  For a fixed prime we prove structure theorems for the kernel and the image of
the map that attaches to any differential modular function its differential
Fourier expansion. The image of this map, which is the ring of differential
Fourier expansions, plays the role of ring of functions on a "differential
Igusa curve". Our constructions are then used to perform an analytic
continuation between isogeny covariant differential modular forms on the
differential Igusa curves belonging to different primes.
