---
tag: XM695ZZ4FJ
kind: statement
role: theorem
name: Theta transformation formula
msc: [11F27, "33E05"]
uses: [2F5CYQJVAF, X6XGGH5FPW, GARGJYSGFG]
about: [2F5CYQJVAF]
sources:
  - url: https://en.wikipedia.org/wiki/Theta_function
---

## Statement

Let $\mathbb{H}$ be the [[upper half-plane|X6XGGH5FPW]], and let $\vartheta(z, \tau) = \sum_{n \in \Z} \exp(\pi i n^2\tau + 2\pi i n z)$ be the Jacobi [[theta function|2F5CYQJVAF]] for $z \in \C$ and $\tau \in \mathbb{H}$.
Then for all $z \in \C$ and $\tau \in \mathbb{H}$:

1. $\vartheta(z, \tau + 1) = \vartheta(z + \tfrac12, \tau)$;
2. $$\vartheta\!\left(\frac{z}{\tau}, -\frac{1}{\tau}\right) = \sqrt{-i\tau}\;\exp\!\left(\frac{\pi i z^2}{\tau}\right)\vartheta(z, \tau),$$
where $\sqrt{-i\tau}$ is the principal branch, which is positive when $\tau$ is on the positive imaginary axis.

## Remarks

Part 1 holds term by term, because $e^{\pi i n^2} = (-1)^n = e^{2\pi i n/2}$.
Proof idea for part 2: for fixed $\tau$, apply the [[Poisson summation formula|GARGJYSGFG]] to the Gaussian $x \mapsto \exp(\pi i x^2\tau + 2\pi i x z)$, whose Fourier transform is again a Gaussian, with $-1/\tau$ in place of $\tau$.
Both sides are holomorphic in $\tau$, so it is enough to check $\tau = it$ with $t > 0$ and continue analytically.
Example: at $z = 0$ and $\tau = it$ the formula reads $\sum_{n} e^{-\pi n^2/t} = \sqrt{t}\sum_n e^{-\pi n^2 t}$, the classical identity behind the functional equation of the Riemann zeta function.
The hypothesis $\tau \in \mathbb{H}$ is needed: for real $\tau$ the series does not converge.

## Explanations

### L1 human

The theta function is a sum of Gaussian terms.
Replacing $\tau$ by $-1/\tau$ turns narrow Gaussians into wide ones, and the formula says the sum changes only by a simple explicit factor.
Together with the shift $\tau \mapsto \tau + 1$ this makes theta a modular object.

### L2 human

The maps $\tau \mapsto \tau + 1$ and $\tau \mapsto -1/\tau$ generate the [[modular group|VEKV1EM77B]], so the formula describes how $\vartheta$ behaves under it.
The theta constant $\vartheta(0, \tau)$ is invariant under $\tau \mapsto \tau + 2$ and picks up the factor $\sqrt{-i\tau}$ under $\tau \mapsto -1/\tau$.
So it is a [[modular form|GWKN5VG78A]] of weight $1/2$ for the subgroup generated by these two maps.
Raising to the fourth power and using part 2 gives the transformation law used in counting sums of four squares.
In the integral $\int_0^\infty (\vartheta(0, it) - 1)\,t^{s/2 - 1}\,dt$, the formula of the example yields Riemann's proof of the functional equation of $\zeta(s)$.