  In previous papers (arxiv:math/0612370 and arxiv:0909.1342) we defined the
C*-algebra and the longitudinal pseudodifferential calculus of any singular
foliation (M,F). Here we construct the analytic index of an elliptic operator
as a KK-theory element, and prove that the same element can be obtained from an
"adiabatic foliation" TF on $M \times \R$, which we introduce here.
