  This paper is concerned with small parameter asymptotics of magnetic quantum
systems. In addition to a semiclassical parameter \eps, the case of small
coupling $\lambda$ to the magnetic vector potential naturally occurs in this
context. Magnetic Weyl calculus is adapted to incorporate both parameters, at
least one of which needs to be small. Of particular interest is the expansion
of the Weyl product which can be used to expand the product of operators in a
small parameter, a technique which is prominent to obtain perturbation
expansions. Three asymptotic expansions for the magnetic Weyl product of two
H\"ormander class symbols are proven: (i) \eps \ll 1 and \lambda \ll 1, (ii)
\eps \ll 1 and \lambda = 1 as well as (iii) \eps = 1 and \lambda \ll 1.
Expansions (i) and (iii) are impossible to obtain with ordinary Weyl calculus.
Furthermore, I relate results derived by ordinary Weyl calculus with those
obtained with magnetic Weyl calculus by one- and two-parameter expansions. To
show the power and versatility of magnetic Weyl calculus, I derive the
semirelativistic Pauli equation as a scaling limit from the Dirac equation up
to errors of 4th order in 1/c.
