---
tag: R49Q9M519Z
kind: concept
name: Torus integral
msc: [32A26, 32A10]
sources:
  - url: https://en.wikipedia.org/wiki/Cauchy%27s_integral_formula
---

## Summary

The torus integral of a function $f$ on $\C^n$ is the iterated contour integral of $f(\zeta) \, d\zeta_1 \cdots d\zeta_n$ over a product of $n$ circles.
This product is the [[distinguished boundary|A6HN207Y2T]] of a [[polydisc|DJBWP7D3M8]].
It is computed by parametrising each circle by an angle and integrating over the cube $[0, 2\pi]^n$.
It appears in the [[Cauchy integral formula on a polydisc|FG5448NZ18]].
It also gives the coefficients of a [[power series in several variables|R3QTCPTJK7]].
For $n = 1$ it is the [[contour integral|06ZGPYTWK8]] over a circle.

## Explanations

### L1 human

In one variable, one integrates a function around a circle by walking around the circle once.
In $n$ variables, each coordinate $\zeta_j$ walks around its own circle, independently of the others.
The points visited form a torus, like the surface of a doughnut when $n = 2$.
For example, the torus integral of $1/(\zeta_1 \zeta_2)$ over the product of two unit circles is $(2\pi i)^2$, the product of two one-variable integrals equal to $2\pi i$.