Dual state vector of nonlinear coherent state and its application in complex P-representation

We introduce the dual vector \z, f (n)). of a single-mode nonlinear coheren t state (NCS) by using the contour integral form of delta -function, and th e explicit form of \z, f (n)),, is obtained in enlarged Hilbert space. The nonlinear Fock states and their contour representation are deduced. A new c ompleteness relation composed of \z, f (n)). and NCS in contour integral fo rm is derived and its application in constructing a complex P-representatio n of density operators is shown.