APPLICATIONS OF COMPLEX VARIABLES
C 1
C 2
C 3
RetImtFigure 25.10 The contours used in the complext-plane to define the functions
Ai(z)andBi(z).though sometimes the sense of traversal is reversed. Consequently there are
relationships connecting Ai and Bi when the rotated variables are used as their
arguments. As two examples,
Ai(z)+ΩAi(Ωz)+Ω^2 Ai(Ω^2 z)=0, (25.37)Bi(z)=i[Ω^2 Ai(Ω^2 z)−ΩAi(Ωz)] =e−πi/^6 Ai(ze−^2 πi/^3 )+eπi/^6 Ai(ze^2 πi/^3 ).
(25.38)Since the only requirements for the integral paths is that they start and end inthe correct sectors, we can distort pathC 1 so that it lies on the imaginary axis
for virtually its whole length and just to the left of the axis at its two ends. This
enables us to obtain an alternative expression for Ai(z), as follows.
Settingt=is,wheresis real and−∞<s<∞, converts the integral represen-tation of Ai(z)to
Ai(z)=1
2 π∫∞−∞exp[i(^13 s^3 +zs)]ds.Now, the exponent in this integral is an odd function ofsand so the imaginary
part of the integrand contributes nothing to the integral. What is left is therefore
Ai(z)=1
π∫∞0cos(^13 s^3 +zs)ds. (25.39)This form shows explicitly that whenzis real, so is Ai(z).
This same representation can also be used to justify the association of the