In mathematics, theta functions are special functions of several complex variables. Fundamentally, they are a family of continuous functions which encode the behavior of discrete multi-dimensional periodic systems, such as crystal lattices or points on a torus. Because they are smooth, they allow the study and manipulation of discrete combinatorial systems using the tools of 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 curves?" and others, including abelian varieties, moduli spaces, quadratic forms, and solitons. Theta functions in two dimensions are functions of two complex arguments. In one choice of parameter, for example, z {\displaystyle z} encodes position on a two-dimensional lattice, and τ {\displaystyle \tau } or q {\displaystyle 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.
Basic example One example of a theta function is
θ ( z , q ) ≡ ∑ n = − ∞ ∞ q n 2 exp ( 2 π i n z ) , {\displaystyle \theta (z,q)\equiv \sum _{n=-\infty }^{\infty }q^{n^{2}}\exp {(2\pi inz)},}
where z {\displaystyle z} and q {\displaystyle q} are complex numbers and | q | < 1 {\displaystyle |q|<1} so that the sum converges. This analytic function can be used to solve a combinatorics problem: in how many different ways can an integer be written as the sum of two squares? When z = 0 {\displaystyle z=0} , we have
θ ( 0 , q ) = ∑ n = − ∞ ∞ q n 2 = 1 + 2 q + 2 q 4 + 2 q 9 + … + 2 q n 2 + … {\displaystyle \theta (0,q)=\sum _{n=-\infty }^{\infty }q^{n^{2}}=1+2q+2q^{4}+2q^{9}+\ldots +2q^{n^{2}}+\ldots }
This is a generating function where the coefficient of q k {\displaystyle q^{k}} represents how many ways there are to write k {\displaystyle k} as a perfect square: when k = 0 {\displaystyle k=0} , there is just one way. When k {\displaystyle k} is any other perfect square, there are two ways: n 2 = ( − n ) 2 {\displaystyle n^{2}=(-n)^{2}} . When k {\displaystyle k} is not a perfect square, there are zero ways. Squaring this generating function, we obtain
θ ( 0 , q ) 2 = ( ∑ m q m 2 ) ( ∑ n q n 2 ) = ∑ m , n q m 2 + n 2 . {\displaystyle \theta (0,q)^{2}={\Bigl (}\sum _{m}q^{m^{2}}{\Bigr )}{\Bigl (}\sum _{n}q^{n^{2}}{\Bigr )}=\sum _{m,n}q^{m^{2}+n^{2}}.}
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