We study the complex-time Segal-Bargmann transform
Bs,ΟKNββ on a compact type Lie group KNβ, where KNβ is
one of the following classical matrix Lie groups: the special orthogonal group
SO(N,R), the special unitary group SU(N), or the
compact symplectic group Sp(N). Our work complements and extends the
results of Driver, Hall, and Kemp on the Segal-Bargman transform for the
unitary group U(N). We provide an effective method of computing the
action of the Segal-Bargmann transform on \emph{trace polynomials}, which
comprise a subspace of smooth functions on KNβ extending the polynomial
functional calculus. Using these results, we show that as Nββ, the
finite-dimensional transform Bs,ΟKNββ has a meaningful limit
Gs,Ο(Ξ²)β (where Ξ² is a parameter associated with
SO(N,R), SU(N), or Sp(N)), which can
be identified as an operator on the space of complex Laurent polynomials