  We address the Mach limit problem for the Euler equations in an exterior
domain with analytic boundary. We first prove the existence of tangential
analytic vector fields for the exterior domain with constant analyticity radii,
and introduce an analytic norm in which we distinguish derivatives taken from
different directions. Then we prove the uniform boundedness of the solutions in
the analytic space on a time interval independent of the Mach number, and Mach
limit holds in the analytic norm. The results extends more generally to Gevrey
initial data with convergence in a Gevrey norm.
