Airy function ( $Ai (Z), Bi (Z)$ )

Named after George Biddell Airy, the standard Airy functions solve the Airy ordinary differential equation. Their Wronskian is

Ai (Z) Bi^{'} (Z) - Ai^{'} (Z) Bi (Z) = 1/ π

, so they form a fundamental solution pair, including for complex

Z

.

Table of contents
- Airy displacement integral Airy function

Airy displacement integral ( $I_{0} (C), I_{1} (C)$ )

For

q = e^{iπ /6}

I_{0} (C) = \int_{0}^{\infty} Ai (q (s - C)) d s

and

I_{1} (C) = \int_{0}^{\infty} s Ai (q (s - C)) d s

. Both converge by Airy function decay. Integrating its differential equation gives

I_{1} - C I_{0} = i q Ai' (- qC)

. These moments convert a decaying vorticity profile into its velocity and displacement for three-layer long-wave shear-flow matching.

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Airy function by

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The Airy function is a special function that arises in various contexts within mathematics and physics, particularly in problems involving differential equations associated with quantum mechanics and wave propagation. The Airy functions are denoted as \( \text{Ai}(x) \) and \( \text{Bi}(x) \), where: - \( \text{Ai}(x) \) is the Airy function of the first kind.

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