Let $T,\Delta$ be two constant coefficient differential (scalar valued) operators on $\mathbb{R}^n$ where $T$ is of order $q$ and $\Delta$ of order $2$. We assume that $T$ and $\Delta$ are homogenous (so in the expressions for these operators all derivatives are of the same order) and $\Delta$ is positive. Consider the integral $$\int_{\mathbb{R}^n}\sigma(T)\sigma(I+\Delta)^{-s}d\xi$$ where $\sigma$ is total symbol (insert $i\xi_k$ in place of $\frac{\partial}{\partial x_k}$). Here $s$ is such that $Re(s)>\frac{n+q}{2}$
How to show that the residuum of this integral for $s=\frac{n+q}{2}$ (viewed as the function of the complex value $s$) is equal to $$const \int_{S^{n-1}} \sigma(T) \sigma(\Delta)^{-\frac{n+q}{2}}d\xi$$ where the integral is over sphere and $const$ is some universal constant depending only from the dimension $n$?
EDIT: One should somehow derive this using polar coordinates but I don't see how.