The Laplace transform of the
semigroup exp(tA) generated by an operator A gives the resolvent of A. An integral
formula is obtained for the Laplace transform of exp(tA + B), where B is another
operator which does not commute with A. The new transform has analytic
continuation to the same domain as the resolvent, but the analytic continuation is
not single-valued. The integral formula is then applied to the joint spectral theory of
noncommutative operators. Explicit compulations with matrices of degree two
illustrate the results.