What are some of your favorite integrals /sci/?

Here is one of mine. Let $G = (V,E)$ be a finite connected graph, and let $f: \mathbb{R}^{|V|} \times \mathbb{R}^{|E|} /rightarrow \mathbb{\R}$ be a smooth function, compactly supported on its restriction to $\mathbb{\R}^V$. We write $f(t,\tilde{t})$ to denote $f(t_1, \dots, t_{|V|}, \tilde{t}_{1}, \dots, \tilde{t}_{|E|})$ this function Cartesian coordinates.


Call a subset [/math] (R,F) \subseteq (V,E) [/math] a rooted forest if the set of edges $F$ contains no cycles, and there is exactly one vertex in $R$ for each connected component of the graph induced by $F$, plus all singleton vertices (see pic).

Let $f^{(R,F)}(t,\tilde{t})$ denote the derivative of $f(t)$ with respect to the variables $t_{i}, \tilde{t}_{ij}$ for all $i \in R$ and $ij \in E$.

Finally, let $z_i = x_i + iy_i, 1 \le i \le |V|$ denote a collection of complex variables, indexed by $V[\math], and set <script type="math/tex"> t_i = |z_i|^2 <script type="math/tex"> and <script type="math/tex"> \tilde{t}_{ij} = |z_i - z_j|^2$. Then:

$f(0,0) = (2\pi)^{-|V|}\sum_{(R,F)} \int_{\mathbb{C}^{|V|} f^{(R,F)} (t,\tilde{t}) \frac{dx_1dy_1\dots dx_{|V|}dy_{|V|},$

where the sum is over all rooted forests on $G$. Lovely isn't it?