  The self-duality of Dirichlet $3$-brane action under the $SL(2,R)$ duality
transformation of type IIB superstring theory is shown in the Hamiltonian form
of the path integral for the partition function by performing the direct
integration with respect to the boundary gauge field. Through the integration
in the phase space the canonical momentum conjugate to the boundary gauge field
can be effectively replaced by the dual gauge field.
