I'm looking for pithy examples of integrals of the form \begin{align*} \int_{\Omega \subset \mathbb{R}^{2}} f(\mathbf{x}) \, \mathrm{d}x \end{align*} where $f \colon \Omega \to \mathbb{C}$, to demonstrate high-dimensional numerical quadrature.
Of course quantum mechanics is the intended use, but I'd rather have something more explicit. In 1D, a great example is the complex Bessel function \begin{align*} J_{n}(z) := \frac{1}{\pi} \int_{0}^{\pi} \cos(n\tau - z\sin(\tau)\, \mathrm{d}\tau \end{align*}