  For slowly varying fields the Yang-Mills Schroedinger functional can be
expanded in terms of local functionals. We show how analyticity in a complex
scale parameter enables the Schroedinger functional for arbitrarily varying
fields to be reconstructed from this expansion. We also construct the form of
the Schroedinger equation that determines the coefficients. Solving this in
powers of the coupling reproduces the results of the `standard' perturbative
solution of the functional Schroedinger equation which we also describe. In
particular the usual result for the beta-function is obtained illustrating how
analyticity enables the effects of rapidly varying fields to be computed from
the behaviour of slowly varying ones.
