  Higher order terms in the effective action of noncommutative gauge theories
exhibit generalizations of the *-product (e.g. *' and *-3). These terms do not
manifestly respect the noncommutative gauge invariance of the tree level
action. In U(1) gauge theories, we note that these generalized *-products occur
in the expansion of some quantities that are invariant under noncommutative
gauge transformations, but contain an infinite number of powers of the
noncommutative gauge field. One example is an open Wilson line. Another is the
expression for a commutative field strength tensor in terms of the
noncommutative gauge field. Seiberg and Witten derived differential equations
that relate commutative and noncommutative gauge transformations, gauge fields
and field strengths. In the U(1) case we solve these equations neglecting terms
of fourth order in the gauge field but keeping all orders in the noncommutative
parameter.
