  In the field of numerical integration, methods specially tuned on oscillating
functions, are of great practical importance. Such methods are needed in
various branches of natural sciences, particularly in physics, since a lot of
physical phenomena exhibit a pronounced oscillatory behavior. Among others,
probably the most important tool used to construct efficient methods for
oscillatory problems is the exponential (trigonometric) fitting. The basic
characteristic of these methods is that their phase lag vanishes at a
predefined frequency. In this work, we introduce a new tool which improves the
behavior of exponentially fitted numerical methods. The new technique is based
on the vanishing of the first derivatives of the phase lag function at the
fitted frequency. It is proved in the text that these methods present improved
characteristics in oscillatory problems.
