  We have obtained a set of coupled differential equations from the continuous
limit of the transfer matrix method. Decoupling such a set of equations yields
an extension to the Wentzel-Kramers-Brillouin (WKB) approximation for the
Schr\"{o}dinger equation with the position-dependent effective mass (PDEM). In
the classically allowed region, the decoupling is to ignore the reflection
resulting from the variations of both the potential and effective mass. By
considering an infinite-well example with the PDEM, it is shown that the
extended WKB approximation can provide not only an estimation to eigenenergies,
but also an analytic form to approximate wavefunctions.
