A Gutzmer formula for the complexification of a Riemann symmetric space. We consider a complex manifold $\mathrm{\Omega }$ and a real Lie group $G$ of holomorphic automorphisms of $\mathrm{\Omega }$. The question we study is, for a holomorphic function $f$ on $\mathrm{\Omega }$, to evaluate the integral of ${\left|f\right|}^2$ over a $G$-orbit by using the harmonic analysis of $G$. When $\mathrm{\Omega }$ is an annulus in the complex plane and $G$ the rotation group, it is solved by a classical formula which is sometimes called Gutzmer’s formula. We establish a generalization of it when $\mathrm{\Omega }$ is...