In his book about Rings of Differential Operators, Björk mentions the following:
Let $f\in C_0^{\infty}(\mathbb{R}^n)$, and let $P(x_1,...,x_n)$ be a real valued and $\geq 0$ polynomial. We define the complex function $\Gamma_f(\lambda) =\int_{\mathbb{R}^n}P^{\lambda}f$. The function $\Gamma_f$ is analytic in the half-space $\Re (\lambda)> 0$.
My work in the matter has to do with some algebraic methods Björk later uses to extend the function $\Gamma_f$, and I do not know the process followed to conclude that this function is analytic, but my question is:
Would this statement still hold if $f$ was any function in the Schwartz's class $\mathcal{S}$?