This book presents Maslov's canonical operator method for finding
asymptotic solutions of pseudo differential equations. The classical WKB
method, so named in honor of its authors: Wentzel, Kramers and
Brillouin, was created for finding quasi classical approximations in
quantum mechanics. The simplicity, obviousness and "physicalness" of
this method quickly made it popular: specialists in mathematical physics
accepted it unequivocally as one of the weapons in their arsenal. The
number of publications which are connected with the WKB method in one
way or another can probably no longer be counted. The alternative name
of the WKB method in diffraction problem- the ray method or the method
of geometric optics - indicates that the approximations in the WKB
method are constructed by means of rays. More precisely, the first
approximation of the WKB method is constructed by means of rays
(isolating the singular part), after which the usual methods of the
(regular) theory of perturbations are applied. However, the ray method
is not applicable at the points of space where the rays focus or form a
caustic. Mathematically this fact expresses itself in the fact that the
amplitude of the waves at such points become infinite.