  In this note, we mainly consider the extended Weyl algebra of two generators
(u,v), that is, the algebra generated by u,v with the fundamental commutation
relation. Weyl algebra is realized on the space of polynomials of u and v by
defining various product depending on a symmetric matrix K called the
expression parameter. Via such expressions and ordinary calculus one can treat
various transcendental elements such as *-exponential functions and elements
obtained by integrations.
