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►Inside the turningpoints, that is, when , there can be a loss of precision by a factor of approximately .
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►WKBJ approximations (§2.7(iii)) for are presented in Hull and Breit (1959) and Seaton and Peach (1962: in Eq.
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►Hull and Breit (1959) and Barnett (1981b) give WKBJ approximations for and in the region inside the turningpoint: .
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►►►Figure 33.3.3:
, with , .
The turningpoint is at .
Magnify►►►Figure 33.3.4:
, with , .
The turningpoint is at .
Magnify
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►►►Figure 33.3.5:
, , and with , .
The turningpoint is at .
Magnify►►►Figure 33.3.6:
, , and with , .
The turningpoint is at (as in Figure 33.3.5).
Magnify
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►Airy functions play an indispensable role in the construction of uniform asymptotic expansions for contour integrals with coalescing saddle points, and for solutions of linear second-order ordinary differential equations with a simple turningpoint.
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►These expansions are uniform with respect to , including the turningpoint
and its neighborhood, and the region of validity often includes cut neighborhoods (§1.10(vi)) of other singularities of the differential equation, especially irregular singularities.
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►
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►PCFs are used as basic approximating functions in the theory of contour integrals with a coalescing saddle point and an algebraic singularity, and in the theory of differential equations with two coalescing turningpoints; see §§2.4(vi) and 2.8(vi).
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►For applications of Whittaker functions to the uniform asymptotic theory of differential equations with a coalescing turningpoint and simple pole see §§2.8(vi) and 18.15(i).
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►The frequent appearances of the Airy functions in both classical and quantum physics is associated with wave equations with turningpoints, for which asymptotic (WKBJ) solutions are exponential on one side and oscillatory on the other.
The Airy functions constitute uniform approximations whose region of validity includes the turningpoint and its neighborhood.
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►This reference provides several examples of applications to problems in quantum mechanics in which Airy functions give uniform asymptotic approximations, valid in the neighborhood of a turningpoint.
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