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The \hbar-dependent KP hierarchy is a formulation of the KP hierarchy thatdepends on the Planck constant \hbar and reduces to the dispersionless KPhierarchy as \hbar -> 0. A recursive construction of its solutions on the basisof a Riemann-Hilbert problem for the pair (L,M) of Lax and Orlov-Schulmanoperators is presented. The Riemann-Hilbert problem is converted to a set ofrecursion relations for the coefficients X_n of an \hbar-expansion of theoperator X = X_0 + \hbar X_1 + \hbar^2 X_2 +... for which the dressing operatorW is expressed in the exponential form W = \exp(X/\hbar). Given the lowestorder term X_0, one can solve the recursion relations to obtain the higherorder terms. The wave function \Psi associated with W turns out to have the WKBform \Psi = \exp(S/\hbar), and the coefficients S_n of the \hbar-expansion S =S_0 + \hbar S_1 + \hbar^2 S_2 +..., too, are determined by a set of recursionrelations. This WKB form is used to show that the associated tau function hasan \hbar-expansion of the form \log\tau = \hbar^{-2}F_0 + \hbar^{-1}F_1 + F_2 +...

hbar-expansion of KP hierarchy: Recursive construction of solutions