Torus integral

Remarks

The factor $i r_j e^{i\theta_j} \, d\theta_j$ is $d\zeta_j$ along the circle $\zeta_j = c_j + r_j e^{i\theta_j}$, so for $n = 1$ this is the contour integral over the positively oriented circle. For continuous $f$, Fubini's theorem makes it equal to the iterated integral of $n$ one-variable contour integrals, in any order. Example: for $f(\zeta) = 1/(\zeta_1 \zeta_2)$ on $T(0, (1, 1))$, the integrand is $e^{-i\theta_1} e^{-i\theta_2} \cdot i e^{i\theta_1} \cdot i e^{i\theta_2} = -1$, so the integral is $-(2\pi)^2 = (2\pi i)^2$. Non-example: $f(\zeta) = 1/(\zeta_1 - 1)$ is not defined at the points of $T(0, (1, 1))$ with $\zeta_1 = 1$. Near $\theta_1 = 0$, $\abs{f(\gamma(\theta))}$ behaves like $1/\abs{\theta_1}$, so it is not integrable and the torus integral is not defined. Other notation: $\oint_{T(c, r)} f(\zeta) \, d\zeta$.