PCFs are solutions of the differential equation
with three distinct standard forms
All solutions are entire functions of and entire functions of or .
For real values of , numerically satisfactory pairs of solutions (§2.7(iv)) of (12.2.2) are and when is positive, or and when is negative. For (12.2.3) and comprise a numerically satisfactory pair, for all . The solutions are treated in §12.14.
In , for and comprise a numerically satisfactory pair of solutions in the half-plane .
When is real the solution is defined by
unless , in which case is undefined. Its importance is that when is negative and is large, and asymptotically have the same envelope (modulus) and are out of phase in the oscillatory interval . Properties of follow immediately from those of via (12.2.21).
In the oscillatory interval we define
where (0), , (0), and are real. or is the modulus and or is the corresponding phase.