\}\backslash bigr)\backslash beta\backslash bigl(\backslash frac\{1\}\{4\}\backslash bigr)\; =\; \backslash int\_\{0\}^\{\backslash infty\}\; \backslash frac\{1\}\{\backslash sqrt\{x^4\; +\; 1\}\}\; \backslash ,\backslash mathrm\{d\}x\; =\; \backslash biggl\backslash \; =\; \backslash frac\{\backslash mathrm\{d\}\}\{\backslash mathrm\{d\}x\}\backslash ,\backslash frac\{1\}\{6\}\backslash sqrt\; 4\{27\}F\backslash biggl2\backslash arctan\backslash biggl(\backslash frac\{\backslash sqrt[4\{3\}\backslash ,x\}\{\backslash sqrt\{x^2\; +\; 1\}\}\backslash biggr);\backslash frac\{1\; \}\{4\}(\backslash sqrt\{6\}\; +\; \backslash sqrt\{2\})\backslash biggr]$ For the value $n\; =\; 4$ that identity appears: $\backslash frac\{1\}\{6\backslash sqrt3\{2\}\}\backslash csc\backslash bigl(\backslash frac\{\backslash pi\}\{6\}\backslash bigr)\backslash beta\backslash bigl(\backslash frac\{1\}\{3\}\; \backslash bigr)\; =\; \backslash int\_\{0\}^\{\backslash infty\}\; \backslash frac\{1\}\{\backslash sqrt\{x^6\; +\; 1\}\}\; \backslash ,\backslash mathrm\{d\}x\; =\; \backslash biggl\backslash \backslash biggr);\backslash frac\{1\}\{4\}(\backslash sqrt\{6\}\; +\; \backslash sqrt\{2\})\backslash biggr]\}\backslash biggr\backslash \}\_\{x\; =\; 0\}^\{x\; =\; \backslash infty\}\; =$ $=\; \backslash frac\{1\}\{6\}\backslash sqrt4\{27\}\; F\backslash bigl2\backslash arctan(\backslash sqrt[4\{3\});\backslash frac\{1\}\{4\}(\backslash sqrt\; \{6\}\; +\; \backslash sqrt\{2\})\backslash bigr]\; =\; \backslash frac\{2\}\{9\}\backslash sqrt4\{27\}\; K\backslash bigl\backslash frac\{1\}\{4\}(\backslash sqrt\{6\}\; =\; \backslash frac\{2\}\{3\}\backslash sqrt4\{3\}\; K\backslash bigl\backslash frac\{1\}\{4\}(\backslash sqrt\{6\}$ This result follows from that equation chain: $\{\backslash color\{ForestGreen\}K\backslash bigl\backslash frac\{1\}\{4\}(\backslash sqrt\{6\}\; =\; \backslash frac\{1\}\{2\backslash sqrt3\{2\}\backslash sqrt4\{3\}\}\backslash beta\backslash bigl(\backslash frac\{1\}\{3\}\backslash bigr)\}$

And the following function has the following elliptic antiderivative: $\backslash frac\{1\}\{\backslash sqrt\{x^8\; +\; 1\}\}\; =$ $=\; \backslash frac\{\backslash mathrm\{d\}\}\{\backslash mathrm\{d\}x\}\backslash ,\backslash frac\{1\}\{4\}\backslash sec\; \backslash bigl(\backslash frac\{\backslash pi\}\{8\}\backslash bigr)F\backslash biggl\backslash \{2\backslash arctan\backslash biggl\backslash frac\{2\backslash cos(\backslash pi/8)\backslash ,x\}\{\backslash sqrt\{x^4;2\backslash sqrt4\{2\}\backslash sin\backslash bigl(\backslash frac\{\backslash pi\}\{8\}\backslash bigr)\; \backslash biggr\backslash \}\; +\; \backslash frac\{1\}\{4\}\backslash sec\backslash bigl(\backslash frac\{\backslash pi\}\{8\}\backslash bigr)F\backslash biggl\backslash \{\backslash arcsin\backslash biggl\backslash frac\{2\backslash cos(\backslash pi/;\backslash tan\backslash bigl(\backslash frac\{\backslash pi\}\{8\}\backslash bigr)\backslash biggr\backslash \}$ For the value $n\; =\; 6$ the following identity appears: $\backslash frac\{1\}\{8\backslash sqrt4\{2\}\}\backslash csc\backslash bigl(\backslash frac\{\backslash pi\}\{8\}\backslash bigr)\backslash beta\backslash bigl(\backslash frac\{3\}\{8\}\; \backslash bigr)\; =\; \backslash int\_\{0\}^\{\backslash infty\}\; \backslash frac\{1\}\{\backslash sqrt\{x^8\; +\; 1\}\}\; \backslash ,\backslash mathrm\{d\}x\; =$ $=\; \backslash biggl\backslash langle\{\backslash color\{blue\}\backslash frac\{1\}\{4\}\backslash sec\backslash bigl(\backslash frac\{\backslash pi\}\{8\}\backslash bigr)F\backslash biggl\backslash \{2\backslash arctan\backslash biggl\backslash frac\{2\backslash cos(\backslash pi/8)\backslash ,x\}\{\backslash sqrt\{x^4;2\backslash sqrt4\{\; 2\}\backslash sin\backslash bigl(\backslash frac\{\backslash pi\}\{8\}\backslash bigr)\backslash biggr\backslash \}\; +\; \backslash frac\{1\}\{4\}\backslash sec\backslash bigl(\backslash frac\{\backslash pi\}\{8\}\backslash bigr)F\; \backslash biggl\backslash \{\backslash arcsin\backslash biggl\backslash frac\{2\backslash cos(\backslash pi/8)\backslash ,x\}\{x^2;\backslash tan\backslash bigl(\backslash frac\{\backslash pi\}\{8\}\; \backslash bigr)\backslash biggr\backslash \}\}\backslash biggr\backslash rangle\_\{x\; =\; 0\}^\{x\; =\; \backslash infty\}\; =$ $=\; \backslash frac\{1\}\{4\}\backslash sec\backslash bigl(\backslash frac\{\backslash pi\}\{8\}\backslash bigr)F\backslash bigl\backslash pi;2\backslash sqrt[4\{2\}\backslash sin\backslash bigl\; (\backslash frac\{\backslash pi\}\{8\}\backslash bigr)\backslash bigr]\; =\; \backslash frac\{1\}\{2\}\backslash sec\backslash bigl(\backslash frac\{\backslash pi\}\{8\}\backslash bigr)K(\backslash sqrt\{2\backslash sqrt\; \{2\}\; -\; 2\}\backslash bigr)\; =\; 2\backslash sin\backslash bigl(\backslash frac\{\backslash pi\}\{8\}\backslash bigr)K(\backslash sqrt\{2\}\; -\; 1)$ This result follows from that equation chain: $\{\backslash color\{ForestGreen\}K(\backslash sqrt\{2\}\; -\; 1)\; =\; \backslash frac\{1\}\{8\}\backslash sqrt4\{2\}\backslash ,(\backslash sqrt\{2\}\; +\; 1)\backslash ,\; \backslash beta\backslash bigl(\backslash frac\{3\}\{8\}\backslash bigr)\}$

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Combination of the integral identities with the nome

The elliptic nome function has these important values:

$q(\backslash tfrac\{1\}\{2\}\backslash sqrt\{2\})\; =\; \backslash exp(-\backslash pi)$

$q\backslash tfrac\{1\}\{4\}(\backslash sqrt\{6\}\; =\; \backslash exp(-\backslash sqrt\{3\}\backslash ,\backslash pi)$

$q(\backslash sqrt\{2\}\; -\; 1)\; =\; \backslash exp(-\backslash sqrt\{2\}\backslash ,\backslash pi)$

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:

>

$\backslash theta\_\{3\}q(k)\; =\; \backslash sqrt\{2\backslash pi^\{-1\}\; K(k)\}$

$\backslash theta\_\{3\}\backslash exp(-\backslash pi)\; =\; \backslash theta\_\{3\}q(\backslash tfrac\{1\}\{2\}\backslash sqrt\{2\})\; =\; \backslash sqrt\{2\backslash pi^\; \{-1\}K(\backslash tfrac\{1\}\{2\}\backslash sqrt\{2\})\}\; =\; 2^\{-1/2\}\backslash pi^\{-1/2\}\backslash beta(\backslash tfrac\{1\}\{4\}\; )^\{1/2\}\; =\; 2^\{-1/4\}\backslash sqrt4\{\backslash pi\}\backslash ,\{\backslash Gamma\backslash bigl(\backslash tfrac\{3\}\{4\}\backslash bigr)\}^\{-1\}$

$\backslash theta\; \_\{3\}\backslash exp(-\backslash sqrt\{3\}\backslash ,\backslash pi\; =\; \backslash theta\; \_\{3\}\backslash bigl\backslash \{q\backslash bigl\backslash tfrac\{1\}\{4\}(\backslash sqrt\backslash bigr\backslash \}\; =\; \backslash sqrt\{2\backslash pi^\{-1\}K\backslash bigl\backslash tfrac\{1\}\{4\}(\backslash sqrt\{6\}\}\; =\; 2^\{-1/6\}3^\{-1/8\}\backslash pi^\{-1/2\}\backslash beta(\backslash tfrac\{1\}\{3\})^\{1/\; 2\}$

$\backslash theta\; \_\{3\}\backslash exp(-\backslash sqrt\{2\}\backslash ,\backslash pi\; =\; \backslash theta\; \_\{3\}q(\backslash sqrt\{2\}\; =\; \backslash sqrt\{2\backslash pi\; ^\{-1\}K(\backslash sqrt\{2\}\; -\; 1)\}\; =\; 2^\{-1/8\}\backslash cos(\backslash tfrac\{1\}\{8\}\backslash pi)\backslash ,\backslash pi^\{-1/2\}\; \backslash beta(\backslash tfrac\{3\}\{8\})^\{1/2\}$

>

$\backslash theta\_\{4\}q(k)\; =\; \backslash sqrt4\{1\; -\; k^2\}\backslash ,\backslash sqrt\{2\backslash pi^\{-1\}\; K(k)\}$

$\backslash theta\_\{4\}\backslash exp(-\backslash sqrt\{2\}\backslash ,\backslash pi)\; =\; \backslash theta\_\{4\}q(\backslash sqrt\{2\}\; =\; \backslash sqrt4\{\; 2\backslash sqrt\{2\}\; -\; 2\}\backslash ,\backslash sqrt\{2\backslash pi^\{-1\}K(\backslash sqrt\{2\}\; -\; 1)\}\; =\; 2^\{-1/4\}\backslash cos(\backslash tfrac\{1\}\; \{8\}\backslash pi)^\{1/2\}\backslash ,\backslash pi^\{-1/2\}\backslash beta(\backslash tfrac\{3\}\{8\})^\{1/2\}$

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Partition sequences and Pochhammer products

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Regular partition number sequence

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

+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:

$\backslash sum\; \_\{k\; =\; 0\}^\backslash infty\; P(k)x^k\; =\; \backslash frac\{1\}\{(x;x)\_\{\backslash infty\}\}\; =\; \backslash theta\_\{3\}(x)^\{-1/6\}\backslash theta\_\{4\}(x)^\{-2/3\}\; \backslash biggl\backslash frac\{\backslash theta\_\{3\}(x)^4^\{-1/24\}$

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

$(x;x)\_\{\backslash infty\}\; =\; 1\; +\; \backslash sum\_\{n\; =\; 1\}^\{\backslash infty\}\; \backslash bigl-$

The following basic definitions apply to the pentagonal numbers and the card house numbers:

$\backslash text\{Fn\}(z)\; =\; \backslash tfrac\{1\}\{2\}z(3z-1)$

$\backslash text\{Kr\}(z)\; =\; \backslash tfrac\{1\}\{2\}z(3z+1)$

As a further application Ramanujan's theta-function identities involving Lambert series one obtains a formula for the third power of the Euler product:

$(x;x)^3\; =\; \backslash prod\_\{n=1\}^\backslash infty\; (1-x^n)^3\; =\; \backslash sum\; \_\{m=0\}^\backslash infty\; (-1)^m(2m\; +1)x^\{m(m+1)/2\}$

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Strict partition number sequence

And the strict partition sequence $Q(n)$ indicates the number of ways in which such a positive integer number $n$ can be split into positive integer summands such that each summand appears at most once and no summand value occurs repeatedly. Exactly the same sequence 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:

+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:

$\backslash sum\; \_\{k\; =\; 0\}^\backslash infty\; Q(k)x^k\; =\; \backslash frac\{1\}\{(x;x^2)\_\{\backslash infty\; \}\}\; =\; \backslash theta\_\{3\}(x)^\{1/6\}\backslash theta\_\{4\}(x)^\{-1/3\}\; \backslash biggl\backslash frac\{\backslash theta\_\{3\}(x)^4^\{1/24\}$

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Overpartition number sequence

The Maclaurin series for the reciprocal of the function has the numbers of over partition sequence as coefficients with a positive sign:

$\backslash frac\{1\}\{\backslash theta\_\{4\}(x)\}\; =\; \backslash prod\_\{n=1\}^\{\backslash infty\}\; \backslash frac\{1\; +\; x^\{n\}\}\{1\; -\; x^\{n\; \}\}\; =\; \backslash sum\_\{k=0\}^\{\backslash infty\}\; \backslash overline\{P\}(k)x^\{k\}$

$\backslash frac\{1\}\{\backslash theta\_\{4\}(x)\}\; =\; 1+2x+4x^2+8x^3+14x^4+24x^5+40x^6+64x^7+100x^\; 8+154x^9+232x^\{10\}\; +\; \backslash dots$

If, for a given number $k$, 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 of the marked partitions depending on $k$ by the overpartition function $\backslash overline\{P\}(k)$ .

First example:

$\backslash overline\{P\}(4)\; =\; 14$

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

(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:

$\backslash overline\{P\}(5)\; =\; 24$

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

(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)

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Relations of the partition number sequences to each other

In the Online Encyclopedia of Integer Sequences (OEIS), the sequence of regular partition numbers $P(n)$ is under the code A000041, the sequence of strict partitions is $Q(n)$ under the code A000009 and the sequence of superpartitions $\backslash overline\{P\}(n)$ under the code A015128. All parent partitions from index $n\; =\; 1$ are even.

The sequence of superpartitions $\backslash overline\{P\}(n)$ can be written with the regular partition sequence P and the strict partition sequence Q can be generated like this:

$\backslash overline\{P\}(n)\; =\; \backslash sum\_\{k=0\}^\{n\}\; P(n\; -\; k)Q(k)$

In the following table of sequences of numbers, this formula should be used as an example:

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 :

$\backslash theta\_\{4\}(x)\; =\; \backslash biggl\backslash sum\_\{k^\{-1\}\; \backslash biggl\backslash sum\_\{k^\{-1\}$

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Notes

• (1964). 9780486612720, Dover Publications. ISBN 9780486612720
• Naum Illyich Akhiezer (1990). 9780821845325, AMS. ISBN 9780821845325
• (1980). 9780387904658, Springer-Verlag. ISBN 9780387904658
  . (for treatment of the Riemann theta)
• David Mumford (1983). 9783764331092, Birkhauser. ISBN 9783764331092
• (1974). 9780683071962, Williams & Wilkins. ISBN 9780683071962
• (history of Jacobi's functions)

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Further reading

• Hershel M. Farkas (2026). 9780387785097, Springer-Verlag. ISBN 9780387785097
• Bruno Schoeneberg (1974). 9783540063827, Springer-Verlag. ISBN 9783540063827

Harry Rauch with Hershel M. Farkas: Theta functions with applications to Riemann Surfaces, Williams and Wilkins, Baltimore MD 1974, .

• Charles Hermite: Sur la résolution de l'Équation du cinquiéme degré Comptes rendus, C. R. Acad. Sci. Paris, Nr. 11, March 1858.

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External links

Categories: Elliptic Functions, Riemann Surfaces, Analytic Functions, Several Complex Variables

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