  We derive a "master" perturbation expansion for the quantum transition
amplitude in a light field between the field-free initial and final atomic
states in the minimal-coupling (MC) "velocity" gauge. The result is used to
prove that the traditional "velocity" and "length" gauge perturbation series
are equivalent infinite series representations or branches of the same
amplitude function, that are equal but in a common domain of convergence (if it
exists). More generally, we show that they constitute only two members of a
one-parameter family of infinitely many branches of the given transition
amplitude.
