= Theta function
{title2=$\Theta(z,\tau)$}
{wiki}
A <theta function> is formed by summing an exponential quadratic expression over a <Euclidean lattice>. A basic one-dimensional example, in period-one spatial normalization, is
$$
\Theta(z,\tau)=\sum_{n\in\mathbb Z}e^{\pi i n^2\tau+2\pi i nz},\qquad\operatorname{Im}\tau>0.
$$
<NIST Digital Library of Mathematical Functions>[Section 20.2.3: theta3 series, with spatial variable rescaled by pi]{ref}
Gaussian decay gives local uniform convergence, so it is an <entire function> of $z$ and a <holomorphic function> of $\tau$. Its value at $z=0$ is the <Jacobi theta function> in the $Q=e^{\pi i\tau}$ convention:
$$
\Theta(0,\tau)=1+2Q+2Q^4+2Q^9+\cdots.
$$
The <theta function with characteristics> shifts the summation lattice and its phase; evaluating at $z=0$ gives a <theta constant>.
These sums translate geometric counting into analytic functions. The coefficient of $Q^n$ in $\Theta(0,\tau)^d$ counts ordered <integer> $d$-tuples with sum of squares $n$. For instance, the coefficient of $Q$ in the square is $4$, from $(\pm1,0)$ and $(0,\pm1)$. <Poisson summation> relates the sum over a <Euclidean lattice> to the sum over its <dual lattice>, producing the <lattice theta functional equation>. Termwise differentiation also gives the <heat equation for theta functions with characteristics>. These connections explain the role of <theta functions> in <modular forms>, <Euclidean lattice> geometry, and diffusion.
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