theta functions

In mathematics, theta functions are special functions of several complex variables. They show up in many topics, including Abelian varieties, moduli spaces, quadratic forms, and solitons. As Grassmann algebras, they appear in quantum field theory.

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 cohomology class of a line bundle on 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, $(e^)^$ should be interpreted as $e^$ (in order to resolve issues of choice of branch).See e.g. https://dlmf.nist.gov/20.1. Note that this is, in general, not equivalent to the usual interpretation $(e^z)^\alpha=e^$ when $z$ is outside the strip $-\pi<\operatornamez\le\pi$. Here, $\operatorname$ denotes the principal branch of the complex logarithm.

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 positive imaginary part. It is given by the formula

:$\begin \vartheta(z; \tau) &= \sum_^\infty \exp \left(\pi i n^2 \tau + 2 \pi i n z\right) \\ &= 1 + 2 \sum_^\infty q^ \cos(2\pi n z) \\ &= \sum_^\infty q^\eta^n \end$

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 :

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

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

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

Thus, in general,

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

for any integers and .

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

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:

:$\vartheta_(z;\tau) = \vartheta(z;\tau)$

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

:\begin
\vartheta_(z;\tau)& = \vartheta \left(z+\tfrac12;\tau\right)\\ pt\vartheta_(z;\tau)& = \exp\left(\tfrac14\pi i \tau + \pi i z\right)\vartheta\left(z + \tfrac12\tau;\tau\right)\\ pt\vartheta_(z;\tau)& = \exp\left(\tfrac14\pi i \tau + \pi i\left(z+\tfrac12\right)\right)\vartheta\left(z+\tfrac12\tau + \tfrac12;\tau\right).
\end

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

:$\begin \theta_1(z;q) &=\theta_1(\pi z,q)= -\vartheta_(z;\tau)\\ \theta_2(z;q) &=\theta_2(\pi z,q)= \vartheta_(z;\tau)\\ \theta_3(z;q) &=\theta_3(\pi z,q)= \vartheta_(z;\tau)\\ \theta_4(z;q) &=\theta_4(\pi z,q)= \vartheta_(z;\tau) \end$

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. Alternatively, we obtain four functions of only, defined on the unit disk $, q, <1$. They are sometimes called theta constants:$\theta_1(q)=0$ for all $q\in\mathbb$ with $, q, <1$.

:$\begin \vartheta_(0;\tau)&=-\theta_1(q)=-\sum_^\infty (-1)^q^ \\ \vartheta_(0;\tau)&=\theta_2(q)=\sum_^\infty q^\\ \vartheta_(0;\tau)&=\theta_3(q)=\sum_^\infty q^\\ \vartheta_(0;\tau)&=\theta_4(q)=\sum_^\infty (-1)^n q^ \end$

with the nome .
Observe that $\theta_1(q)=0$.
These can be used to define a variety of modular forms, and to parametrize certain curves; in particular, the Jacobi identity is

:$\theta_2(q)^4 + \theta_4(q)^4 = \theta_3(q)^4$

or equivalently,

:$\vartheta_(0;\tau)^4 + \vartheta_(0;\tau)^4 = \vartheta_(0;\tau)^4$

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 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

:$\alpha = (-i \tau)^\frac12 \exp\left(\frac i z^2 \right).$

Then

:\begin
\vartheta_\!\left(\frac; \frac\right)& = \alpha\,\vartheta_(z; \tau)\quad&
\vartheta_\!\left(\frac; \frac\right)& = \alpha\,\vartheta_(z; \tau)\\ pt\vartheta_\!\left(\frac; \frac\right)& = \alpha\,\vartheta_(z; \tau)\quad&
\vartheta_\!\left(\frac; \frac\right)& = -i\alpha\,\vartheta_(z; \tau).
\end

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

:\begin
\vartheta_(w, q)& = \sum_^\infty \left(w^2\right)^n q^\quad&
\vartheta_(w, q)& = \sum_^\infty (-1)^n \left(w^2\right)^n q^\\ pt\vartheta_(w, q)& = \sum_^\infty \left(w^2\right)^ q^\quad&
\vartheta_(w, q)& = i \sum_^\infty (-1)^n \left(w^2\right)^ q^.
\end

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 -adic numbers.

Product representations

The Jacobi triple product (a special case of the Macdonald identities) tells us that for complex numbers and with and we have
:$\prod_^\infty \left( 1 - q^\right) \left( 1 + w^2 q^\right) \left( 1 + w^q^\right) = \sum_^\infty w^q^.$

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

:$\vartheta(z; \tau) = \sum_^\infty \exp(\pi i \tau n^2) \exp(2\pi i z n) = \sum_^\infty w^q^.$

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

:$\vartheta(z; \tau) = \prod_^\infty \big( 1 - \exp(2m \pi i \tau)\big) \Big( 1 + \exp\big((2m-1) \pi i \tau + 2 \pi i z\big)\Big) \Big( 1 + \exp\big((2m-1) \pi i \tau - 2 \pi i z\big)\Big).$

In terms of and :
:$\begin \vartheta(z; \tau) &= \prod_^\infty \left( 1 - q^\right) \left( 1 + q^w^2\right) \left( 1 + \frac\right) \\ &= \left(q^2;q^2\right)_\infty\,\left(-w^2q;q^2\right)_\infty\,\left(-\frac;q^2\right)_\infty \\ &= \left(q^2;q^2\right)_\infty\,\theta\left(-w^2q;q^2\right) \end$

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

:$\prod_^\infty \left( 1 - q^\right) \Big( 1 + \left(w^2+w^\right)q^+q^\Big),$

which we may also write as

:$\vartheta(z\mid q) = \prod_^\infty \left( 1 - q^\right) \left( 1 + 2 \cos(2 \pi z)q^+q^\right).$

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

:\begin
\vartheta_(z\mid q) &= \prod_^\infty \left( 1 - q^\right) \left( 1 - 2 \cos(2 \pi z)q^+q^\right),\\ pt\vartheta_(z\mid q) &= 2 q^\frac14\cos(\pi z)\prod_^\infty \left( 1 - q^\right) \left( 1 + 2 \cos(2 \pi z)q^+q^\right),\\ pt\vartheta_(z\mid q) &= -2 q^\frac14\sin(\pi z)\prod_^\infty \left( 1 - q^\right)\left( 1 - 2 \cos(2 \pi z)q^+q^\right).
\end
In particular, $\lim_\frac = \cos(\pi z),\quad \lim_\frac = \sin(\pi z)$so we may interpret them as one-parameter deformations of the periodic functions $\sin, \cos$, 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:

:\begin
\vartheta_ (z; \tau) &= -i\int_^ e^ \frac \mathrmu; \\ pt\vartheta_ (z; \tau) &= -i\int_^ e^ \frac \mathrmu; \\ pt\vartheta_ (z; \tau) &= -ie^ \int_^ e^ \frac \mathrmu; \\ pt\vartheta_ (z; \tau) &= e^ \int_^ e^ \frac \mathrmu.
\end

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). Define,

:$\quad \varphi(q) =\vartheta_(0;\tau) =\theta_3(0;q)=\sum_^\infty q^$

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

:\begin
\varphi\left(e^ \right) &= \frac = \sqrt2\,\eta\left(\sqrt\right)\\
\varphi\left(e^\right) &= \frac \frac\\
\varphi\left(e^\right) &= \frac \frac\\
\varphi\left(e^\right) &= \frac \frac\\
\varphi\left(e^\right) &= \frac \sqrt\\
\varphi\left(e^\right) &= \frac \frac\\
\varphi\left(e^\right) &= \frac \frac\\
\varphi\left(e^\right) &= \frac \frac\\
\varphi\left(e^\right) &= \frac \frac\\
\varphi\left(e^\right) &= \frac \frac\\
\varphi\left(e^\right) &= \frac \frac\\
\varphi\left(e^\right) &= \frac \frac\\
\varphi\left(e^\right) &= \frac \sqrt\\
\varphi\left(e^\right) &= \frac \frac\\
\varphi\left(e^\right) &= \frac \frac\\
2\varphi\left(e^\right) &= \varphi\left(e^\right) + \frac \frac\\
\varphi\left(e^\right) &= \frac \frac\\
2\varphi\left(e^\right) &= \varphi\left(e^\right) + \frac \sqrt\\
6\varphi\left(e^\right) &= 3\varphi\left(e^\right) + 2\varphi\left(e^\right) - \varphi\left(e^\right) + \frac \sqrt \end

Note that the following modular identities hold:

:\begin
2\varphi\left(q^4\right) &= \varphi(q)+\sqrt\\
3\varphi\left(q^9\right) &= \varphi(q)+\sqrt \end

Equianharmonic values

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

:\begin
\varphi\left(\exp( -\sqrt\,\pi)\right) &=& \pi^^2^3^ \\
\varphi\left(\exp(-2\sqrt\,\pi)\right) &=& \pi^^2^3^\cos(\tfrac\pi) \\
\varphi\left(\exp(-3\sqrt\,\pi)\right) &=& \pi^^2^3^(\sqrt 1) \\
\varphi\left(\exp(-4\sqrt\,\pi)\right) &=& \pi^^2^3^\Bigl(1+\sqrt\Bigr) \\
\varphi\left(\exp(-5\sqrt\,\pi)\right) &=& \pi^^2^3^\sin(\tfrac\pi)(\tfrac\sqrt \tfrac\sqrt \tfrac\sqrt+1)
\end

Further values

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

:\begin
\varphi\left(\exp( -\sqrt\,\pi)\right) &=& \pi^\Gamma\left(\tfrac\right)^2^ \\
\varphi\left(\exp(-2\sqrt\,\pi)\right) &=& \pi^\Gamma\left(\tfrac\right)^2^\Bigl(1+\sqrt\Bigr) \\
\varphi\left(\exp(-3\sqrt\,\pi)\right) &=& \pi^\Gamma\left(\tfrac\right)^2^3^(\sqrt+1)\sqrt \\
\varphi\left(\exp(-4\sqrt\,\pi)\right) &=& \pi^\Gamma\left(\tfrac\right)^2^\Bigl(1+\sqrt Bigr) \\
\varphi\left(\exp(-5\sqrt\,\pi)\right) &=& \pi^\Gamma\left(\tfrac\right)^2^ \\
&& \cdot \biggl(\tfrac\sqrt\cos(\tfrac\pi)\dfrac
-\tfrac(\sqrt+\sqrt)\sin(\tfrac\pi)\biggr) \\
\varphi\left(\exp( -\sqrt\,\pi)\right) &=& \pi^\Gamma\left(\tfrac\right)^2^3^\sqrt \\
\varphi\left(\exp(-\tfrac\sqrt\,\pi)\right) &=& \pi^\Gamma\left(\tfrac\right)^2^3^\sin(\tfrac\pi)
\end

Some series identities

The next two series identities were proved by István Mező:

:\begin
\theta_4^2(q)&=iq^\sum_^\infty q^\theta_1\left(\frac\ln q,q\right),\\ pt\theta_4^2(q)&=\sum_^\infty q^\theta_4\left(\frac,q\right).
\end

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

:\begin
\sqrt\cdot\frac&=i\sum_^\infty e^ \theta_1 \left(\frac(2k-1),e^\right),\\ pt\sqrt\cdot\frac&=\sum_^\infty\frac
\end

Zeros of the Jacobi theta functions

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

:\begin
\vartheta(z;\tau) = \vartheta_(z;\tau) &= 0 \quad &\Longleftrightarrow&& \quad z &= m + n \tau + \frac + \frac
\\ pt\vartheta_(z;\tau) &= 0 \quad &\Longleftrightarrow&& \quad z &= m + n \tau
\\ pt\vartheta_(z;\tau) &= 0 \quad &\Longleftrightarrow&& \quad z &= m + n \tau + \frac
\\ pt\vartheta_(z;\tau) &= 0 \quad &\Longleftrightarrow&& \quad z &= m + n \tau + \frac
\end
where , are arbitrary integers.

Relation to the Riemann zeta function

The relation

:$\vartheta\left(0;-\frac\right)=\left(-i\tau\right)^\frac12 \vartheta(0;\tau)$

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

:$\Gamma\left(\frac\right) \pi^ \zeta(s) = \frac\int_0^\infty\bigl(\vartheta(0;it)-1\bigr)t^\frac\frac$

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.

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

:$\wp(z;\tau) = -\big(\log \vartheta_(z;\tau)\big)'' + c$

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

Relation to the ''q''-gamma function

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

:$\left(\Gamma_(x)\Gamma_(1-x)\right)^=\frac \theta_4\left(\frac(1-2x)\log q,\frac\right).$

Relations to Dedekind eta function

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

:\begin
\theta_2(q) = \vartheta_(0;\tau) &= \frac,\\ pt\theta_3(q) = \vartheta_(0;\tau) &= \frac = \frac, \\ pt\theta_4(q) = \vartheta_(0;\tau) &= \frac,
\end

and,

:$\theta_2(q)\,\theta_3(q)\,\theta_4(q) = 2\eta^3(\tau).$

See also the Weber modular functions.

Elliptic modulus

The elliptic modulus is
:$k(\tau) = \frac$
and the complementary elliptic modulus is
:$k'(\tau) = \frac$

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

:$\vartheta (x;it)=1+2\sum_^\infty \exp\left(-\pi n^2 t\right) \cos(2\pi nx)$

which solves the heat equation

:$\frac \vartheta(x;it)=\frac \frac \vartheta(x;it).$

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

:$\lim_ \vartheta(x;it)=\sum_^\infty \delta(x-n)$.

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

:$\theta_F (z)= \sum_ e^$

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

:$\hat_F (z) = \sum_^\infty R_F(k) e^,$

the numbers are called the ''representation numbers'' of the form.

Theta series of a Dirichlet character

For a primitive Dirichlet character modulo and then

:$\theta_\chi(z) = \frac12\sum_^\infty \chi(n) n^\nu e^$

is a weight modular form of level and character
:$\chi(d) \left(\frac\right)^\nu,$
which meansShimura, On modular forms of half integral weight

:$\theta_\chi\left(\frac\right) = \chi(d) \left(\frac\right)^\nu \left(\frac\right)^\theta_\chi(z)$

whenever

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

Ramanujan theta function

Riemann theta function

Let

:$\mathbb_n=\left\$

the set of symmetric square matrices whose imaginary part is positive definite. $\mathbb_n$ 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 ; for , . The -dimensional analogue of the congruence subgroups is played by

:$\ker \big\.$

Then, given , the Riemann theta function is defined as

:$\theta (z,\tau)=\sum_ \exp\left(2\pi i \left(\tfrac12 m^\mathsf \tau m +m^\mathsf z \right)\right).$

Here, is an -dimensional complex vector, and the superscript T denotes the transpose. The Jacobi theta function is then a special case, with and where 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 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 $\mathbb^n \times \mathbb_n$.

The functional equation is

:$\theta (z+a+\tau b, \tau) = \exp 2\pi i \left(-b^\mathsfz-\tfrac12 b^\mathsf\tau b\right) \theta (z,\tau)$

which holds for all vectors , and for all and .

Poincaré series

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

Theta function coefficients

If and are positive integers, any arithmetical function and , then

:$\sum_^\infty \chi(n)q^ = \sum_^\infty q^n \sum_ \chi(d).$

The general case, where and are any arithmetical functions, and is strictly increasing with , is
:$\sum_^\infty \chi(n)q^ = \sum_^\infty q^n \sum_ \sum_ \chi(\delta)\mu\left(\frac\right).$

Notes

References

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*. ''(for treatment of the Riemann theta)''
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* ''(history of Jacobi's functions)''

Further reading

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

External links

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Elliptic functions
Riemann surfaces
Analytic functions
Several complex variables