The Airy wave functions AiryAi and AiryBi are linearly independent solutions for w in the equation . Specifically,

where 0F1 is the generalized hypergeometric function,  and .

The two argument forms are used to represent the derivatives, so AiryAi(1, x) = D(AiryAi)(x) and AiryBi(1, x) = D(AiryBi)(x). Note that all higher derivatives can be written in terms of the 0'th and 1st derivatives.

Note also that  is the 3rd derivative of  evaluated at , and not the 3rd derivative of .

The Airy functions are related to Bessel functions of order  for  (see the examples below).