Definitions
For real values of x, the Airy function is defined by the improper integral
which converges because the positive and negative parts of the rapid oscillations tend to cancel one another out (as can be checked by integration by parts).
By differentiating under the integration sign, we find that y = Ai(x) satisfies the differential equation
This equation has two linearly independent solutions. The standard choice for the other solution is the Airy function of the second kind, denoted Bi(x). It is defined as the solution with the same amplitude of oscillation as Ai(x) as x goes to −∞ which differs in phase by π / 2:
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