Working definition
DLMF 9.2.1 gives Airy equation as the second derivative of w equal to z times w, and records that all its solutions are entire functions. That is a sharper statement than it looks: the equation changes character at the origin, oscillating on one side and growing or decaying on the other, yet its solutions have no singularity anywhere. The standard solutions are named at 9.2.2, and their values at the origin are given at 9.2.3 through 9.2.6 in terms of the gamma function at one third and two thirds, which is why a chapter on a differential equation depends on one about a factorial. Table 9.2.1 records that Ai and Bi are the numerically satisfactory pair on the whole real line, with other pairs preferred in other sectors.
Notation
Ai of z and Bi of z for the standard pairAi prime and Bi prime for their derivativesthe gamma function at one third and two thirds in the initial valuesAssumptions
- All solutions are entire, so unlike the Bessel case there is no branch point and no cut to respect.
- Which pair is numerically satisfactory depends on the sector, and Table 9.2.1 gives the real line rather than the whole plane.
- The values at the origin are exact expressions in the gamma function rather than decimal approximations.
Invariants
- Airy equation, DLMF 9.2.1
- The values at the origin in terms of the gamma function, DLMF 9.2.3 to 9.2.6
- Ai and Bi are numerically satisfactory on the real line, DLMF Table 9.2.1
Reproducible procedure
- Use Ai and Bi on the real line, per Table 9.2.1, and check the table before carrying that choice into the complex plane.
- Take the initial values from 9.2.3 to 9.2.6 in gamma form rather than from rounded decimals, so precision is set by the gamma evaluation and not by the transcription.
- Expect different behaviour either side of the origin, since the equation changes character there even though the solutions do not.
Error and boundary controls
- Bi grows rapidly for positive argument while Ai decays, so a scheme that computes one accurately can lose the other entirely to cancellation.
- Numerical satisfactoriness is a property of a pair in a region, not of a solution: the same functions can be a poor basis in a sector where the table names a different pair.
- The chapter defines the solutions and tabulates the choice; it makes no accuracy claim about any evaluation method.
What this does not establish
The chapter gives the equation, its solutions and where each pair is usable. It does not establish that any implementation is accurate, and entirety of the solutions is not stability of a computation.
Explicit applications
0 cross-domain bridges
This foundational concept currently supports related concepts; a direct domain bridge is scheduled for a later registry version.
Authoritative references
- [1]DLMF Chapter 9: Airy and Related Functions · National Institute of Standards and Technology
Establishes: Airy equation, its standard solutions and their values at the origin in terms of the gamma function, and which solution pairs are numerically satisfactory in which region.
Boundary: A reference giving the solutions and their analytic character. Numerical satisfactoriness is tabulated by region rather than settled once, and the chapter makes no accuracy claim about any evaluation scheme.
Direct answer
- DLMF 9.2.1 gives Airy equation as the second derivative of w equal to z times w, and records that all its solutions are entire functions. That is a sharper statement than it looks: the equation changes character at the origin, oscillating on one side and growing or decaying on the other, yet its solutions have no singularity anywhere. The standard solutions are named at 9.2.2, and their values at the origin are given at 9.2.3 through 9.2.6 in terms of the gamma function at one third and two thirds, which is why a chapter on a differential equation depends on one about a factorial. Table 9.2.1 records that Ai and Bi are the numerically satisfactory pair on the whole real line, with other pairs preferred in other sectors.
Mechanism and method
- Use Ai and Bi on the real line, per Table 9.2.1, and check the table before carrying that choice into the complex plane.
- Take the initial values from 9.2.3 to 9.2.6 in gamma form rather than from rounded decimals, so precision is set by the gamma evaluation and not by the transcription.
- Expect different behaviour either side of the origin, since the equation changes character there even though the solutions do not.
What is measured
- Airy equation, DLMF 9.2.1
- The values at the origin in terms of the gamma function, DLMF 9.2.3 to 9.2.6
- Ai and Bi are numerically satisfactory on the real line, DLMF Table 9.2.1
Limitations
- Bi grows rapidly for positive argument while Ai decays, so a scheme that computes one accurately can lose the other entirely to cancellation.
- Numerical satisfactoriness is a property of a pair in a region, not of a solution: the same functions can be a poor basis in a sector where the table names a different pair.
- The chapter defines the solutions and tabulates the choice; it makes no accuracy claim about any evaluation method.
- All solutions are entire, so unlike the Bessel case there is no branch point and no cut to respect.
- Which pair is numerically satisfactory depends on the sector, and Table 9.2.1 gives the real line rather than the whole plane.
- The values at the origin are exact expressions in the gamma function rather than decimal approximations.
What this does not establish
- The chapter gives the equation, its solutions and where each pair is usable. It does not establish that any implementation is accurate, and entirety of the solutions is not stability of a computation.