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Let $(M, \omega, J)$ be a K\"ahler manifold and K its group of hamiltoniansymplectomorphisms. The complexification of K introduced by Donadson is not agroup, only a "formal Lie group". However it still makes sense to talk aboutthe exponential map in the complexification. In this note we show how toconstruct geometrically the exponential map (for small time), in case theinitial data are real-analytic. The construction is motivated by, but does notuse, semiclassical analysis.

The exponential map of the complexification of {\em Ham} in the real-analytic case