Going back to the poles of the mass-spring system in Eq.(F.56), we see that, as the imaginary part of the two poles, , approach zero, they come together at to create a repeated pole. The same thing happens at since both poles go to ``the point at infinity''.
It is a well known fact from linear systems theory that two poles at the same point in the plane can correspond to an impulse-response component of the form , in addition to the component produced by a single pole at . In the discrete-time case, a double pole at can give rise to an impulse-response component of the form . This is the fundamental source of the linearly growing internal states of the wave digital sine oscillator at dc and . It is interesting to note, however, that such modes are always unobservable at any physical output such as the mass force or spring force that is not actually linearly growing.