In 1838, G. B. Airy investigated the simple-looking differential equation:
This is quite similar to the differential equation for the hyperbolic sine and hyperbolic cosine functions, which has the general solution . Airy built two partial solutions and for the first equation in the form of a power series . These solutions were named the Airy functions. Much later, H. Jeffreys (1928–1942) investigated these functions more deeply. The current notations Ai and Bi were proposed by J. C. P. Miller (1946).
The Airy functions and are the special solutions of the differential equation:
satisfying the following initial conditions:
These functions have different equivalent representations in the form of series or generalized hypergeometric functions. The hypergeometric representation can be conveniently used as a definition of the Airy functions.