Airy Function - Complex Arguments

Complex Arguments

We can extend the definition of the Airy function to the complex plane by

where the integral is over a path starting at the point at infinity with argument -(1/3)π and ending at the point at infinity with argument (1/3)π. Alternatively, we can use the differential equation to extend Ai(x) and Bi(x) to entire functions on the complex plane.

The asymptotic formula for Ai(x) is still valid in the complex plane if the principal value of x2/3 is taken and x is bounded away from the negative real axis. The formula for Bi(x) is valid provided x is in the sector {x∈C : |arg x| < (1/3)π−δ} for some positive δ. Finally, the formulae for Ai(−x) and Bi(−x) are valid if x is in the sector {x∈C : |arg x| < (2/3)π−δ}.

It follows from the asymptotic behaviour of the Airy functions that both Ai(x) and Bi(x) have an infinity of zeros on the negative real axis. The function Ai(x) has no other zeros in the complex plane, while the function Bi(x) also has infinitely many zeros in the sector {z∈C : (1/3)π < |arg z| < (1/2)π}.