  Electric-magnetic dualities are equivalence between strong and weak coupling
constants. A standard example is the exchange of electric and magnetic fields
in an abelian gauge theory. We show three methods to perform electric-magnetic
dualities in the case of the non-commutative $U(1)$ gauge theory. The first
method is to use covariant field strengths to be the electric and magnetic
fields. We find an invariant form of an equation of motion after performing the
electric-magnetic duality. The second method is to use the Seiberg-Witten map
to rewrite the non-commutative $U(1)$ gauge theory in terms of abelian field
strength. The third method is to use the large Neveu Schwarz-Neveu Schwarz
(NS-NS) background limit (non-commutativity parameter only has one degree of
freedom) to consider the non-commutative $U(1)$ gauge theory or D3-brane. In
this limit, we introduce or dualize a new one-form gauge potential to get a
D3-brane in a large Ramond-Ramond (R-R) background via field redefinition. We
also use perturbation to study the equivalence between two D3-brane theories.
Comparison of these methods in the non-commutative $U(1)$ gauge theory gives
different physical implications. The comparison reflects the differences
between the non-abelian and non-commutative gauge theories in the
electric-magnetic dualities. For a complete study, we also extend our studies
to the simplest abelian and non-abelian $p$-form gauge theories, and a
non-commutative theory with the non-abelian structure.
