Let M be a complex manifold of dimension n with smooth connected boundary X. Assume that M‾ admits a holomorphic S1-action preserving the boundary X and the S1-action is transversal on X. We show that the ∂‾-Neumann Laplacian on M is transversally elliptic and as a consequence, the m-th Fourier component of the q-th Dolbeault cohomology group Hm q(M‾) is finite dimensional, for every m∈Z and every q=0,1,…,n. This enables us to define ∑j=0 n(−1)jdimHm j(M‾) the m-th Fourier component of the Euler characteristic on M and to study large m-behavior of Hm q(M‾). In this paper, we establish an index formula for ∑j=0 n(−1)jdimHm j(M‾) and Morse inequalities for Hm q(M‾).