# Theta function

Jacobi's theta function θ1 with nome q = eiπτ = 0.1e0.1iπ: {\displaystyle {\begin{aligned}\theta _{1}(z,q)&=2q^{\frac {1}{4}}\sum _{n=0}^{\infty }(-1)^{n}q^{n(n+1)}\sin(2n+1)z\\&=\sum _{n=-\infty }^{\infty }(-1)^{n-{\frac {1}{2}}}q^{\left(n+{\frac {1}{2}}\right)^{2}}e^{(2n+1)iz}.\end{aligned}}}

In mathematics, theta functions are special functions of several complex variables. They are important in many areas, including the theories of Abelian varieties and moduli spaces, and of quadratic forms. They have also been applied to soliton theory. When generalized to a Grassmann algebra, they also appear in quantum field theory.[1]

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 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 cohomology class of a line bundle on a complex torus, a condition of descent.

Throughout this article, ${\displaystyle (e^{\pi i\tau })^{\alpha }}$ should be interpreted as ${\displaystyle e^{\alpha \pi i\tau }}$ (in order to resolve issues of choice of branch).[note 1]

## 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 z and τ, where z can be any complex number and τ is the half-period ratio, confined to the upper half-plane, which means it has positive imaginary part. It is given by the formula

{\displaystyle {\begin{aligned}\vartheta (z;\tau )&=\sum _{n=-\infty }^{\infty }\exp \left(\pi in^{2}\tau +2\pi inz\right)\\&=1+2\sum _{n=1}^{\infty }q^{n^{2}}\cos(2\pi nz)\\&=\sum _{n=-\infty }^{\infty }q^{n^{2}}\eta ^{n}\end{aligned}}}

where q = exp(πiτ) is the nome and η = exp(2πiz). It is a Jacobi form. The restriction ensures that it is an absolutely convergent series. At fixed τ, this is a Fourier series for a 1-periodic entire function of z. Accordingly, the theta function is 1-periodic in z:

${\displaystyle \vartheta (z+1;\tau )=\vartheta (z;\tau ).}$

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

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

Thus, in general,

${\displaystyle \vartheta (z+a+b\tau ;\tau )=\exp \left(-\pi ib^{2}\tau -2\pi ibz\right)\vartheta (z;\tau )}$

for any integers a and b.

For any fixed ${\displaystyle \tau }$, the function is an entire function on the complex plane, so by Liouville's theorem, it cannot be doubly periodic in ${\displaystyle 1,\tau }$ unless it is constant, and so the best we could do is to make it periodic in ${\displaystyle 1}$ and quasi-periodic in ${\displaystyle \tau }$. Indeed, since

${\displaystyle \left|{\frac {\vartheta (z+a+b\tau ;\tau )}{\vartheta (z;\tau )}}\right|=\exp \left(\pi (a^{2}\Im (\tau )+2a\Im (z))\right)}$
and ${\displaystyle \Im (\tau )>0}$, the function ${\displaystyle \vartheta (z,\tau )}$ 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:[2]

Theorem — If ${\displaystyle f:\mathbb {C} \to \mathbb {C} }$ is entire and nonconstant, and satisfies the functional equations ${\displaystyle {\begin{cases}f(z+1)=f(z)\\f(z+\tau )=e^{az+2\pi ib}f(z)\end{cases}}}$ for some constant ${\displaystyle a,b\in \mathbb {C} }$.

If ${\displaystyle a=0}$, then ${\displaystyle b=\tau }$ and ${\displaystyle f(z)=e^{2\pi iz}}$. If ${\displaystyle a=-2\pi i}$, then ${\displaystyle f(z)=C\vartheta (z+{\frac {1}{2}}\tau +b,\tau )}$ for some nonzero ${\displaystyle C\in \mathbb {C} }$.

Theta function θ1 with different nome q = eiπτ. The black dot in the right-hand picture indicates how q changes with τ.
Theta function θ1 with different nome q = eiπτ. The black dot in the right-hand picture indicates how q changes with 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:

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

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

{\displaystyle {\begin{aligned}\vartheta _{01}(z;\tau )&=\vartheta \left(z+{\tfrac {1}{2}};\tau \right)\\[3pt]\vartheta _{10}(z;\tau )&=\exp \left({\tfrac {1}{4}}\pi i\tau +\pi iz\right)\vartheta \left(z+{\tfrac {1}{2}}\tau ;\tau \right)\\[3pt]\vartheta _{11}(z;\tau )&=\exp \left({\tfrac {1}{4}}\pi i\tau +\pi i\left(z+{\tfrac {1}{2}}\right)\right)\vartheta \left(z+{\tfrac {1}{2}}\tau +{\tfrac {1}{2}};\tau \right).\end{aligned}}}

This notation follows Riemann and Mumford; Jacobi's original formulation was in terms of the nome q = eiπτ rather than τ. In Jacobi's notation the θ-functions are written:

{\displaystyle {\begin{aligned}\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{aligned}}}
Jacobi theta 1
Jacobi theta 2
Jacobi theta 3
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 z = 0 in the above theta functions, we obtain four functions of τ only, defined on the upper half-plane. Alternatively, we obtain four functions of q only, defined on the unit disk ${\displaystyle |q|<1}$. They are sometimes called theta constants:[note 2]

{\displaystyle {\begin{aligned}\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{aligned}}}

with the nome q = eiπτ. These can be used to define a variety of modular forms, and to parametrize certain curves; in particular, the Jacobi identity is

${\displaystyle \theta _{2}(q)^{4}+\theta _{4}(q)^{4}=\theta _{3}(q)^{4}}$

or equivalently,

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

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 ττ + 1 and τ ↦ −1/τ. Equations for the first transform are easily found since adding one to τ in the exponent has the same effect as adding 1/2 to z (nn2 mod 2). For the second, let

${\displaystyle \alpha =(-i\tau )^{\frac {1}{2}}\exp \left({\frac {\pi }{\tau }}iz^{2}\right).}$

Then

{\displaystyle {\begin{aligned}\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{aligned}}}

## Theta functions in terms of the nome

Instead of expressing the Theta functions in terms of z and τ, we may express them in terms of arguments w and the nome q, where w = eπiz and q = eπiτ. In this form, the functions become

{\displaystyle {\begin{aligned}\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+{\frac {1}{2}}}q^{\left(n+{\frac {1}{2}}\right)^{2}}\quad &\vartheta _{11}(w,q)&=i\sum _{n=-\infty }^{\infty }(-1)^{n}\left(w^{2}\right)^{n+{\frac {1}{2}}}q^{\left(n+{\frac {1}{2}}\right)^{2}}.\end{aligned}}}

We see that the theta functions can also be defined in terms of w and q, 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 numbers.

## Product representations

The Jacobi triple product (a special case of the Macdonald identities) tells us that for complex numbers w and q with |q| < 1 and w ≠ 0 we have

${\displaystyle \prod _{m=1}^{\infty }\left(1-q^{2m}\right)\left(1+w^{2}q^{2m-1}\right)\left(1+w^{-2}q^{2m-1}\right)=\sum _{n=-\infty }^{\infty }w^{2n}q^{n^{2}}.}$

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 q = eπiτ (noting some authors instead set q = e2πiτ) and take w = eπiz then

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

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

${\displaystyle \vartheta (z;\tau )=\prod _{m=1}^{\infty }{\big (}1-\exp(2m\pi i\tau ){\big )}{\Big (}1+\exp {\big (}(2m-1)\pi i\tau +2\pi iz{\big )}{\Big )}{\Big (}1+\exp {\big (}(2m-1)\pi i\tau -2\pi iz{\big )}{\Big )}.}$

In terms of w and q:

{\displaystyle {\begin{aligned}\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^{2}q;q^{2}\right)_{\infty }\,\left(-{\frac {q}{w^{2}}};q^{2}\right)_{\infty }\\&=\left(q^{2};q^{2}\right)_{\infty }\,\theta \left(-w^{2}q;q^{2}\right)\end{aligned}}}

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

${\displaystyle \prod _{m=1}^{\infty }\left(1-q^{2m}\right){\Big (}1+\left(w^{2}+w^{-2}\right)q^{2m-1}+q^{4m-2}{\Big )},}$

which we may also write as

${\displaystyle \vartheta (z\mid q)=\prod _{m=1}^{\infty }\left(1-q^{2m}\right)\left(1+2\cos(2\pi z)q^{2m-1}+q^{4m-2}\right).}$

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

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

In particular,

${\displaystyle \lim _{q\to 0}{\frac {\vartheta _{01}(z\mid q)}{2q^{\frac {1}{4}}}}=\cos(\pi z),\quad \lim _{q\to 0}{\frac {-\vartheta _{11}(z\mid q)}{2q^{-{\frac {1}{4}}}}}=\sin(\pi z)}$
so we may interpret them as one-parameter deformations of the periodic functions ${\displaystyle \sin ,\cos }$, 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:

{\displaystyle {\begin{aligned}\vartheta _{00}(z;\tau )&=-i\int _{i-\infty }^{i+\infty }e^{i\pi \tau u^{2}}{\frac {\cos(2uz+\pi u)}{\sin(\pi u)}}\mathrm {d} u;\\[6pt]\vartheta _{01}(z;\tau )&=-i\int _{i-\infty }^{i+\infty }e^{i\pi \tau u^{2}}{\frac {\cos(2uz)}{\sin(\pi u)}}\mathrm {d} u;\\[6pt]\vartheta _{10}(z;\tau )&=-ie^{iz+{\frac {1}{4}}i\pi \tau }\int _{i-\infty }^{i+\infty }e^{i\pi \tau u^{2}}{\frac {\cos(2uz+\pi u+\pi \tau u)}{\sin(\pi u)}}\mathrm {d} u;\\[6pt]\vartheta _{11}(z;\tau )&=e^{iz+{\frac {1}{4}}i\pi \tau }\int _{i-\infty }^{i+\infty }e^{i\pi \tau u^{2}}{\frac {\cos(2uz+\pi \tau u)}{\sin(\pi u)}}\mathrm {d} u.\end{aligned}}}

## Explicit values

### 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).[3] Define,

${\displaystyle \quad \varphi (q)=\vartheta _{00}(0;\tau )=\theta _{3}(0;q)=\sum _{n=-\infty }^{\infty }q^{n^{2}}}$

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

{\displaystyle {\begin{aligned}\varphi \left(e^{-\pi }\right)&={\frac {\sqrt[{4}]{\pi }}{\Gamma \left({\frac {3}{4}}\right)}}={\sqrt {2}}\,\eta {\big (}{\sqrt {-1}}{\big )}\\[8pt]\varphi \left(e^{-2\pi }\right)&={\frac {\sqrt[{4}]{\pi }}{\Gamma \left({\frac {3}{4}}\right)}}{\frac {\sqrt[{4}]{6+4{\sqrt {2}}}}{2}}\\[8pt]\varphi \left(e^{-3\pi }\right)&={\frac {\sqrt[{4}]{\pi }}{\Gamma \left({\frac {3}{4}}\right)}}{\frac {\sqrt[{4}]{27+18{\sqrt {3}}}}{3}}\\[8pt]\varphi \left(e^{-4\pi }\right)&={\frac {\sqrt[{4}]{\pi }}{\Gamma \left({\frac {3}{4}}\right)}}{\frac {{\sqrt[{4}]{8}}+2}{4}}\\[8pt]\varphi \left(e^{-5\pi }\right)&={\frac {\sqrt[{4}]{\pi }}{\Gamma \left({\frac {3}{4}}\right)}}{\frac {\sqrt[{4}]{225+100{\sqrt {5}}}}{5}}\\[8pt]\varphi \left(e^{-6\pi }\right)&={\frac {\sqrt[{4}]{\pi }}{\Gamma \left({\frac {3}{4}}\right)}}{\frac {\sqrt {{\sqrt[{4}]{1}}+{\sqrt[{4}]{3}}+{\sqrt[{4}]{4}}+{\sqrt[{4}]{9}}}}{\sqrt[{8}]{1728}}}={\frac {\sqrt[{4}]{\pi }}{\Gamma \left({\frac {3}{4}}\right)}}{\frac {{\sqrt[{3}]{3{\sqrt {2}}+3{\sqrt[{4}]{3}}+2{\sqrt {3}}-{\sqrt[{4}]{27}}+{\sqrt[{4}]{1728}}-4}}\cdot {\sqrt[{8}]{243}}}{6{\sqrt[{6}]{1+{\sqrt {6}}-{\sqrt {2}}-{\sqrt {3}}}}}}\\[8pt]\varphi \left(e^{-7\pi }\right)&={\frac {\sqrt[{4}]{\pi }}{\Gamma \left({\frac {3}{4}}\right)}}{\sqrt {{\frac {{\sqrt {13+{\sqrt {7}}}}+{\sqrt {7+3{\sqrt {7}}}}}{14}}\cdot {\sqrt[{8}]{28}}}}={\frac {\sqrt[{4}]{\pi }}{\Gamma \left({\frac {3}{4}}\right)}}{\frac {\sqrt[{4}]{7+4{\sqrt {7}}+5{\sqrt[{4}]{28}}+{\sqrt[{4}]{1372}}}}{\sqrt {7}}}\\[8pt]\varphi \left(e^{-8\pi }\right)&={\frac {\sqrt[{4}]{\pi }}{\Gamma \left({\frac {3}{4}}\right)}}{\frac {{\sqrt[{8}]{128}}+{\sqrt {2+{\sqrt {2}}}}}{4}}\\[8pt]\varphi \left(e^{-9\pi }\right)&={\frac {\sqrt[{4}]{\pi }}{\Gamma \left({\frac {3}{4}}\right)}}{\frac {\left(1+\left(1+{\sqrt {3}}\right){\sqrt[{3}]{2-{\sqrt {3}}}}\right)}{3}}\\[8pt]\varphi \left(e^{-10\pi }\right)&={\frac {\sqrt[{4}]{\pi }}{\Gamma \left({\frac {3}{4}}\right)}}{\frac {\sqrt {{\sqrt[{4}]{64}}+{\sqrt[{4}]{80}}+{\sqrt[{4}]{81}}+{\sqrt[{4}]{100}}}}{\sqrt[{4}]{200}}}\\[10pt]\varphi \left(e^{-12\pi }\right)&={\frac {\sqrt[{4}]{\pi }}{\Gamma \left({\frac {3}{4}}\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}}}}={\frac {\sqrt[{4}]{\pi }}{\Gamma \left({\frac {3}{4}}\right)}}{\frac {{\sqrt {{\sqrt[{4}]{1}}+{\sqrt[{4}]{9}}}}+{\sqrt {{\sqrt[{4}]{3}}+{\sqrt[{4}]{4}}}}}{2{\sqrt[{8}]{108}}}}\\[8pt]\varphi \left(e^{-16\pi }\right)&={\frac {\sqrt[{4}]{\pi }}{\Gamma \left({\frac {3}{4}}\right)}}{\frac {\left(4+{\sqrt[{4}]{128}}+{\sqrt[{4}]{1024{\sqrt[{4}]{8}}+1024{\sqrt[{4}]{2}}}}\right)}{16}}\\[8pt]2\varphi \left(e^{-20\pi }\right)&=\varphi \left(e^{-5\pi }\right)+{\frac {\sqrt[{4}]{\pi }}{\Gamma \left({\frac {3}{4}}\right)}}{\sqrt {\frac {3+2{\sqrt[{4}]{5}}}{5{\sqrt {2}}}}}\\[8pt]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({\frac {3}{4}}\right)}}{\sqrt[{12}]{2-{\sqrt {3}}}}{\sqrt {1+{\sqrt {3}}+{\sqrt[{4}]{12}}}}\end{aligned}}}

### Equianharmonic values

The mathematician Bruce Berndt found out further values[4] of the theta function:

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

### Further values

Many values of the theta function[5] and especially of the shown phi function can be represented in terms of the gamma function:

${\displaystyle {\begin{array}{lll}\varphi \left(\exp(-{\sqrt {2}}\,\pi )\right)&=&\pi ^{-1/2}\Gamma \left({\tfrac {9}{8}}\right){\Gamma \left({\tfrac {5}{4}}\right)}^{-1/2}2^{7/8}\\\varphi \left(\exp(-2{\sqrt {2}}\,\pi )\right)&=&\pi ^{-1/2}\Gamma \left({\tfrac {9}{8}}\right){\Gamma \left({\tfrac {5}{4}}\right)}^{-1/2}2^{1/8}{\Bigl (}1+{\sqrt {{\sqrt {2}}-1}}{\Bigr )}\\\varphi \left(\exp(-3{\sqrt {2}}\,\pi )\right)&=&\pi ^{-1/2}\Gamma \left({\tfrac {9}{8}}\right){\Gamma \left({\tfrac {5}{4}}\right)}^{-1/2}2^{3/8}3^{-1/2}({\sqrt {3}}+1){\sqrt {\tan({\tfrac {5}{24}}\pi )}}\\\varphi \left(\exp(-4{\sqrt {2}}\,\pi )\right)&=&\pi ^{-1/2}\Gamma \left({\tfrac {9}{8}}\right){\Gamma \left({\tfrac {5}{4}}\right)}^{-1/2}2^{-1/8}{\Bigl (}1+{\sqrt[{4}]{2{\sqrt {2}}-2}}{\Bigr )}\\\varphi \left(\exp(-5{\sqrt {2}}\,\pi )\right)&=&\pi ^{-1/2}\Gamma \left({\tfrac {9}{8}}\right){\Gamma \left({\tfrac {5}{4}}\right)}^{-1/2}2^{15/8}\\&&\cdot {\biggl (}{\tfrac {1}{5}}{\sqrt {3}}\cos({\tfrac {1}{10}}\pi )\operatorname {coth} \left({\tfrac {1}{4}}\operatorname {arcosh} (2)+{\tfrac {1}{6}}\operatorname {artanh} ({\tfrac {1}{9}}{\sqrt {6}})\right)-{\tfrac {1}{5}}({\sqrt {5}}+{\sqrt {2}})\sin({\tfrac {1}{5}}\pi ){\biggr )}\end{array}}}$

## Some series identities

The next two series identities were proved by István Mező:[6]

{\displaystyle {\begin{aligned}\theta _{4}^{2}(q)&=iq^{\frac {1}{4}}\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{aligned}}}

These relations hold for all 0 < q < 1. Specializing the values of q, we have the next parameter free sums

{\displaystyle {\begin{aligned}{\sqrt {\frac {\pi {\sqrt {e^{\pi }}}}{2}}}\cdot {\frac {1}{\Gamma ^{2}\left({\frac {3}{4}}\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),\\[6pt]{\sqrt {\frac {\pi }{2}}}\cdot {\frac {1}{\Gamma ^{2}\left({\frac {3}{4}}\right)}}&=\sum _{k=-\infty }^{\infty }{\frac {\theta _{4}\left(ik\pi ,e^{-\pi }\right)}{e^{2\pi k^{2}}}}\end{aligned}}}

## Zeros of the Jacobi theta functions

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

{\displaystyle {\begin{aligned}\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{aligned}}}

where m, n are arbitrary integers.

## Relation to the Riemann zeta function

The relation

${\displaystyle \vartheta \left(0;-{\frac {1}{\tau }}\right)=\left(-i\tau \right)^{\frac {1}{2}}\vartheta (0;\tau )}$

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

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

which can be shown to be invariant under substitution of s by 1 − s. The corresponding integral for 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) 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

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

where the second derivative is with respect to z and the constant c is defined so that the Laurent expansion of ℘(z) at 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 Jackson q-gamma function via the relation[7]

${\displaystyle \left(\Gamma _{q^{2}}(x)\Gamma _{q^{2}}(1-x)\right)^{-1}={\frac {q^{2x(1-x)}}{\left(q^{-2};q^{-2}\right)_{\infty }^{3}\left(q^{2}-1\right)}}\theta _{4}\left({\frac {1}{2i}}(1-2x)\log q,{\frac {1}{q}}\right).}$

## Relations to Dedekind eta function

Let η(τ) be the Dedekind eta function, and the argument of the theta function as the nome q = eπiτ. Then,

{\displaystyle {\begin{aligned}\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{aligned}}}

and,

${\displaystyle \theta _{2}(q)\,\theta _{3}(q)\,\theta _{4}(q)=2\eta ^{3}(\tau ).}$

## Elliptic modulus

The elliptic modulus is

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

and the complementary elliptic modulus is

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

## 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.[8] Taking z = x to be real and τ = it with t real and positive, we can write

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

which solves the heat equation

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

This theta-function solution is 1-periodic in x, and as t → 0 it approaches the periodic delta function, or Dirac comb, in the sense of distributions

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

General solutions of the spatially periodic initial value problem for the heat equation may be obtained by convolving the initial data at 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 F is a quadratic form in n variables, then the theta function associated with F is

${\displaystyle \theta _{F}(z)=\sum _{m\in \mathbb {Z} ^{n}}e^{2\pi izF(m)}}$

with the sum extending over the lattice of integers ${\displaystyle \mathbb {Z} ^{n}}$. This theta function is a modular form of weight n/2 (on an appropriately defined subgroup) of the modular group. In the Fourier expansion,

${\displaystyle {\hat {\theta }}_{F}(z)=\sum _{k=0}^{\infty }R_{F}(k)e^{2\pi ikz},}$

the numbers RF(k) are called the representation numbers of the form.

### Theta series of a Dirichlet character

For χ a primitive Dirichlet character modulo q and ν = 1 − χ(−1)/2 then

${\displaystyle \theta _{\chi }(z)={\frac {1}{2}}\sum _{n=-\infty }^{\infty }\chi (n)n^{\nu }e^{2i\pi n^{2}z}}$

is a weight 1/2 + ν modular form of level 4q2 and character

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

which means[9]

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

whenever

${\displaystyle a,b,c,d\in \mathbb {Z} ^{4},ad-bc=1,c\equiv 0{\bmod {4}}q^{2}.}$

### Riemann theta function

Let

${\displaystyle \mathbb {H} _{n}=\left\{F\in M(n,\mathbb {C} )\,{\big |}\,F=F^{\mathsf {T}}\,,\,\operatorname {Im} F>0\right\}}$

the set of symmetric square matrices whose imaginary part is positive definite. ${\displaystyle \mathbb {H} _{n}}$ is called the Siegel upper half-space and is the multi-dimensional analog of the upper half-plane. The n-dimensional analogue of the modular group is the symplectic group Sp(2n,${\displaystyle \mathbb {Z} }$); for n = 1, Sp(2,${\displaystyle \mathbb {Z} }$) = SL(2,${\displaystyle \mathbb {Z} }$). The n-dimensional analogue of the congruence subgroups is played by

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

Then, given τ${\displaystyle \mathbb {H} _{n}}$, the Riemann theta function is defined as

${\displaystyle \theta (z,\tau )=\sum _{m\in \mathbb {Z} ^{n}}\exp \left(2\pi i\left({\tfrac {1}{2}}m^{\mathsf {T}}\tau m+m^{\mathsf {T}}z\right)\right).}$

Here, z${\displaystyle \mathbb {C} ^{n}}$ is an n-dimensional complex vector, and the superscript T denotes the transpose. The Jacobi theta function is then a special case, with n = 1 and τ${\displaystyle \mathbb {H} }$ where ${\displaystyle \mathbb {H} }$ 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 Riemann surfaces, as well as other auxiliary objects that figure prominently in their function theory, by taking τ 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 ${\displaystyle \mathbb {C} ^{n}\times \mathbb {H} _{n}}$.

The functional equation is

${\displaystyle \theta (z+a+\tau b,\tau )=\exp 2\pi i\left(-b^{\mathsf {T}}z-{\tfrac {1}{2}}b^{\mathsf {T}}\tau b\right)\theta (z,\tau )}$

which holds for all vectors a, b${\displaystyle \mathbb {Z} ^{n}}$, and for all z${\displaystyle \mathbb {C} ^{n}}$ and τ${\displaystyle \mathbb {H} _{n}}$.

### Poincaré series

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

### Generalized theta functions

There are in general higher-order non-quadratic theta functions. They have the form[10]

${\displaystyle \theta _{\chi }^{\{\kappa \}}(z)=\sum _{n=1}^{\infty }\chi (n)q^{n^{\kappa }},}$

where q = e2πiz. The variable z lies in the upper half plane, χ(n) is any arithmetical function and κ is an integer greater than 1.

The restriction κ = 2 and

${\displaystyle \chi (n)\rightarrow {\frac {1}{2}}\chi (n)n^{\nu },}$

with χ a Dirichlet character is related to the classically theta series of a Dirichlet character θχ(z). Some properties are

${\displaystyle \exp \left(\theta _{\chi }^{\{\kappa \}}(z)\right)=e^{q\chi (1)}\prod _{n=1}^{\infty }\left(1-q^{n}\right)^{-\chi {\bigl (}n_{\kappa }(n){\bigr )}\epsilon {\bigl (}n_{\kappa }^{*}(n){\bigr )}}.}$

Here ε(n) = μ(n)/n when n ≠ 0, and zero otherwise. The arithmetical function μ(n) is the Moebius μ function. The arithmetical functions nκ(n) and n
κ
(n)
are evaluated from the prime factorization of n by decomposing it into a power of κ and its κ-free part. This is as follows:

{\displaystyle {\begin{aligned}n&=p_{1}^{a_{1}}p_{2}^{a_{2}}\ldots p_{t}^{a_{t}}\\[3pt]&=\left(p_{k_{1}}^{b_{k_{1}}}p_{k_{2}}^{b_{k_{2}}}\ldots p_{k_{s}}^{b_{k_{s}}}\right)^{\kappa }p_{j_{1}}^{c_{j_{1}}}p_{j_{2}}^{c_{j_{2}}}\ldots p_{j_{r}}^{c_{j_{r}}}.\end{aligned}}}

Here, bk1, bk2,..., bks and cj1, cj2,..., cjs are positive integers with all cj < κ. This decomposition is unique and by convection we set

${\displaystyle n_{\kappa }(n)={\begin{cases}0&{\text{if }}p_{k_{1}}^{b_{k_{1}}}p_{k_{2}}^{b_{k_{2}}}\ldots p_{k_{s}}^{b_{k_{s}}}=1,\\[3pt]p_{k_{1}}^{b_{k_{1}}}p_{k_{2}}^{b_{k_{2}}}\ldots p_{k_{s}}^{b_{k_{s}}}&{\text{otherwise}}.\end{cases}}}$

The function n
κ
(n)
can also evaluated as

${\displaystyle n_{\kappa }^{*}(n)={\frac {n}{{\bigl (}n_{\kappa }(n){\bigr )}^{\kappa }}}}$

when nκ(n) ≠ 0. Setting

${\displaystyle C_{\kappa }(\chi ;n):=\mu (n)+\chi {\bigl (}n_{\kappa }(n){\bigr )}\mu \left(n_{\kappa }^{*}(n)\right),}$

then Cκ(χ;n) is multiplicative whenever χ(n) is multiplicative.

Another property is

${\displaystyle \exp \left(2\pi i\int _{z}^{i\infty }\theta _{\chi }^{\{\kappa \}}(w)\,\mathrm {d} w\right)=e^{{\bigl (}1-\chi (1){\bigr )}q}\prod _{n=1}^{\infty }\left(1-q^{n}\right)^{\frac {C_{\kappa }(\chi ;n)}{n}}.}$

### Theta function coefficients

If a and b are positive integers, χ(n) any arithmetical function and |q| < 1, then[10]

${\displaystyle \sum _{n=1}^{\infty }\chi (n)q^{an^{2}+bn}=\sum _{n=1}^{\infty }q^{n}\sum _{\stackrel {d|n}{ad^{2}+bd=n}}\chi (d).}$

The general case, where f(n) and χ(n) are any arithmetical functions, and f(n) : ${\displaystyle \mathbb {N} }$${\displaystyle \mathbb {N} }$ is strictly increasing with f(0) = 0, is[10]

${\displaystyle \sum _{n=1}^{\infty }\chi (n)q^{f(n)}=\sum _{n=1}^{\infty }q^{n}\sum _{d|n}\sum _{f(\delta )|d}\chi (\delta )\mu \left({\frac {d}{f(\delta )}}\right).}$

## Notes

1. ^ See e.g. https://dlmf.nist.gov/20.1. Note that this is, in general, not equivalent to the usual interpretation ${\displaystyle (e^{z})^{\alpha }=e^{\alpha \operatorname {Log} e^{z}}}$ when ${\displaystyle z}$ is outside the strip ${\displaystyle -\pi <\operatorname {Im} z\leq \pi }$. Here, ${\displaystyle \operatorname {Log} }$ denotes the principal branch of the complex logarithm.
2. ^ ${\displaystyle \theta _{1}(q)=0}$ for all ${\displaystyle q\in \mathbb {C} }$ with ${\displaystyle |q|<1}$.

## References

• Abramowitz, Milton; Stegun, Irene A. (1964). Handbook of Mathematical Functions. New York: Dover Publications. sec. 16.27ff. ISBN 978-0-486-61272-0.
• Akhiezer, Naum Illyich (1990) [1970]. Elements of the Theory of Elliptic Functions. AMS Translations of Mathematical Monographs. Vol. 79. Providence, RI: AMS. ISBN 978-0-8218-4532-5.
• Farkas, Hershel M.; Kra, Irwin (1980). Riemann Surfaces. New York: Springer-Verlag. ch. 6. ISBN 978-0-387-90465-8.. (for treatment of the Riemann theta)
• Hardy, G. H.; Wright, E. M. (1959). An Introduction to the Theory of Numbers (4th ed.). Oxford: Clarendon Press.
• Mumford, David (1983). Tata Lectures on Theta I. Boston: Birkhauser. ISBN 978-3-7643-3109-2.
• Pierpont, James (1959). Functions of a Complex Variable. New York: Dover Publications.
• Rauch, Harry E.; Farkas, Hershel M. (1974). Theta Functions with Applications to Riemann Surfaces. Baltimore: Williams & Wilkins. ISBN 978-0-683-07196-2.
• Reinhardt, William P.; Walker, Peter L. (2010), "Theta Functions", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, MR 2723248
• Whittaker, E. T.; Watson, G. N. (1927). A Course in Modern Analysis (4th ed.). Cambridge: Cambridge University Press. ch. 21. (history of Jacobi's θ functions)