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}.
Read more about this topic: Airy Function
Famous quotes containing the words complex and/or arguments:
“By “object” is meant some element in the complex whole that is defined in abstraction from the whole of which it is a distinction.”
—John Dewey (1859–1952)
“There is no assurance of the great fact in question [namely, immortality]. All the arguments are mere probabilities, analogies, fancies, whims. We believe, or disbelieve, or are in doubt according to our own make-up—to accidents, to education, to environment. For myself, I do not reach either faith or belief ... that I—the conscious person talking to you—will meet you in the world beyond—you being yourself a conscious person—the same person now reading what I say.”
—Rutherford Birchard Hayes (1822–1893)