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:
“We must open our eyes and see that modern civilization has become so complex and the lives of civilized men so interwoven with the lives of other men in other countries as to make it impossible to be in this world and out of it.”
—Franklin D. Roosevelt (1882–1945)
“The conclusion suggested by these arguments might be called the paradox of theorizing. It asserts that if the terms and the general principles of a scientific theory serve their purpose, i. e., if they establish the definite connections among observable phenomena, then they can be dispensed with since any chain of laws and interpretive statements establishing such a connection should then be replaceable by a law which directly links observational antecedents to observational consequents.”
—C.G. (Carl Gustav)